> ######################## COMPUTATION OF THE PAPER#################################### The sum of divisors function and the Riemann hypothesis############> read "colHR.mpl": # contains the procedures> read "colHR.m": # contains the registered data:# lisCA2, list of 6704 CA numbers of type 2, [eps,q,alfaq,p,xi,
sigma(N)/N,logN]# lisCA, list of the 47 first CA numbers, [eps,q,alfaq,p,xi,
sigma(N)/N,N]# lisSA, list of SA numbers < 10^50, [M,sigma(M)/M]# lisctilda, computes the min and the max of ctilda(N) for all
CA number N# between the jth and the (j+1)th CA number of type 2# [i,pmin,ctildamin,ctildamax,pmax] cf. Sect. 5.2# lisalpha, computes the min and the max of alpha(N) for all CA
number N# between the jth and the (j+1)th CA number of type 2# [i,pmin,alphamin,alphamax,pmax] cf. Sect. 6.2# the two real numbers ega:=exp(evalf(gamma)), tau:=evalf(2+gamma-log(4*Pi)); cf. (1.5)> Digits:=20: # Computation of n_0, cf. (1.8)
fsolve(-2*(sqrt(2)-1)+tau+3.789/log(log(n0))+0.026*log(log(n0))/log(n0)^(1/6),n0);
3.9852277562711315731 1080
(1)
> plot([-2*(sqrt(2)-1)+3.789/log(log(10^t))+0.026*log(log(10^t))/log(10^t)^(1/6),-tau,tau],t=40..90,color=["blue","red","red"]); # cf.
(1.8)

0.06

0.04

0.02

0

50

60

70

80

90

t

0.02

0.04

0.06
> ######################## Proof of Lemma 2.1 ####################
n:='n': t:='t': int(1/(t^2*log(t)^n)+n/(t^2*log(t)^(n+1)),t);1
t ln t n> Digits:=10: J:='J': A:='A': for n from 1 to 5 do J[n]:=A[n]-n*J[n+1] od:
-A[1]+A[4]+2*J[1]+J[2]+2*J[4]+J[5], -3+17/log(10.^8);
A1 3 A4 A2 2 A3 17 A5 85 J6 , 2.077124226> J:='J': A:='A': for n from 1 to 4 do J[n]:=A[n]-n*J[n+1] od:-A[1]-A[4]+2*J[1]+J[2]-2*J[4]-J[5];A1 9 A4 A2 2 A3 31 J5
>

(2)

(3)

(4)

2*log(10.^8)^4/10^16;2.302781322 10 11> y:='y': a:='a': f:=-1/(y*log(y))+1/(y*log(y)^2)-2/(y*log(y)^3)+a/(y*log(y)^4);
f1:=normal(diff(f,y)): "df/dy = ",f1;
Y:='Y': hY:=subs(log(y)=Y,numer(f1)): h1Y:=diff(hY,Y):"h(y) = ",hY, "dh/dy = ",h1Y;
subs(a=13,Y=log(19),h1Y), evalf(subs(a=13,Y=log(19),h1Y));(5)

subs(a=13,Y=log(19),hY),evalf(subs(a=13,Y=log(19),hY));
2*log(10.^8)^4/10^16;
1y ln yf1
y ln y"df/dy = ","h(y) = ", Y4

ln y 4aY2
2

2

y ln y6Y

4 ln 19 3ln 19 47 ln 19y ln y 4

6 ln y4ay ln ya ln ya3

5

4 a, "dh/dy = ", 4 Y 3

a

67, 95.1098566
52, 2.55298765

2.302781322 10 11> ######################## Product g(y) (cf. Lemma 2.1)#######################(6)# The negative step function y -> g(y) = sum(log(1-1/p^2),p >= y)# is constant on [p',p'') where p' < p'' are two consecutive primes
and g(p'') > g(p')
Digits:=50: fongy(24318); # for y = 24318, computes g(y) = g(p''),a(p'), a(y) and a(p'')"p=", 24317, "np=", 24329"a(p'')=comin=", 9.121231627, "a(y)=", 9.583870096, "a(p')=comax=", 9.625920922, "g(y)=",6

3.710643530 10> Digits:=50: ti:=time(): calcullogPy(100000,19300,19400,24250,24350); "time = ",time()-ti;# To prove (2.13) and (2.14), execute calcullogPy(10^8+7,19300,
19400,24250,24350);# the running time is about 6 hours"p' = ", 19301, "p'' = ", 19309, "a(p'') = comin = ", 3.5645756, "a(p') = comax = ", 3.9630903,"g(p'') = gnp = ", 4.80454331416 10 6
"p' = ", 19309, "p'' = ", 19319, "a(p'') = comin = ", 3.5577752, "a(p') = comax = ", 4.0557258,"g(p'') = gnp = ", 4.80186118039 10 6
"p' = ", 19319, "p'' = ", 19333, "a(p'') = comin = ", 3.3518364, "a(p') = comax = ", 4.0487741,"g(p'') = gnp = ", 4.79918182257 10 6
"p' = ", 19333, "p'' = ", 19373, "a(p'') = comin = ", 1.8508293, "a(p') = comax = ", 3.8426238,"g(p'') = gnp = ", 4.79650634386 10 6
"p' = ", 19373, "p'' = ", 19379, "a(p'') = comin = ", 2.0422065, "a(p') = comax = ", 2.3410138,"g(p'') = gnp = ", 4.79384190203 10 6
"p' = ", 19379, "p'' = ", 19381, "a(p'') = comin = ", 2.4327548, "a(p') = comax = ", 2.5323007,"g(p'') = gnp = ", 4.79117910984 10 6
"p' = ", 19381, "p'' = ", 19387, "a(p'') = comin = ", 2.6243507, "a(p') = comax = ", 2.9228189,"g(p'') = gnp = ", 4.78851686718 10 6
"p' = ", 19387, "p'' = ", 19391, "a(p'') = comin = ", 2.9154515, "a(p') = comax = ", 3.1143246,(7)

"g(p'') = gnp = ", 4.78585627213 10 6
"p' = ", 19391, "p'' = ", 19403, "a(p'') = comin = ", 2.8089975, "a(p') = comax = ", 3.4053653,"g(p'') = gnp = ", 4.78319677462 10 6
"p' = ", 24251, "p'' = ", 24281, "a(p'') = comin = ", 10.283947, "a(p') = comax = ", 11.545085,"g(p'') = gnp = ", 3.71403083208 10 6
"p' = ", 24281, "p'' = ", 24317, "a(p'') = comin = ", 9.1981693, "a(p') = comax = ", 10.712081,"g(p'') = gnp = ", 3.71233467190 10 6
"p' = ", 24317, "p'' = ", 24329, "a(p'') = comin = ", 9.1212316, "a(p') = comax = ", 9.6259209,"g(p'') = gnp = ", 3.71064353016 10 6
"p' = ", 24329, "p'' = ", 24337, "a(p'') = comin = ", 9.2124964, "a(p') = comax = ", 9.5488558,"g(p'') = gnp = ", 3.70895405627 10 6
"p' = ", 24337, "p'' = ", 24359, "a(p'') = comin = ", 8.7151904, "a(p') = comax = ", 9.6400356,"g(p'') = gnp = ", 3.70726569292 10 6"*************** Table of minimal records for a(y) ************""for y = p'' = ", 19373, "a(y) = comini = ", 2., "a(p'') = ", 1.8508293"for y = p'' = ", 3299, "a(y) = comini = ", 1.8508293, "a(p'') = ", 1.8041609"for y = p'' = ", 2657, "a(y) = comini = ", 1.8041609, "a(p'') = ", 1.5679327"for y = p'' = ", 1427, "a(y) = comini = ", 1.5679327, "a(p'') = ", 0.77877493"for y = p'' = ", 1423, "a(y) = comini = ", 0.77877493, "a(p'') = ", 0.087998269"*************** Table of maximal records for a(y) ************""for yp'' = ", 60317, "a(y) = comaxi = ", 9., "p' = ", 60293, "a(p') = ", 9.0669708"for yp'' = ", 60271, "a(y) = comaxi = ", 9.0669708, "p' = ", 60259, "a(p') = ", 9.0852552"for yp'' = ", 60251, "a(y) = comaxi = ", 9.0852552, "p' = ", 60223, "a(p') = ", 9.1477572"for yp'' = ", 60209, "a(y) = comaxi = ", 9.1477572, "p' = ", 60169, "a(p') = ", 9.6068201"for yp'' = ", 59863, "a(y) = comaxi = ", 9.6068201, "p' = ", 59833, "a(p') = ", 9.9510113"for yp'' = ", 59833, "a(y) = comaxi = ", 9.9510113, "p' = ", 59809, "a(p') = ", 10.237190"for yp'' = ", 59809, "a(y) = comaxi = ", 10.237190, "p' = ", 59797, "a(p') = ", 10.257915"for yp'' = ", 24317, "a(y) = comaxi = ", 10.257915, "p' = ", 24281, "a(p') = ", 10.712081"for yp'' = ", 24281, "a(y) = comaxi = ", 10.712081, "p' = ", 24251, "a(p') = ", 11.545085"for yp'' = ", 24151, "a(y) = comaxi = ", 11.545085, "p' = ", 24137, "a(p') = ", 11.628057"for yp'' = ", 6389, "a(y) = comaxi = ", 11.628057, "p' = ", 6379, "a(p') = ", 11.649848"for yp'' = ", 4327, "a(y) = comaxi = ", 11.649848, "p' = ", 4297, "a(p') = ", 11.680765"for yp'' = ", 2999, "a(y) = comaxi = ", 11.680765, "p' = ", 2971, "a(p') = ", 11.771144"for yp'' = ", 2879, "a(y) = comaxi = ", 11.771144, "p' = ", 2861, "a(p') = ", 12.558278"for yp'' = ", 2819, "a(y) = comaxi = ", 12.558278, "p' = ", 2803, "a(p') = ", 12.785982"for yp'' = ", 1693, "a(y) = comaxi = ", 12.785982, "p' = ", 1669, "a(p') = ", 13.288559"for yp'' = ", 1657, "a(y)= comaxi = ", 13.288559, "p' = ", 1637, "a(p') = ", 13.680096(8)

"for y"for y"for yp'' = ", 1637, "a(y)p'' = ", 293, "a(y)p'' = ", 127, "a(y)= comaxi = ", 13.680096, "p' = ", 1627, "a(p') = ", 14.241810= comaxi = ", 14.241810, "p' = ", 283, "a(p') = ", 14.885060= comaxi = ", 14.885060, "p' = ", 113, "a(p') = ", 15.067897"time = ", 63.251> ############################ Proof of Lemma 2.2, (2.15)####################
Digits:=10: eta:=7.5*10.^(-7): c:=0.945:(1+eta)*(3+1/log(1418.))/2-c,
0.624/log(1418.);0.623900145, 0.08598591250> ############################ Proof of Lemma 2.3, (2.16)####################
Digits:=10: 1+1/log(89693.)^3; lempipi(89693):
1.000674235"x = ", 89693, "quomax = ", 1.000674"p=", 89689, "pp=", 89753, "pp-p = ", 64, "pp/p = ", 1.00071358"p=", 79699, "pp=", 79757, "pp-p = ", 58, "pp/p = ", 1.00072774"p=", 60539, "pp=", 60589, "pp-p = ", 50, "pp/p = ", 1.00082591"p=", 59281, "pp=", 59333, "pp-p = ", 52, "pp/p = ", 1.00087718"p=", 58831, "pp=", 58889, "pp-p = ", 58, "pp/p = ", 1.00098587"p=", 48679, "pp=", 48731, "pp-p = ", 52, "pp/p = ", 1.00106822"p=", 45893, "pp=", 45943, "pp-p = ", 50, "pp/p = ", 1.00108949"p=", 44293, "pp=", 44351, "pp-p = ", 58, "pp/p = ", 1.00130946"p=", 43331, "pp=", 43391, "pp-p = ", 60, "pp/p = ", 1.00138469"p=", 35677, "pp=", 35729, "pp-p = ", 52, "pp/p = ", 1.00145752"p=", 35617, "pp=", 35671, "pp-p = ", 54, "pp/p = ", 1.00151613"p=", 34061, "pp=", 34123, "pp-p = ", 62, "pp/p = ", 1.00182026"p=", 31397, "pp=", 31469, "pp-p = ", 72, "pp/p = ", 1.00229321"p=", 19609, "pp=", 19661, "pp-p = ", 52, "pp/p = ", 1.00265184"p=", 15683, "pp=", 15727, "pp-p = ", 44, "pp/p = ", 1.00280559"p=", 11743, "pp=", 11777, "pp-p = ", 34, "pp/p = ", 1.00289534"p=", 10799, "pp=", 10831, "pp-p = ", 32, "pp/p = ", 1.00296324"p=", 9973, "pp=", 10007, "pp-p = ", 34, "pp/p = ", 1.00340920"p=", 9551, "pp=", 9587, "pp-p = ", 36, "pp/p = ", 1.00376924"p=", 8467, "pp=", 8501, "pp-p = ", 34, "pp/p = ", 1.00401559"p=", 7253, "pp=", 7283, "pp-p = ", 30, "pp/p = ", 1.00413622"p=", 6917, "pp=", 6947, "pp-p = ", 30, "pp/p = ", 1.00433714"p=", 6491, "pp=", 6521, "pp-p = ", 30, "pp/p = ", 1.00462178"p=", 5953, "pp=", 5981, "pp-p = ", 28, "pp/p = ", 1.00470351"p=", 5749, "pp=", 5779, "pp-p = ", 30, "pp/p = ", 1.00521830

(8)

(9)

"p=", 5591, "pp=", 5623, "pp-p = ", 32, "pp/p = ", 1.00572348"p=", 4831, "pp=", 4861, "pp-p = ", 30, "pp/p = ", 1.00620989"p=", 4297, "pp=", 4327, "pp-p = ", 30, "pp/p = ", 1.00698162"p=", 3271, "pp=", 3299, "pp-p = ", 28, "pp/p = ", 1.00856007"p=", 2971, "pp=", 2999, "pp-p = ", 28, "pp/p = ", 1.00942444"p=", 2477, "pp=", 2503, "pp-p = ", 26, "pp/p = ", 1.01049657"p=", 2179, "pp=", 2203, "pp-p = ", 24, "pp/p = ", 1.01101423"p=", 1951, "pp=", 1973, "pp-p = ", 22, "pp/p = ", 1.01127627"p=", 1669, "pp=", 1693, "pp-p = ", 24, "pp/p = ", 1.01437987"p=", 1327, "pp=", 1361, "pp-p = ", 34, "pp/p = ", 1.02562170"p=", 523, "pp=", 541, "pp-p = ", 18, "pp/p = ", 1.03441683"p=", 317, "pp=", 331, "pp-p = ", 14, "pp/p = ", 1.04416404"p=", 293, "pp=", 307, "pp-p = ", 14, "pp/p = ", 1.04778157"p=", 211, "pp=", 223, "pp-p = ", 12, "pp/p = ", 1.05687204"p=", 199, "pp=", 211, "pp-p = ", 12, "pp/p = ", 1.06030151"p=", 139, "pp=", 149, "pp-p = ", 10, "pp/p = ", 1.07194245"p=", 113, "pp=", 127, "pp-p = ", 14, "pp/p = ", 1.12389381"p=", 47, "pp=", 53, "pp-p = ", 6, "pp/p = ", 1.12765957"p=", 31, "pp=", 37, "pp-p = ", 6, "pp/p = ", 1.19354839"p=", 23, "pp=", 29, "pp-p = ", 6, "pp/p = ", 1.26086957"p=", 13, "pp=", 17, "pp-p = ", 4, "pp/p = ", 1.30769231"p=", 7, "pp=", 11, "pp-p = ", 4, "pp/p = ", 1.57142857"p=", 3, "pp=", 5, "pp-p = ", 2, "pp/p = ", 1.66666667> ############################### Function S_1, Sect. 3#################################
assume(x > 1): f:=int(log(1-1/t^2)/(2*t^2),t=1..x): series(f,x=infinity,14);# proof of Lemma 3.2
1
1
1
1
1
1
1
ln 2
1
O
3
5
7
9
11
13
6 x~
20 x~
42 x~
72 x~
110 x~
156 x~
x~14> Digits:=30:constrlisteS1(10000): nops(lisS1);# computes a list [p^m,p,psi(p^m),integral(psi(t)/t^2,t=2..p^m)]
for p^m <= 9973
# allows the computation of S_1(t) by Lemma 3.2 for t <= 9973
for i from 1 to 5 do print(lisS1[i]); od:

(10)

(11)

1280
2, 2, 0.693147180559945309417232121458, 0.
3, 3, 1.79175946922805500081247735838, 0.115524530093324218236205353577
4, 2, 2.48490664978800031022970947984, 0.264837819195662134970578466775
5, 5, 4.09434456222210068483046881307, 0.389083151685062150482063940767(12)

7, 7, 6.04025471127741398993582155651, 0.623045698097753618186662158657> S1num(1000,0);# computation of S_1(1000)
0.00030696356355675791527871692644> liste:=[]: for x from 3. by 0.005 to 17 do liste:=[op(liste),[x,S1num(x,0)]]; od:plot(liste,style=point); # curve S_1(x), 3 <= x <= 17

(12)
(13)

0.004
0.002
0

4

6

8

10

12

14

16

0.002
0.004
0.006
0.008
0.010
0.012
> Digits:=10:y0:=73.2: # proof of Lemma 3.4, (3.3)
evalf(1/(8*Pi)+(1/sqrt(y0)/log(y0)^2)*(log(2*Pi)+1/2/(y0^2-1))),
0.0462/log(73.2)^3; # proof of Lemma 3.4, (3.4)
0.05144398748, 0.0005838482773(14)> Digits:=10:(1/1.072)*(1-log(128.)^2/(8*evalf(Pi)*sqrt(128.))),log(1-1/128.^2)/256,
0.16+0.855-1-2.4*10^(-7); # proof of Lemma 3.5
0.8556020264, 2.384260262 10 7 , 0.01499976000(15)> liste:=[]: for x from 2. by 0.001 to 6 do liste:=[op(liste),[x,0.16*log(x)+S1num(x,0)]]; od:plot(liste,style=point); # curve h_1(x), 2 <= x <= 6, cf. (3.8)

0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
2

3

> ##################################

4

5

6

Functions G and H : Proof of Lemma 3.6

Digits:=10: u:='u': v:='v': t:='t': g1:=u*log(t)+v/sqrt(t)/log(t);g11:=normal(diff(g1,t));num1:=expand(numer(g11)/t^(3/2)/log(t)^2); evalf(subs(u=0.8,v=3.3,
t=6.,num1));
g12:=normal(diff(g11,t));num2:=-expand(numer(g12)/t^(3/2)/log(t)^3); evalf(subs(u=0.95,v=
3.3,t=10.,num2));v
g1
u ln tt ln t2 u t3g11

ln t 2
5

2t

num1

3

g12

2

4t

2u

2

ln t

3

v t ln t
2

ln t

2vt

2

v

2v

t ln t0.0088186568

t ln t 2u

3 t ln t
7

4t

2

2

v

ln t

8 v t ln t

8vt

3

(16)

num2

3v

4u

8v

8v
2

t ln tt ln tt ln t 30.1819217966
g2:=-a/sqrt(t)+v/sqrt(t)/log(t);(16)> a:='a': v:='v': t:='t':g21:=
normal(diff(g2,t));num1:=expand(numer(g21)/log(t)^2);evalf(subs(a=0.75,v=0.6,t=6.,num1));
g22:=normal(diff(g21,t));num2:=-expand(numer(g22)/log(t)^3);
evalf(subs(a=0.75,v=0.6,t=10.,num2));av
g2
tt ln ta ln t 2

g21

v ln t
3

2

2t

num1

a

ln t

2v
2

v

2v

ln t

ln t 20.0413486582
3 ln t 3 a

g22
num2

3a

3 ln t 2 v
4 t5
3v
ln t

2

8 v ln t

ln t 3
8v

8v

8v

2

ln t
ln t 30.1697512951
g3:=u*log(t)+c*log(t)/t^(2/3);(17)> u:='u': c:='c': t:='t':g31:=
normal(diff(g3,t));g31modif:=(1/(3*t))*(3*u-2*c*(log(t)-3/2)/t^(2/3));"g31 - g31modif = ",simplify(g31-g31modif);(1/(2/3))^1*exp(-1(2/3)*(3/2)); # cf.(1.12)
evalf(subs(u=0.04,c=0.1,3*u-2*c*3/2/exp(2.)));
c ln tg3
u ln tt2 3
3 u t5g31

32 c ln t t
3 t8

3ct

3

2 c ln t
3u

3
2

t2 3
3t
"g31 - g31modif = ", 0

g31modif

3e 2
20.07939941504(18)> g32:=normal(diff(g31,t)); g32modif:=-(1/(9*t^2))*(9*u-10*c*(log(t)-21/10)/t^(2/3));"g32 - g32modif = ",simplify(g32-g32modif);(1/(2/3))^1*exp(-1-(2/3)

>*(21/10));
evalf(subs(u=0.04,c=0.1,9*u-10*c*3/2*exp(-2.4)));9 u t5 3 10 c ln t t 21 c t
g32
9 t11 321
10 c ln t
10
9u
2 3
t
g32modif2
9t"g32 - g32modif = ", 0
3e

12
5

2
0.2239230701(19)> ################# Section 4.1, Definition of CA numbers #########
superabondant(10^3,0); # list of SA numbers <= 1000 [M,sigma(M)/M],
cf. (4.2)
nops(lisSA), lisSA[323..324]; # lisSA is the list of the 324 SA
numbers < 10^50
3
7
7
5
91
31
14
1, 1 , 2,, 4,, 6, 2 , 12,
, 24,
, 36,
, 48,
, 60,
, 120,
2
4
3
2
36
12
5
91
31
13
403
24
3 , 180,, 240,, 360,, 720,, 840,
30
10
4
120
7
324,

43561581874692815930555271574104476867717507424000,(20)44182896271258530223226880000,
5325436922711235449158398689
51220761105408036313949605037683285987316190048000,
288762620528362675297526061465600
34759126794536233776656868243103> t:='t': u:='u': F(t,u); # cf. (4.5)
t 1.
ln 1.
u 1.
t
1. t
ln t
> Digits:=7: liseps(20); # The set E_i, cf. Section 4.1, (4.7) and(4.8)# liseps(xmax) computes the elements >= F(xmax,1) of E_i, 1, 0 , 0.5849625, 2, 1 , 0.2618593, 3, 1 , 0.2223929, 2, 2 , 0.1132828, 5, 1 ,0.09953625, 2, 3 , 0.07285775, 3, 2 , 0.06862152, 7, 1 , 0.04730525, 2, 4 ,0.03628653, 11, 1 , 0.02889254, 13, 1 , 0.02308357, 2, 5 , 0.02304524, 3, 3 ,0.02037326, 5, 2 , 0.02017457, 17, 1 , 0.01742053, 19, 1> p:='p': eps:='eps':

(21)

(22)

>

mu:=proc(p,eps) evalf(log((p^(1+eps)-1)/(p^eps-1))/log(p)-1) end: #cf (4.11)
mu(p,eps);
A:=F(p,mu(p,eps)):
A:= simplify(A,assume(p > 2 and eps > 0)); # cf. (4.12)
peps 1. 1.
lnpeps 1.
1.
ln p
A
eps
(23)
> Digits:=50: calcullogN(7.1,2):# calcullogN(xi,2) computes the CA number N of parameter epsilon =F(xi,1),# cf. (4.11) and (4.19). xi can be real or integer. Choose Digits
large# computes xiN, called xi(N) in the paper cf. Definition 4.3# computes xi_0[k] with xi_0[1] = xiN cf. (4.15) and (4.16)# epsmin and epsmax are the smallest and the largest parameter of N
# epsmax is also eps_i such that N_{eps_{i-1}} < N <= N_{eps_i},
cf. Definition 4.3# F(xiN,1) = epsmax"k=", 1, "xi0[k] = ", 7., "pp = ", 7, "np = ", 11"k=", 2, "xi0[k] = ", 3.07030931523062, "pp = ", 3, "np = ", 5"k=", 3, "xi0[k] = ", 2.20978372619847, "pp = ", 2, "np = ", 3"k=", 4, "xi0[k] = ", 1.84794020465102, "pp = ", 1, "np = ", 2"xi = ", 7.1, "eps = F(xi,1) = ", 0.0672259722864226, "xiN = ", 7.
"epsmax = ", 7, 1 , 0.0686215613240665, "epsmin = ", 2, 4 , 0.0473057147783567"N = ", 2520, "prime factors of N = ", 2 3 3 2 5 7"q = ", 3, "expq = ", 2, "p = ", 7, "K(N) = ", 3
"sigNsN = ", 3.71428571428571, "logN = ", 7.83201418050547, "loglogN = ",(24)
2.05821971581205> ca(47,false,2): # list of the CA numbers with largest prime factor<= 17, cf. Figure 2# ca(17,true,2) returns only the CA numbers of type2"i=", 1, 1, 0 , "epsq = ", , "xi = ", 1., "p = ", 1, "sigma(N)/N = ", 1, "N = ", 1.
33"i=", 2, 2, 1 , "eps = ", 0.58496, "N = ", 2, 2 , "sigma(N)/N = ", ,, "xi = ", 222"i=", 3, 3, 1 , "eps = ", 0.26186, "N = ", 6, 2 , 3 , "sigma(N)/N = ", 2, 2 , "xi = ", 377"i=", 4, 2, 2 , "eps = ", 0.22239, "N = ", 12, 2 2 , 3 , "sigma(N)/N = ", ,, "xi = ",3
3
3.294014
2 7"i=", 5, 5, 1 , "eps = ", 0.11328, "N = ", 60, 2 2 , 3 , 5 , "sigma(N)/N = ",,,55"xi = ", 53"i=", 6, 2, 3 , "eps = ", 0.099536, "N = ", 120, 2 , 3 , 5 , "sigma(N)/N = ", 3, 3 ,"xi = ", 5.4434

13
,
4

13

"i=", 8, 7, 1 , "eps = ", 0.068622, "N = ", 2520, 2 3 , 3 2 , 5 , 7 , "sigma(N)/N = ",26,7"i=", 7, 3, 2 , "eps = ", 0.072858, "N = ", 360, 2 3 , 3 2 , 5 , "sigma(N)/N = ",2 2,

"xi = ", 6.7175

2

13
7

, "xi = ", 7"i=", 9, 2, 4 , "eps = ", 0.047306, "N = ", 5040, 2 4 , 3 2 , 5 , 7 , "sigma(N)/N = ",

403
,
105

13 31
, "xi = ", 9.0874
3 5 7"i=", 10, 11, 1 , "eps = ", 0.036287, "N = ", 55440, 2 4 , 3 2 , 5 , "...", 11 ,"sigma(N)/N = ",2 2 13 31, "xi = ", 11
5 7 111612,
385"i=", 11, 13, 1 , "eps = ", 0.028893, "N = ", 720720, 2 4 , 3 2 , 5 , "...", 13 ,248
"sigma(N)/N = ",,55

2 3 31, "xi = ", 13
5 11"i=", 12, 2, 5 , "eps = ", 0.023084, "N = ", 1441440, 2 5 , 3 2 , 5 , "...", 13 ,"sigma(N)/N = ",

252
,
55

2 2 3 2 7
, "xi = ", 15.362
5 11"i=", 13, 3, 3 , "eps = ", 0.023045, "N = ", 4324320, 2 5 , 3 3 , 5 , "...", 13 ,672
"sigma(N)/N = ",,143

2 5 3 7, "xi = ", 15.382
11 13"i=", 14, 5, 2 , "eps = ", 0.020373, "N = ", 21621600, 2 5 , 3 3 , 5 2 , 7 , "...", 13 ,"sigma(N)/N = ",

3472
,
715

2 4 7
5 11

31
, "xi = ", 16.875
13"i=", 15, 17, 1 , "eps = ", 0.020174, "N = ", 367567200, 2 5 , 3 3 , 5 2 , 7 , "...", 17 ,62496
"sigma(N)/N = ",,12155

2 5 3 2 7
5 11 1331, "xi = ", 1717"i=", 16, 19, 1 , "eps = ", 0.017420, "N = ", 6983776800, 2 5 , 3 3 , 5 2 , 7 , "...", 19 ,"sigma(N)/N = ",249984,46189

2 7 3 2 7 31
, "xi = ", 19
11 13 17 19"i=", 17, 23, 1 , "eps = ", 0.013573, "xi = ", 23, "N = ", 2 5 , 3 3 , 5 2 , 7 , "...", 23 ,5999616"sigma(N)/N = ",,1062347

2 10 3 3 7 31
, "xi = ", 23
11 13 17 19 23"i=", 18, 2, 6 , "eps = ", 0.011405, "N = ", 2 6 , 3 3 , 5 2 , 7 , "...", 23 , "sigma(N)/N = ",6047232,
1062347

2 9 3 31 127, "xi = ", 26.316
11 13 17 19 23"i=", 19, 29, 1 , "eps = ", 0.010068, "xi = ", 29, "N = ", 2 6 , 3 3 , 5 2 , 7 , "...", 29 ,2 10 3 2 5 31 127, "xi = ", 29
11 13 17 19 23 29

181416960"sigma(N)/N = ",,30808063"i=", 20, 31, 1 , "eps = ", 0.0092454, "xi = ", 31, "N = ", 2 6 , 3 3 , 5 2 , 7 , "...", 31 ,"sigma(N)/N = ",187269120,
30808063

2 15 3 2 5 127
13 17 19 23

1129, "xi = ", 31"i=", 21, 7, 2 , "eps = ", 0.0090958, "N = ", 2 6 , 3 3 , 5 2 , 7 2 , 11 , "...", 31 ,
70225920"sigma(N)/N = ",,113503397

2 12 3 3 5
11 13 17127
, "xi = ", 31.399
23 29"i=", 22, 3, 4 , "eps = ", 0.0075539, "N = ", 2 6 , 3 4 , 5 2 , 7 2 , 11 , "...", 31 ,"sigma(N)/N = ",6437376,1031849

2 9 3 2 11 127, "xi = ", 36.346
7 13 17 23 29"i=", 23, 37, 1 , "eps = ", 0.0073854, "xi = ", 37, "N = ", 2 6 , 3 4 , 5 2 , 7 2 , 11 , "...",
244620288
37 , "sigma(N)/N = ",,38178413

2 10 3 2 11 19 127, "xi = ", 37
7 13 17 23 29 37"i=", 24, 41, 1 , "eps = ", 0.0064890, "xi = ", 41, "N = ", 2 6 , 3 4 , 5 2 , 7 2 , 11 , "...",
41 , "sigma(N)/N = ",1467721728,
223616419

2 11 3 3 11 19 127, "xi = ", 41
13 17 23 29 37 41"i=", 25, 43, 1 , "eps = ", 0.0061123, "xi = ", 43, "N = ", 2 6 , 3 4 , 5 2 , 7 2 , 11 , "...",
64579756032
43 , "sigma(N)/N = ",,9615506017

2 13 3 3 11 2 19
13 17 23 29 37

127, "xi = ", 4341 43"i=", 26, 2, 7 , "eps = ", 0.0056688, "N = ", 2 7 , 3 4 , 5 2 , 7 2 , 11 , "...", 43 ,"sigma(N)/N = ",3813765120,
565618001

2 12 3 4 5 11 2 19
, "xi = ", 45.665
13 23 29 37 41 43"i=", 27, 47, 1 , "eps = ", 0.0054682, "xi = ", 47, "N = ", 2 7 , 3 4 , 5 2 , 7 2 , 11 , "...",
47 , "sigma(N)/N = ",183060725760,
26584046047

16

13

2
23

3
29

5

5 11
37 41

2

19
43

47

(25)

, "xi = ",47
> ################ Enumeration of CA numbers, Section 4.2 ###########
Digits:=7: liseps2(20,2); # computation of CA numbers N of type 2
with xi(N) < 20
# [eps,q,alfaq,p,xi,sigNsN,logN]"ecartmin between xi of type 2 and the nearest integer = ", 0.087453, "qmini = ", 2, 4
0.2223929, 2, 2, 3, 3.294007 , 0.09953625, 2, 3, 5, 5.443355 , 0.07285775, 3, 2, 5,
6.717536 , 0.04730525, 2, 4, 7, 9.087453 , 0.02308357, 2, 5, 13, 15.36249 ,

(26)

(26)
0.02304524, 3, 3, 13, 15.38163 , 0.02037326, 5, 2, 13, 16.87465> Digits:=7: cantype2(20,1,0): # computes the list of CA numbers oftype 2# for xi < 20
[epsilon,q,expq,p,xi,sigma(N)/N,log
N]# To generate the list lisCA2 of the 6704 first CA
numbers of type 2# execute lisCA2:=cantype2(2000700000,100,0);# with Digits:=70: (running time about 8 hours)# or read "colHR.m" that contains the list lisCA2"j = ", 1, "T[j] = ", 0.2223929, 2, 2, 3, 3.294007, 2.333333, 2.484906 , "time = ", 0.
"j = ", 2, "T[j] = ", 0.09953625, 2, 3, 5, 5.443355, 3.000000, 4.787491 , "time = ", 0.003"j = ", 3, "T[j] = ", 0.07285775, 3, 2, 5, 6.717536, 3.250000, 5.886103 , "time = ", 0.007"j = ", 4, "T[j] = ", 0.04730525, 2, 4, 7, 9.087453, 3.838097, 8.525160 , "time = ", 0.014"j = ", 5, "T[j] = ", 0.02308357, 2, 5, 13, 15.36249, 4.581821, 14.18116 , "time = ", 0.019"j = ", 6, "T[j] = ", 0.02304524, 3, 3, 13, 15.38163, 4.699304, 15.27977 , "time = ", 0.025(27)> u:=lisCA2[6704]: nops(lisCA2), "log(N) = ",u[7];Digits:=12: [evalf(u[1]),op(u[2..4]),evalf(op(u[5..7]))];
6704, "log(N) = ",2.000538105793703852349682445053652672670513348656314985773618694201452 109
2.33401008331 1011, 62297, 2, 2000530391, 2.00053042579 109 , 38.1446306163,(28)

2.00053810579 109> ############## Smallest difference between two consecutive
epsilon_i's (cf. Sect. 4.2)
Digits:=50: mineps21(1,6702,lisCA2); F(54371,2)-F(1524427141,1);# Difference between an epsilon_i of type 2 and an epsilon_i oftype 1"mineps21=", 2.577682050 10 23 , "epsmin = ", 54371, 2 , " - ", 1524427141, 1
2.5776820504686902084901443275911145306 10 23(29)> mineps22(1,6701,lisCA2); # difference between two epsilons of type2
"mineps22=", 1.631095481 10 16 , " = ", 3, 1559 , " - ", 2, 50723(30)> # difference between two epsilon_i's of type 1
Fp1:=log(1+1/t)/log(t):Fp11:=normal(diff(Fp1,t)):print("Fp1 = ",Fp1,"Fp11 = ",Fp11);
Fp12:=normal(diff(Fp11,t)); # Fp11 is negative and incresing-2*evalf(subs(t=2.*10^9,Fp11)); # Mean value theorem
1
t 1
t 1
ln 1
ln
t ln tlnttt"Fp1 = ",, "Fp11 = ",2
ln t
t t 1 ln t
Fp12

1
t2 t
t
2 ln

t

ln

1 2 ln t 3
1
t2 ln t 2t1tt2 ln tln t lnt

2 ln t 2 t
1t

2 t ln t

t

2 ln
4 ln

1

t ln t

t
t

1
t

t

2 ln t

(31)

2 ln

t

1
t

2.4436705592317371115910909345816611595141022110508 10 20(31)> p:=prevprime(2*10^9):p1:=nextprime(p): di:=p1-p: "p = ",p,"p1 = ",p1,"di = ",di;while di > 2 do
p1:=p; p:=prevprime(p); di:=p1-p;
od: # p and p1 are the largest twin primes < 2*10^9print("p=",p, "F(p,1) = ",evalf(F(p,1)));
print("p1=",p1, "F(p1,1) = ",evalf(F(p1,1)));
print("F(p,1) - F(p1,1) = ",evalf(F(p,1)-F(p1,1)));"p = ", 1999999973, "p1 = ", 2000000011, "di = ", 38"p=", 1999999871, "F(p,1) = ",2.3346581578443740557780187848356621683046465398533 10"p1=", 1999999873, "F(p1,1) = ",11

2.3346581554007031761280770496517654815836121713394 10

11"F(p,1) - F(p1,1) = ", 2.4436708796499417351838966867210343685139 10 20> ############ Asymptotic expansion of xi_2, Section 4.3 ########(32)

devxi2(3,1);# computes the asymptotic expansion of xi_2 (cf. Proposition 4.7
and (4.27))# sqrt(2*xi)*(1+co[1]/t+ ... +co[nmax]/t^nmax) + O(1/t^(nmax+1))# "t" is equal to log(xi), "a" to log(2), "imp" is a printing
parametera1"asy=", 1
O 2
2tt"asy=", 1"asy=", 1

1
a
2

a
2t

3 a2
8

1
a
2

a
2t

O

2

t

1
t3

3 a2
8

t2

1
a
2

2
1
a
2
2

3 2
a
8

1
a
2

3a
2

5 a3
16

3

t
1

1
a
2
t

1
a
2

3 2
a
8
t2

1
a
2

a2

O

1
t4

5 3
a
16

t3

(33)

> devxik(3,k,0);# computes the asymptotic expansion of xi_k (not given in thepaper)# (k*xi)^(1/k)*(1+co[1]/t+ ... +co[nmax]/t^nmax) + O(1/t^(nmax+1))# "t" is equal to log(xi), "a" to log(k)(34)

# "imp" is a printing parametera
k 1 a2a
3 k 2 a2
2 k2 3 k
k
k
a
2 k2
2 k2
6 k3
1
kt
t2
t3
> ################## upper bound of xi_2, Section 4.3##################

1 a3
(34)

Digits:=10: xi:='xi': beta:='beta':
z:=sqrt(2*xi)*(1-beta/log(xi)); # cf. Proposition 4.7, (4.29)
z

2

1

(35)
ln
> t:='t': A:=collect(normal(2*(t-beta)^2*((a+t)/2-beta/(t-beta))),[t,beta]);
2
2
A
t3
a 2
t2
2a 2
t
a 2
(36)
> B:=evalf(subs(beta=0.323,a=log(2.),x0=10.^9,A-t^3-log(x0)^3/(2*x0)));B0.0471471806 t2 0.9894440787 t 0.2809689024(37)> Brac:=solve(B = 0,t); exp(Brac[1]),exp(Brac[2]);Brac
20.69836665, 0.2879164322
9.754082410 108 , 1.333645850(38)
xi:=10.^9+7: t:='t':xi2:=fsolve(F(t,2)=F(xi,1),> Digits:=30:t);solve(xi2=sqrt(2*xi)*(1-beta/log(xi)),beta);
244023.5388108009996569921476413
0.323360649773548837912893005787> ############### Computation of N0 and xi0, cf. Section 4.4##############

Digits:=50: calcullogN(10^9+7,2): # cf. (4.36) and (4.37)"k=", 1, "xi0[k] = ", 1.000000007 109 , "pp = ", 1000000007, "np = ", 1000000009"k=", 2, "xi0[k] = ", 44023.5388108010, "pp = ", 44021, "np = ", 44027"k=", 3, "xi0[k] = ", 1418.38081890335, "pp = ", 1409, "np = ", 1423
"k=", 4, "xi0[k] = ", 247.382205897601, "pp = ", 241, "np = ", 251"k=", 5, "xi0[k] = ", 85.6248328688747, "pp = ", 83, "np = ", 89"k=", 6, "xi0[k] = ", 41.9055559251791, "pp = ", 41, "np = ", 43"k=", 7, "xi0[k] = ", 25.0431586873868, "pp = ", 23, "np = ", 29"k=", 8, "xi0[k] = ", 16.9729134094723, "pp = ", 13, "np = ", 17"k=", 9, "xi0[k] = ", 12.5175409933509, "pp = ", 11, "np = ", 13"k=", 10, "xi0[k] = ", 9.79795805483709, "pp = ", 7, "np = ", 11"k=", 11, "xi0[k] = ", 8.01061571828594, "pp = ", 7, "np = ", 11"k=", 12, "xi0[k] = ", 6.76795496572222, "pp = ", 5, "np = ", 7"k=", 13, "xi0[k] = ", 5.86506102880587, "pp = ", 5, "np = ", 7(39)

"k=", 14, "xi0[k] = ", 5.18545963462664, "pp = ", 5, "np = ", 7"k=", 15, "xi0[k] = ", 4.65896848724528, "pp = ", 3, "np = ", 5"k=", 16, "xi0[k] = ", 4.24122002838259, "pp = ", 3, "np = ", 5"k=", 17, "xi0[k] = ", 3.90301827869997, "pp = ", 3, "np = ", 5"k=", 18, "xi0[k] = ", 3.62448665807836, "pp = ", 3, "np = ", 5"k=", 19, "xi0[k] = ", 3.39169307626448, "pp = ", 3, "np = ", 5"k=", 20, "xi0[k] = ", 3.19462248738160, "pp = ", 3, "np = ", 5"k=", 21, "xi0[k] = ", 3.02591503456320, "pp = ", 3, "np = ", 5"k=", 22, "xi0[k] = ", 2.88005599510308, "pp = ", 2, "np = ", 3"k=", 23, "xi0[k] = ", 2.75284137279059, "pp = ", 2, "np = ", 3"k=", 24, "xi0[k] = ", 2.64101669040753, "pp = ", 2, "np = ", 3"k=", 25, "xi0[k] = ", 2.54202750288495, "pp = ", 2, "np = ", 3"k=", 26, "xi0[k] = ", 2.45384368850374, "pp = ", 2, "np = ", 3"k=", 27, "xi0[k] = ", 2.37483350469800, "pp = ", 2, "np = ", 3"k=", 28, "xi0[k] = ", 2.30367186118659, "pp = ", 2, "np = ", 3"k=", 29, "xi0[k] = ", 2.23927253465124, "pp = ", 2, "np = ", 3"k=", 30, "xi0[k] = ", 2.18073740435309, "pp = ", 2, "np = ", 3"k=", 31, "xi0[k] = ", 2.12731796681198, "pp = ", 2, "np = ", 3"k=", 32, "xi0[k] = ", 2.07838582871057, "pp = ", 2, "np = ", 3"k=", 33, "xi0[k] = ", 2.03340984655939, "pp = ", 2, "np = ", 3"k=", 34, "xi0[k] = ", 1.99193824400876, "pp = ", 1, "np = ", 2"xi = ", 1000000007, "eps = F(xi,1) = ", 4.82549420554828 10"epsmax = ", 1000000007, 1 , 4.82549420554828 10
4.82549419543158 101111, "xiN = ", 1.000000007 109, "epsmin = ", 1000000009, 1 ,11"N = ", 6.42931756558252 10434300801 , "prime factors of N = ", 2 33 , 3 21 , 5 14 , 7 11 ,11 9 , 13 8 , 17 7 , 19 7 , 23 7 , 29 6 , " ... ", 41 6 , 43 5 , " ... ", 83 5 , 89 4 , " ... ",241 4 , 251 3 , " ... ", 1409 3 , 1423 2 , " ... ", 44021 2 , 44027 , " ... ", 1000000007"q = ", 44021, "expq = ", 2, "p = ", 1000000007, "K(N) = ", 33"sigNsN = ", 36.9096185662946, "logN = ", 1.00001455211884 109 , "loglogN = ",20.7232803889594> Digits:=10: log(2.)-log(33.)/33, 1/0.587; # Proof of Lemma 4.9
0.5871924060, 1.703577513> ############################## Section 4.5###############################
Digits:=20:xi0:=10^9+7: T3(xi0,33,1.71,1): # Proof of Lemma 4.10,(4.40)"k0 = ", 4, "T[k0] = ", 9.59966902532, " = ", 1.4422496, "", 8.15742, "exp(w u/v) = ",940719.

(40)

(41)

"k0 = ", 5, "T[k0] = ", 4.43046730712, " = ", 1.6937363, "18754.7"k0 = ", 6, "T[k0] = ", 3.07824135687, " = ", 1.7807913, "7509.84"k0 = ", 7, "T[k0] = ", 2.57388833657, " = ", 1.8234190, "6366.34"k0 = ", 8, "T[k0] = ", 2.34068860669, " = ", 1.8489133, "7285.05"k0 = ", 9, "T[k0] = ", 2.21644240823, " = ", 1.8662069, "9685.26"k0 = ", 10, "T[k0] = ", 2.14333993845, " = ", 1.8789721, "14029.0"k0 = ", 11, "T[k0] = ", 2.09709318304, " = ", 1.8889721, "21438.6"k0 = ", 12, "T[k0] = ", 2.06618867319, " = ", 1.8971540, "33955.3"k0 = ", 13, "T[k0] = ", 2.04464154228, " = ", 1.9040712, "55132.8"k0 = ", 14, "T[k0] = ", 2.02910662990, " = ", 1.9100691, "91171.7"k0 = ", 15, "T[k0] = ", 2.01760155185, " = ", 1.9153745, "152833.
"k0 = ", 16, "T[k0] = ", 2.00889364064, " = ", 1.9201433, "258901.
"k0 = ", 17, "T[k0] = ", 2.00218498785, " = ", 1.9244860, "442236.
"k0 = ", 18, "T[k0] = ", 1.99694123102, " = ", 1.9284835, "760324.
"k0 = ", 19, "T[k0] = ", 1.99279392180, " = ", 1.9321966, "", 2.73673, "exp(wu/v) = ",", 1.29745, "exp(wu/v) = ",", 0.750469, "exp(wu/v) = ",", 0.491775, "exp(wu/v) = ",", 0.350235, "exp(wu/v) = ",", 0.264368, "exp(wu/v) = ",", 0.208121, "exp(wu/v) = ",", 0.169035, "exp(wu/v) = ",", 0.140570, "exp(wu/v) = ",", 0.119038, "exp(wu/v) = ",", 0.102227, "exp(wu/v) = ",", 0.0887503, "exp(wu/v) = ",", 0.0776990, "exp(wu/v) = ",", 0.0684577, "exp(wu/v) = ",", 0.0605973, "exp(wu/v) = ",1.31428 10"k0 = ", 20, "T[k0] = ", 1.98948252283, " = ", 1.9356719, "", 0.0538107, "exp(wu/v) = ",2.28186 106"k0 = ", 21, "T[k0] = ", 1.98681870231, " = ", 1.9389457, "", 0.0478730, "exp(wu/v) = ",3.97645 106"k0 = ", 22, "T[k0] = ", 1.98466366106, " = ", 1.9420469, "", 0.0426168, "exp(wu/v) = ",", 0.0379145, "exp(wu/v) = ",6

6.95103 106"k0 = ", 23, "T[k0] = ", 1.98291333576, " = ", 1.9449988, "1.21824 107"k0 = ", 24, "T[k0] = ", 1.98148850466, " = ", 1.9478205, "", 0.0336680, "exp(wu/v) = ",2.14000 107"k0 = ", 25, "T[k0] = ", 1.98032802622, " = ", 1.9505277, "", 0.0298004, "exp(wu/v) = ",3.76634 10"k0 = ", 26, "T[k0] = ", 1.97938412773, " = ", 1.9531333, "", 0.0262508, "exp(wu/v) = ",6.63995 107"k0 = ", 27, "T[k0] = ", 1.97861906386, " = ", 1.9556485, "", 0.0229705, "exp(wu/v) = ",1.17236 108"k0 = ", 28, "T[k0] = ", 1.97800270874, " = ", 1.9580827, "", 0.0199200, "exp(wu/v) = ",2.07263 108"k0 = ", 29, "T[k0] = ", 1.97751079519, " = ", 1.9604438, "", 0.0170670, "exp(wu/v) = ",3.66827 10"k0 = ", 30, "T[k0] = ", 1.97712361000, " = ", 1.9627387, "", 0.0143849, "exp(wu/v) = ",6.49946 108"k0 = ", 31, "T[k0] = ", 1.97683617392, " = ", 1.9649735, "", 0.0118627, "exp(wu/v) = ",1.15262 109"k0 = ", 32, "T[k0] = ", 1.97686421205, " = ", 1.9671534, "", 0.00971082,78u/v) = ", 2.04569 109"mink = ", 31, "Tmin = ", 1.9768361739216531769> ################### proof of Lemma 4.11 ##################"exp(w(42)

Digits:=10: xi0:=10^9+7.:
sqrt(2.)*(log(2.)-0.323),
sqrt(2.)-0.524/log(xi0), # Proof ofthe lower bound of (4.43)
2.76/(1+log(2.)/log(xi0)); # Proof of the lower bound of (4.43)
0.5234671627, 1.388927972, 2.670671960(43)> sqrt(2.)*(1-log(2.)/2/log(xi0)),2*log(1.39); # Proof of theupper bound of (4.43)
xi:='xi':
u:=log(2.)/2/log(xi):u0:=log(2.)/2/log(xi0);
v:=0.658/log(xi):v0:=0.658/log(xi0);
1+u/(1-u0)-(1-v0)*v,
sqrt(2.)*numer(u/(1-u0)-(1-v0)*v);
1.390562412, 0.6586074942
u00.01672388864
v00.03175175212
0.2846391174
1, 0.4025405001(44)ln> ##################### Section 4.6, Benefit ####################
p:='p': ap:='ap': eps:='eps':
B:=log((p^(ap+1)-1)/(p^(t+1)-1))+(1+eps)*(t-ap)*log(p);(4.49)# cf.

"dB/dt = ",normal(diff(B,t));# proof of Lemma 4.13
"d^2 B/dt^2 = ",normal(diff(B,t$2));
pap 1 1Bln
eps 1 t ap ln p
pt 1 1"dB/dt = ",ln p

eps pt
t

1
1

p

t

"d^2 B/dt^2 = ",

1

1
1

p

eps
ln p 2(45)
2
pt 1 1> Digits:=10: N:=lisCA[23][7]: eps0:=lisCA[23][1]:"N = ",imprifac(N),"eps0 =",evalf(eps0,5);"N = ", 2 6 , 3 4 , 5 2 , 7 2 , 11 , "...", 37 , "eps0 =", 0.0073854(46)> plot(subs(p=37,ap=1,eps=eps0,B), t=0..1.2); # plot of B_{37,1,eps0}
(t)

0.004
0.002
0

0.2

0.4

0.6

0.8

1

1.2

t

0.002
0.004
0.006
0.008
0.010
> plot(subs(p=3,ap=4,eps=eps0,B), t=3..4.3);(t)# plot of B_{3,4,eps0}0.00100.0005

0

3.2

3.4

3.6

3.8

4

4.2

t

0.00050.0010> N:=lisCA[23][7]: eps:=lisCA[23][1]; benmax:=0.01;
bensmall(N,eps,benmax,N/2,N,4); # cf. proposition 4.14
# computation of integers n s.t. ben_{eps}(n) <= benmax# here, there are 172 such n's and 14 satisfy N/2 <= n <= N [n,sigma(n),ben_eps(n)]eps
0.007385448161
benmax0.01"N = ", 224403121196654400, "nops(lisben)=", 172, "nops(lisbenshort) = ", 14112201560598327200, 7.132476067 1017 , 0.0027859770 , 121298984430624000,
7.661256126 1017 , 0.009805792 , 124331459041389600, 7.883263020 1017 ,0.0061148200 , 128230354969516800, 8.119621877 1017 , 0.007678421 ,
130396408262920800, 8.258656498 1017 , 0.0075746080 , 149602080797769600,
9.543651562 1017 , 0.0013749824 , 160287943711896000, 1.015503747 1018 ,0.008783688 , 165775278721852800, 1.054824646 1018 , 0.0047038254 ,
173861877683894400, 1.105054391 1018 , 0.0061636134 , 187002600997212000,
1.186527610 1018 , 0.0084265640 , 190035075607977600, 1.205513882 1018 ,(47)

0.0087566394 , 192345532454275200, 1.218908480 1018 , 0.0098807944 ,
219060189739591200, 1.389886593 1018 , 0.0096280360 , 224403121196654400,
1437816604247654400, 0.
> map(w->w[1]/N,lisbenshort);# list of the quotients n/N with N/2 <= n <= N and ben_{eps}(n) <=0.01
1 20 41 4 43 2 5 82
86 5 94 6 41
,
,
, ,
, , ,
,
, ,
, ,
,1
(48)
2 37 74 7 74 3 7 111 111 6 111 7 42
> map(w->w[1]/N,lisben);# list of all the quotients n/N with ben_{eps}(n) <= 0.01
1
1
1
1
1
1
2
1
1
1
1
1
,
,
,
,
,
,
,
,
,
,
,
,
(49)
24087 8029 6882 3441 3219 2294 3441 1554 1147 1073 777 651
2
1
2
1
1
1
4
1
5
2
1
1
1
3
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1147 518 777 259 222 217 777 186 777 259 111 93 87 259
41
1
4
1
2
5
2
5
6
1
41
1
1
1
,
,
,
,
,
,
,
,
,
,
,
,
,
,
3441 74 259 62 111 259 93 222 259 42 1554 37 31 29
41
4
10
3
5
1
41
2
43
2
5
1
8
41
3
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1147 111 259 74 111 21 777 37 777 31 74 14 111 518 37
10
2
82
4
86
5 1 41643 1 20
41
4
43
,
,
,
,
,
, ,
,
,
, ,
,
,
,
,
111 21 777 37 777 37 7 259 37 259 6 111 222 21 222
8
5 10 2 82 12 86 1 41
43
47 3 41 1 20 41 4
,
,
, ,
,
,
, ,
,
,
, ,
, ,
,
, ,
37 21 37 7 259 37 259 3 111 111 111 7 93 2 37 74 7
43 2 5 82
86 5 94 6 41
41 43 47 41 4 10 164 3
, , ,
,
, ,
, ,
, 1,
,
,
,
, ,
,
, ,
74 3 7 111 111 6 111 7 42
37 37 37 31 3 7 111 2
172 5 205 41
43 82 86 5 94 8 41
123 10 82
86
, ,
,
, 2,
,
,
, ,
, ,
, 3,
,
,
, 4,
,
111 3 111 21
21 37 37 2 37 3 14
37
3 21
21
164 172
205 4143 246 20 41 43
410 82
86 41
,
, 5,
,
, 6,
,
,
,
,
, 8, 10,
,
, 12,
,
,
37
37
37
7
7
37
3
6
6
37
7
7
3
43 47 1763
41 43 82 86 94 3526
1763 164 172
,
,
, 20,
,
,
,
,
,
, 41, 43, 47,
,
,
,
3
3
111
2
2
3
3
3
111
37
3
3
205
3526
1763 3526, 82, 86, 94,, 123, 164, 172, 205, 246, 410,,
, 1763, 3526
3
373
3
> ############################ Section 4.8 Lemma 4.16#################
Digits:= 10: eta:=7.5*10^(-7): 1.9769*(1+eta), # cf. (4.63),(4.64)
and (4.55)
2.^(1/4)*log(10.^38)^2/(5*10.^37)^(1/12)/32/evalf(Pi),1.977 +
0.066;
1.976901483, 0.06536953335, 2.043(50)> "log(t)/t^(1/4) is decreasing for t > ",exp(1./(1/4)); # cf. (1.12)
0.89*(10.^9)^(1/4),
log(10.^9);# proof of the lower bound of(4.55)
1.94*2.^(1/4),
2.31/(10.^9)^(1/12),
0.945-0.411;(51)

"log(t)/t^(1/4) is decreasing for t ", 54.59815003
158.2668675, 20.72326584
2.307061803, 0.4107825437, 0.534(51)> 1/8/evalf(Pi)+sqrt(2.)/log(10.^9)^2 +2.043/(10.^9)^(1/6)/log(10.^9)^2,
1/8/evalf(Pi),0.0433*log(10.^9)^2/sqrt(10.^9); # proof of(4.66)
0.04323222393, 0.03978873576, 0.0005880365121(52)> (1-0.00059)-1/1.0006; # this difference is positive, proof of thelower bound of (4.56)
9.6402 10 6(53)
> 1+0.0006/log(10.^9), # proof of (4.57)
1-log(1.0006)/log(10.^9)-1/1.00003; # this difference is positive
1.000028953, 1.05481696 10 6(54)
> 0.0433*sqrt(1.0006)*1.00003^2;# proof of (4.58)
0.04331558687(55)> t:='t': int((log(t)+2)/(2*t^(3/2)*log(t)^2),t), # proof of (4.59)
0.044*1.0006^(3/2)/2/log(10.^9)*(1+2/log(10.^9)),0.0012*log(10.^9)^2/(10.^9)^(1/3);
1, 0.001165112285, 0.0005153444965(56)
t ln t> t:='t': int((log(t)+4)/(2*t^(3/2)*log(t)^3),t), # proof of (4.60)
0.044*(1.0006)^(3/2)/2/log(10.^9)^2*(1+4/log(10.^9)),0.000062*log(10.^9)^2/1000;
1, 0.00006117087221, 0.00002662613232(57)
t ln t 2> t:='t': int((log(t)+1)/(t^2*log(t)^2),t), # proof of (4.61)
0.044*1.0006^2/log(10.^9)*(1+1/log(10.^9)),0.0023*log(10.^9)^2/(10.^9)^(1/3),
0.00099*1.0006^(7/6);
1, 0.002228344812, 0.0009877436183, 0.0009906930346(58)
ln t t> 1.0006^(1/3)*log(10.^9)^3/1000, # proof of (4.62)
0.0521*0.044*(1.0006)^(7/6)*8.91;
8.901463748, 0.02043958241(59)> ################# Proof of Lemma 4.17 #################
Digits:=10:(4.68)

sqrt(2.)*log(2.)/2,

0.001*sqrt(2.); # proof of

0.4901290716, 0.001414213562(60)> sqrt(2.)+2.043/10.^(9/6),
1.48*sqrt(1.0006),# estimate of u
1.49/sqrt(10.^9),
2*(1-0.00005),# estimate of u_0
1.49^2/1.9999;
1.478818895, 1.480443933, 0.00004711793713, 1.99990, 1.110105505(61)> 2.043+1.12/1000,# estimate of utilda
sqrt(2.)+2.05/10.^(9/6),
1.48*sqrt(1.0006),
1.49/sqrt(10.^9)/log(10.^9), 2*(1-0.000003);# estimate ofutilda_0
2.044120000, 1.479040254, 1.480443933, 2.273673343 10 6 , 1.999994(62)

> 2.05+1.49^2/1.999994/log(10.^9)/1000,
2.06*1.0006/log(10.^9),
0.1+0.001*sqrt(2.), sqrt(2.)*0.323; # proof of (4.69)
2.050053566, 0.09946482451, 0.1014142136, 0.4567909805> ######################## Section 5, proof of Proposition 5.1#################(63)

Digits:=15: 1.26*3.^(1/3)/(1-1/(10.^9+7))/log(1418.); # estimateof U_1, cf. (5.5)
0.250411159228125(64)> 0.086/(1-1/1418.^3); # estimate of U_2, cf. (5.6)
0.0860000000301627(65)> 2*log(44023.)^4/44023; # estimate of U_4
0.593829976589628(66)> 10.26+0.6+0.6, log(44023.)-11.46/2; # estimate of U_3 - U_4
11.46, 4.96246750359880(67)> 2.829+0.524; # proof of (5.7)
3.353
(68)
> 2/log(44023.),0.4+0.81*2.67; # proof of (5.8)
0.187047564028308, 2.5627(69)> 2.062-0.456; # proof of (5.9)
1.606
(70)
> 0.055-0.491; # proof of (5.10)
0.436
(71)
> 3.353+0.436; # proof of (5.11)
3.789
(72)
> 2.5627-1.606, 0.251+0.087+0.11; # proof of (5.12)
0.9567, 0.448(73)> 0.0015*(1.0006)^(2/3), -0.448*1.0006^(2/3); # proof of (5.13) and(5.14)
0.00150059994001599, 0.448179182084776(74)> (2*sqrt(2.)-2)*0.00052+3.789*0.000027+0.021, # proof of (5.15)(2*sqrt(2.)-2)*0.00052+0.9567*0.000027+0.021; # proof of (5.16)
0.0215330851048680, 0.0214566130048680(75)> 0.00151+0.0216; # proof of (5.17)-(5.18)
0.02311(76)> -0.449-0.0215; # proof of (5.19)
0.4705(77)> 2*sqrt(2.)-2+tau, # proof of (5.20)
2*sqrt(2.)-2-tau-3.789/log(10.^9)-0.2311*log(10.^9)/10.^(9/6); #proof of (5.20)
0.874618542678442, 0.447951612621684(78)> t:='t': normal(diff(log(log(t))/t^(1/3),t)), log(log(18.))*log(18.); # proof of (5.22)
1.0006*(evalf(gamma)+log(log(10.^9)))/10^(9/3),
0.4705+0.0037;
ln ln t ln t
3
, 3.06779760342970
3 ln t t4 3
0.00361063777109819, 0.4742(79)

> 0.875^2/2/(1000*log(10.^9)^2); # end of the proof of Proposition
5.1
8.91394015583440 10 7
> ########################### Proof of Corollary 5.2###################
1.0006*exp(evalf(gamma))/10^(9/3);
0.00178214106144099> ########################### Proof of Lemma 5.3#######################
Digits:=10: log(55440.);

(80)

(81)

10.92305663(82)> Digits:=50: calculdec(1,10,1,1): # computation of ctilda(N) for N
CA >= 12# If M < M' are two consecutive CA numbers of type 2 then displays
c = ctilda(N)# for all CA numbers N s.t. M <= N < M'
# To prove Lemma 5.3, execute calculdec(1,6703,100,0): running timeabout 72 h.
# The result is the list lisctilda.
"q = ", 2, " al = ", 2, " p = ", 3, " sigNsN = ", 2.333333333, " logN = ", 2.484906650, " c = ",3.398180969"q = ", 2, " al = ", 2, " p = ", 5, " sigNsN = ", 2.800000000, " logN = ", 4.094344562, " c = ",1.332112699"******j=", 1, "pmin=", 3, "cmin=", 3.398180969, "cmax=", 1.332112699, "pmax=", 5,"temps=", 0.029"q = ", 2, " al = ", 3, " p = ", 5, " sigNsN = ", 3.000000000, " logN = ", 4.787491743, " c = ",1.066027196"******j=", 2, "pmin=", 5, "cmin=", 1.066027196, "cmax=", 1.066027196, "pmax=", 5,"temps=", 0.044"q = ", 3, " al = ", 2, " p = ", 5, " sigNsN = ", 3.250000000, " logN = ", 5.886104031, " c = ",0.8614476326"q = ", 3, " al = ", 2, " p = ", 7, " sigNsN = ", 3.714285714, " logN = ", 7.832014181, " c = ",0.6094796865"******j=", 3, "pmin=", 5, "cmin=", 0.8614476326, "cmax=", 0.6094796865, "pmax=", 7,"temps=", 0.068"q = ", 2, " al = ", 4, " p = ", 7, " sigNsN = ", 3.838095238, " logN = ", 8.525161361, " c = ",0.5726809252"q = ", 2, " al = ", 4, " p = ", 11, " sigNsN = ", 4.187012987, " logN = ", 10.92305663, " c = ",0.5248805923"q = ", 2, " al = ", 4, " p = ", 13, " sigNsN = ", 4.509090909, " logN = ", 13.48800599, " c = ",0.4976624367"******j=", 4, "pmin=", 7, "cmin=", 0.5726809252, "cmax=", 0.4976624367, "pmax=",
13, "temps=", 0.112"q = ", 2, " al = ", 5, " p = ", 13, " sigNsN = ", 4.581818182, " logN = ", 14.18115317, " c = ",0.5005433268"******j=", 5, "pmin=", 13, "cmin=", 0.5005433268, "cmax=", 0.5005433268, "pmax=",
13, "temps=", 0.129"q = ", 3, " al = ", 3, " p = ", 13, " sigNsN = ", 4.699300699, " logN = ", 15.27976546, " c = ",0.4964581984"******j=", 6, "pmin=", 13, "cmin=", 0.4964581984, "cmax=", 0.4964581984, "pmax=",
13, "temps=", 0.146"q = ", 5, " al = ", 2, " p = ", 13, " sigNsN = ", 4.855944056, " logN = ", 16.88920337, " c = ",0.4974482771"q = ", 5, " al = ", 2, " p = ", 17, " sigNsN = ", 5.141587824, " logN = ", 19.72241672, " c = ",0.4508392373"q = ", 5, " al = ", 2, " p = ", 19, " sigNsN = ", 5.412197709, " logN = ", 22.66685570, " c = ",0.3938637821"q = ", 5, " al = ", 2, " p = ", 23, " sigNsN = ", 5.647510653, " logN = ", 25.80234991, " c = ",0.3678054302"******j=", 7, "pmin=", 13, "cmin=", 0.4974482771, "cmax=", 0.3678054302, "pmax=",
23, "temps=", 0.201"q = ", 2, " al = ", 6, " p = ", 23, " sigNsN = ", 5.692332166, " logN = ", 26.49549709, " c = ",0.3679646547"q = ", 2, " al = ", 6, " p = ", 29, " sigNsN = ", 5.888619483, " logN = ", 29.86279292, " c = ",0.3810822121"q = ", 2, " al = ", 6, " p = ", 31, " sigNsN = ", 6.078574950, " logN = ", 33.29678013, " c = ",0.3789034836"******j=", 8, "pmin=", 29, "cmin=", 0.3810822121, "cmax=", 0.3679646547, "pmax=",
23, "temps=", 0.254"q = ", 7, " al = ", 2, " p = ", 31, " sigNsN = ", 6.187120931, " logN = ", 35.24269028, " c = ",0.3629582112"******j=", 9, "pmin=", 31, "cmin=", 0.3629582112, "cmax=", 0.3629582112, "pmax=",
31, "temps=", 0.272"q = ", 3, " al = ", 4, " p = ", 31, " sigNsN = ", 6.238680272, " logN = ", 36.34130256, " c = ",0.3665518795"q = ", 3, " al = ", 4, " p = ", 37, " sigNsN = ", 6.407293252, " logN = ", 39.95222048, " c = ",0.3626491413"q = ", 3, " al = ", 4, " p = ", 41, " sigNsN = ", 6.563568698, " logN = ", 43.66579254, " c = ",0.3640020265"q = ", 3, " al = ", 4, " p = ", 43, " sigNsN = ", 6.716209830, " logN = ", 47.42699266, " c = ",0.3526741340"******j=", 10, "pmin=", 31, "cmin=", 0.3665518795, "cmax=", 0.3526741340, "pmax=",
43, "temps=", 0.325"jmaxabs = ", 10, "pmaxabs = ", 43, "cmaxabs = ",0.3526741340, "total time = ", 0.331

(83)

(83)
> Digits:=50: maxi:=-infinity: # find the maximum of ctilda(N)for j from 1 to 6703 do
lisj:=lisctilda[j];if lisj[4] > maxi then maxi:=lisj[4]; pmaxi:=lisj[5]; fi;
od: "maxi = ",evalf(maxi,20),"pmaxi = ",pmaxi; calcullogN(pmaxi,1):"maxi = ", 0.098377958771985568059, "pmaxi = ", 504457601"xi = ", 504457601, "eps = F(xi,1) = ", 9.89234846896803 10"epsmax = ", 504457601, 1 , 9.89234846896803 10
9.89234723366078 101111, "xiN = ", 5.04457601 108, "epsmin = ", 504457661, 1 ,11"N = ", 6.34735810272381 10219087008 , "prime factors of N = ", 2 32 , 3 20 , 5 13 , 7 11 ,11 9 , 13 8 , 17 7 , 19 7 , 23 6 , " ... ", 37 6 , 41 5 , " ... ", 73 5 , 79 4 , " ... ",199 4 , 211 3 , " ... ", 1123 3 , 1129 2 , " ... ", 31249 2 , 31253 , " ... ", 504457601"q = ", 31249, "expq = ", 2, "p = ", 504457601, "K(N) = ", 32"sigNsN = ", 35.6908715732266, "logN = ", 5.04466480537506 108 , "loglogN = ",(84)
20.0390119545943> nops(lisctilda), lisctilda[3694];
6703, 3694, 504413407, 0.098379723870700230905406315925520229988418716365788,(85)
0.098377958771985568058597609108698278088850350228342, 504457601> calcc:=proc(n) local signsn,logn,lln,sqlogn,a,b,c; global ega,tau;signsn:=numtheory[sigma](n)/n; a:=2*(sqrt(2.)-1)+tau; b:=3.789;
logn:=log(evalf(n));
lln:=log(logn);
sqlogn:=sqrt(logn);
c:=solve(signsn/ega=lln-a/sqlogn+b/sqlogn/lln+t*lln/logn^(2/3),t);
end: # computation of ctilda(n)> cmax:=-infinity:
ti:=time():# computation of ctilda(n) for 2<= n <= 55440
for n from 2 to 55440 do
c:=calcc(n);if c > cmax then cmax:=c; nmax:=n; fi;
od: "cmax = ",evalf(cmax,10), "nmax = ",nmax,time()-ti;"cmax = ", 0.5248805923, "nmax = ", 55440, 230.754(86)> ############################# Proof of Lemma 6.1#####################
a:='a': t:='t': f:=log(t)-a/sqrt(t): f1:=diff(f,t):t): f1,f2;
1
a
1
3a
,
3
2
2
t
2t
t
4 t5 2> ############################# Proof of Lemma 6.2#####################

f2:=diff(f1,(87)> Digits:=10:log(log(8.))-1/sqrt(log(8.));0.0386310224(88)> a:='a':eps:='eps': g:=eps*t-log(log(t)-a/sqrt(t)): g1:=diff(g,t):g2:=diff(g1,t):g,g1,g2;(89)eps tln ln t

a
t

, eps

1
t
ln t

a
2 t3

2

a

,

1

3a

t2

4 t5 2
a

ln t

t

t

1
t
ln t

2

a
2 t3

2

a

2

(89)

t
> # example of computation of an upper bound of ben(n) by (6.2)
Np:=lisCA[15][7]: Npf:=evalf(Np): # Np < N are two consecutive CA
numbers
N:=lisCA[16][7]: sigNsN:=lisCA[16][6]: eps0:=lisCA[16][1]: # eps0
is a parameter of N
Nf:=evalf(N): sigNsNf:=evalf(sigNsN): a0:=0.59:"Np = ",Np,"N = ",N,"sigNsN = ",sigNsN,"a = ",a0,"eps = ",evalf(eps0,6);
B:=subs(a=a0,eps=eps0,g)-gamma-eps0*log(Nf)+log(sigNsNf): # cf.
(6.2)
plot(B,t=log(Np)..log(N)); # B(log(n) is an upper bound of ben_eps0(n)# for an n such that alpha(n) <= a0"Np = ", 367567200, "N = ", 6983776800, "sigNsN = ",0.0174204

249984, "a = ", 0.59, "eps = ",46189

0.0138
0.01360.01340.01320.01300.01280.01260.0124
20

20.5

21

21.5

22

22.5

t
> t0:=fsolve(subs(a=a0,eps=eps0,g1)=0,t=log(Np)..log(N));"mingt0 = ",evalf(subs(t=t0,B)),"maxgNp = ",evalf(subs(t=log(Npf),B)),"maxgN = ",evalf(subs(t=log(Nf),B));
t0
20.96756253"mingt0 = ", 0.0122664121, "maxgNp = ", 0.0132123741, "maxgN = ", 0.0138346331(90)> ################# proof of Lemma 6.3 #####################
Digits:=10:/10.^(9/6);-2*(sqrt(2.)-1)+tau+3.789/log(10.^9)+0.026*log(10.^9)

0.5823592218(91)> Digits:=50: maxi:=-infinity: # find the maximum of alpha(N)for j from 1 to 6703 do
lisj:=lisalpha[j];if lisj[4] > maxi then maxi:=lisj[4]; pmaxi:=lisj[5]; fi;
od: "maxi = ",evalf(maxi,20),"pmaxi = ",pmaxi;
calcullogN(pmaxi,
1):"maxi = ", 0.92019027627094120029, "pmaxi = ", 1019
"xi = ", 1019, "eps = F(xi,1) = ", 0.000141610069575099, "xiN = ", 1019.
"epsmax = ", 1019, 1 , 0.000141610069575099, "epsmin = ", 1021, 1 ,0.000141292813097974(92)

"N = ", 1.35299473675479 10450 , "prime factors of N = ", 2 12 , 3 7 , 5 5 , 7 4 , 11 3 ,13 3 , 17 2 , " ... ", 41 2 , 43 , " ... ", 1019
"q = ", 41, "expq = ", 2, "p = ", 1019, "K(N) = ", 12"sigNsN = ", 12.3160966345326, "logN = ", 1036.46561230645, "loglogN = ",(92)
6.94357175458302> Digits:=50: calculdealpha(1,12,1,2):# computes alpha(N) for the 34 first CA numbers N but N = 1,2,6# To prove Lemma 6.3 execute calculdealpha(1,6703,100,0): running
time about 76h"N = ", 12, " p = ", 3, " sigNsN = ", 2.333333333, " logN = ", 2.484906650, " alpha(N) = ",0.6302865758"N = ", 60, " p = ", 5, " sigNsN = ", 2.800000000, " logN = ", 4.094344562, " alpha(N) = ",0.3287697545"******j=", 1, "pmin=", 3, "alphamin=", 0.6302865758, "alphamax=", 0.3287697545,"pmax=", 5, "temps=", 0.023"N = ", 120, " p = ", 5, " sigNsN = ", 3.000000000, " logN = ", 4.787491743, " alpha(N) = ",0.2590015407"******j=", 2, "pmin=", 5, "alphamin=", 0.2590015407, "alphamax=", 0.2590015407,"pmax=", 5, "temps=", 0.040"N = ", 360, " p = ", 5, " sigNsN = ", 3.250000000, " logN = ", 5.886104031, " alpha(N) = ",0.1265202179"N = ", 2520, " p = ", 7, " sigNsN = ", 3.714285714, " logN = ", 7.832014181, " alpha(N) = ",0.07612462493"******j=", 3, "pmin=", 5, "alphamin=", 0.1265202179, "alphamax=", 0.07612462493,"pmax=", 7, "temps=", 0.065"N = ", 5040, " p = ", 7, " sigNsN = ", 3.838095238, " logN = ", 8.525161361, " alpha(N) = ",0.03478348953"N = ", 55440, " p = ", 11, " sigNsN = ", 4.187012987, " logN = ", 10.92305663," alpha(N) = ", 0.1323247680"N = ", 720720, " p = ", 13, " sigNsN = ", 4.509090909, " logN = ", 13.48800599," alpha(N) = ", 0.2575558825"******j=", 4, "pmin=", 7, "alphamin=", 0.03478348953, "alphamax=", 0.2575558825,"pmax=", 13, "temps=", 0.098"N = ", 1441440, " p = ", 13, " sigNsN = ", 4.581818182, " logN = ", 14.18115317," alpha(N) = ", 0.2990357813"******j=", 5, "pmin=", 13, "alphamin=", 0.2990357813, "alphamax=", 0.2990357813,"pmax=", 13, "temps=", 0.113"N = ", 4324320, " p = ", 13, " sigNsN = ", 4.699300699, " logN = ", 15.27976546," alpha(N) = ", 0.3442304680"******j=", 6, "pmin=", 13, "alphamin=", 0.3442304680, "alphamax=", 0.3442304680,

"pmax=", 13, "temps=", 0.125"N = ", 21621600, " p = ", 13, " sigNsN = ", 4.855944056, " logN = ", 16.88920337," alpha(N) = ", 0.4120280169"N = ", 367567200, " p = ", 17, " sigNsN = ", 5.141587824, " logN = ", 19.72241672," alpha(N) = ", 0.4217284365"N = ", 6983776800, " p = ", 19, " sigNsN = ", 5.412197709, " logN = ", 22.66685570," alpha(N) = ", 0.3912282441"N = ", 1.606268664 1011 , " p = ", 23, " sigNsN = ", 5.647510653, " logN = ", 25.80234991," alpha(N) = ", 0.4044234073"******j=", 7, "pmin=", 19, "alphamin=", 0.3912282441, "alphamax=", 0.4217284365,"pmax=", 17, "temps=", 0.193"N = ", 3.212537328 1011 , " p = ", 23, " sigNsN = ", 5.692332166, " logN = ", 26.49549709," alpha(N) = ", 0.4167364287"N = ", 9.316358251 1012 , " p = ", 29, " sigNsN = ", 5.888619483, " logN = ", 29.86279292," alpha(N) = ", 0.4939642677"N = ", 2.888071058 1014 , " p = ", 31, " sigNsN = ", 6.078574950, " logN = ", 33.29678013," alpha(N) = ", 0.5342589702"******j=", 8, "pmin=", 23, "alphamin=", 0.4167364287, "alphamax=", 0.5342589702,"pmax=", 31, "temps=", 0.240"N = ", 2.021649741 1015 , " p = ", 31, " sigNsN = ", 6.187120931, " logN = ", 35.24269028," alpha(N) = ", 0.5250314374"******j=", 9, "pmin=", 31, "alphamin=", 0.5250314374, "alphamax=", 0.5250314374,"pmax=", 31, "temps=", 0.259"N = ", 6.064949222 1015 , " p = ", 31, " sigNsN = ", 6.238680272, " logN = ", 36.34130256," alpha(N) = ", 0.5436913001"N = ", 2.244031212 1017 , " p = ", 37, " sigNsN = ", 6.407293252, " logN = ", 39.95222048," alpha(N) = ", 0.5704418439"N = ", 9.200527969 1018 , " p = ", 41, " sigNsN = ", 6.563568698, " logN = ", 43.66579254," alpha(N) = ", 0.6038870108"N = ", 3.956227027 1020 , " p = ", 43, " sigNsN = ", 6.716209830, " logN = ", 47.42699266," alpha(N) = ", 0.6081793561"******j=", 10, "pmin=", 31, "alphamin=", 0.5436913001, "alphamax=", 0.6081793561,"pmax=", 43, "temps=", 0.321"N = ", 7.912454053 1020 , " p = ", 43, " sigNsN = ", 6.742651601, " logN = ", 48.12013984," alpha(N) = ", 0.6102717490"N = ", 3.718853405 1022 , " p = ", 47, " sigNsN = ", 6.886112273, " logN = ", 51.97028744,

" alpha(N) = ", 0.6084367998"N = ", 1.970992305 1024 , " p = ", 53, " sigNsN = ", 7.016038920, " logN = ", 55.94057936," alpha(N) = ", 0.6362562023"N = ", 1.162885460 1026 , " p = ", 59, " sigNsN = ", 7.134954834, " logN = ", 60.01811680," alpha(N) = ", 0.6868487398"******j=", 11, "pmin=", 47, "alphamin=", 0.6084367998, "alphamax=", 0.6868487398,"pmax=", 59, "temps=", 0.371"N = ", 5.814427299 1026 , " p = ", 59, " sigNsN = ", 7.180986801, " logN = ", 61.62755471," alpha(N) = ", 0.7008449240"N = ", 3.546800652 1028 , " p = ", 61, " sigNsN = ", 7.298707896, " logN = ", 65.73842858," alpha(N) = ", 0.7115100315"N = ", 2.376356437 1030 , " p = ", 67, " sigNsN = ", 7.407643834, " logN = ", 69.94312120," alpha(N) = ", 0.7408999030"N = ", 1.687213070 1032 , " p = ", 71, " sigNsN = ", 7.511976846, " logN = ", 74.20580107," alpha(N) = ", 0.7681497915"N = ", 1.231665541 1034 , " p = ", 73, " sigNsN = ", 7.614880639, " logN = ", 78.49626051," alpha(N) = ", 0.7761552501"******j=", 12, "pmin=", 59, "alphamin=", 0.7008449240, "alphamax=", 0.7761552501,"pmax=", 73, "temps=", 0.446"jmaxabs = ", 12, "pmaxabs = ", 73, "alphamaxabs = ", 0.7761552501, "total time = ", 0.452(93)> lisalpha[11]; # lisalpha[j] = [j,pmin,alphamin,alphamax,pmax]# alphamin is the minimum of alpha(N) for lisCA2[j]<= N < lisCA2[j+1]# alphamax is the maximum of alpha(N) for lisCA2[j]<= N < lisCA2[j+1]# N is a CA number. lisCA2[j] is the jth CA number oftype 2
11, 47, 0.60843679983847274863932478827521260192380048034959,(94)
0.68684873978232983923997858799284619471553635588955, 59> calculdealpha(34,35,1,2); # displays the largest value of alpha(N)for a CA number N <= N0# this value has been obtained by
calculdealpha(1,6703,100,0):"N = ", 4.810464883 10406 , " p = ", 919, " sigNsN = ", 12.13320312, " logN = ", 936.4203415," alpha(N) = ", 0.9107618420"N = ", 4.468921876 10409 , " p = ", 929, " sigNsN = ", 12.14626362, " logN = ", 943.2544502," alpha(N) = ", 0.9121958641"******j=", 34, "pmin=", 919, "alphamin=", 0.9107618420, "alphamax=", 0.9121958641,"pmax=", 929, "temps=", 0.018"N = ", 1.832257969 10411 , " p = ", 929, " sigNsN = ", 12.15331720, " logN = ", 946.9680223,

" alpha(N) = ", 0.9130342331"N = ", 1.716825717 10414 , " p = ", 937, " sigNsN = ", 12.16628766, " logN = ", 953.8107056," alpha(N) = ", 0.9137797139"N = ", 1.615533000 10417 , " p = ", 941, " sigNsN = ", 12.17921676, " logN = ", 960.6576487," alpha(N) = ", 0.9137589944"N = ", 1.529909751 10420 , " p = ", 947, " sigNsN = ", 12.19207760, " logN = ", 967.5109478," alpha(N) = ", 0.9135225521"N = ", 1.458003992 10423 , " p = ", 953, " sigNsN = ", 12.20487097, " logN = ", 974.3705627," alpha(N) = ", 0.9130715450"N = ", 1.409889861 10426 , " p = ", 967, " sigNsN = ", 12.21749234, " logN = ", 981.2447612," alpha(N) = ", 0.9145279524"N = ", 1.369003055 10429 , " p = ", 971, " sigNsN = ", 12.23007472, " logN = ", 988.1230877," alpha(N) = ", 0.9152394938"N = ", 1.337515985 10432 , " p = ", 977, " sigNsN = ", 12.24259271, " logN = ", 995.0075743," alpha(N) = ", 0.9157326808"N = ", 1.314778213 10435 , " p = ", 983, " sigNsN = ", 12.25504703, " logN = ", 1001.898183," alpha(N) = ", 0.9160087452"N = ", 1.302945209 10438 , " p = ", 991, " sigNsN = ", 12.26741337, " logN = ", 1008.796898," alpha(N) = ", 0.9165793916"N = ", 1.299036373 10441 , " p = ", 997, " sigNsN = ", 12.27971770, " logN = ", 1015.701649," alpha(N) = ", 0.9169330864"N = ", 1.310727701 10444 , " p = ", 1009, " sigNsN = ", 12.29188789, " logN = ",1022.618364, " alpha(N) = ", 0.9185673900"N = ", 1.327767161 10447 , " p = ", 1013, " sigNsN = ", 12.30402203, " logN = ",1029.539035, " alpha(N) = ", 0.9194876162"N = ", 1.352994737 10450 , " p = ", 1019, " sigNsN = ", 12.31609663, " logN = ",1036.465612, " alpha(N) = ", 0.9201902763"N = ", 1.381407626 10453 , " p = ", 1021, " sigNsN = ", 12.32815941, " logN = ",1043.394150, " alpha(N) = ", 0.9197001193"******j=", 35, "pmin=", 929, "alphamin=", 0.9130342331, "alphamax=", 0.9201902763,"pmax=", 1019, "temps=", 0.197"jmaxabs = ", 35, "pmaxabs = ", 1019, "alphamaxabs = ", 0.9201902763, "total time = ", 0.203
34, 919, 0.91076184195392898648346320327782052459342719285114,
0.91219586413522343596286513552897764580685101456257, 929 , 35, 929,
0.91303423312912968562194027786121982011541907103530,(95)

0.92019027627094120028705021722227056223156281666467, 1019
> ############### proof of Corollary 1.2 (i), (1.10), Section 6.2###########
Digits:=15: N_eps_24:=lisCA[24][7]: "The 24th CA number is",N_eps_24, " = ";ifactor(N_eps_24)," =",evalf(N_eps_24,4);"the common parameter of N_eps_24 and N_eps_23 is F(41,1) = ",evalf(F(41,1));"L_alpha(N_eps_24,0.582) = ",Lalpha(24,0.582,0);"The 24th CA number is", 9200527969062830400, " = "2 6 3 4 5 2 7 2 11 13 17 19 23 29 31 37 41 , " =", 9.201 1018
"the common parameter of N_eps_24 and N_eps_23 is F(41,1) = ", 0.00648904912742070"L_alpha(N_eps_24,0.582) = ",(96)
> ############### proof of Corollary 1.2 (ii), Figure 1, Section 6.3###########
Digits:=15: L_hat:=Lhat(9,24,0.582,2):"L^hat = ", map(u->u[2],sort(L_hat,proc(u,v) u[2]< v[2] end));# compute the list L^{hat} defined in 6.2# L_ben is the list L_ben(N,0.582) and L_alpha is L_alpha(N,0.582)"i = ", 24, "Np = ", 224403121196654400, "N = ", 9200527969062830400, "alp = ", 0.582"nops(L_ben) = ", 3, "nops(L_alpha) = ", 0"i = ", 23, "Np = ", 6064949221531200, "N = ", 224403121196654400, "alp = ", 0.582"nops(L_ben) = ", 5, "nops(L_alpha) = ", 3"i = ", 22, "Np = ", 2021649740510400, "N = ", 6064949221531200, "alp = ", 0.582"nops(L_ben) = ", 3, "nops(L_alpha) = ", 2"i = ", 21, "Np = ", 288807105787200, "N = ", 2021649740510400, "alp = ", 0.582"nops(L_ben) = ", 5, "nops(L_alpha) = ", 4"i = ", 20, "Np = ", 9316358251200, "N = ", 288807105787200, "alp = ", 0.582"nops(L_ben) = ", 18, "nops(L_alpha) = ", 9"i = ", 19, "Np = ", 321253732800, "N = ", 9316358251200, "alp = ", 0.582"nops(L_ben) = ", 21, "nops(L_alpha) = ", 12"i = ", 18, "Np = ", 160626866400, "N = ", 321253732800, "alp = ", 0.582"nops(L_ben) = ", 2, "nops(L_alpha) = ", 1"i = ", 17, "Np = ", 6983776800, "N = ", 160626866400, "alp = ", 0.582"nops(L_ben) = ", 16, "nops(L_alpha) = ", 11"i = ", 16, "Np = ", 367567200, "N = ", 6983776800, "alp = ", 0.582"nops(L_ben) = ", 14, "nops(L_alpha) = ", 12"i = ", 15, "Np = ", 21621600, "N = ", 367567200, "alp = ", 0.582"nops(L_ben) = ", 24, "nops(L_alpha) = ", 16"i = ", 14, "Np = ", 4324320, "N = ", 21621600, "alp = ", 0.582"nops(L_ben) = ", 17, "nops(L_alpha) = ", 11"i = ", 13, "Np = ", 1441440, "N = ", 4324320, "alp = ", 0.582

"nops(L_ben) = ", 9, "nops(L_alpha) = ", 8"i = ", 12, "Np = ", 720720, "N = ", 1441440, "alp = ", 0.582"nops(L_ben) = ", 9, "nops(L_alpha) = ", 7"i = ", 11, "Np = ", 55440, "N = ", 720720, "alp = ", 0.582"nops(L_ben) = ", 54, "nops(L_alpha) = ", 32"i = ", 10, "Np = ", 5040, "N = ", 55440, "alp = ", 0.582"nops(L_ben) = ", 64, "nops(L_alpha) = ", 32"i = ", 9, "Np = ", 2520, "N = ", 5040, "alp = ", 0.582"nops(L_ben) = ", 13, "nops(L_alpha) = ", 1
"nops(L_hat) = ", 161, "number of CA, SA, other = ", 15, 58, 88"L^hat = ", 5040, 6300, 6720, 7560, 7920, 8400, 9240, 9360, 10080, 10920, 12600, 13860,
15120, 15840, 16380, 16800, 17640, 18480, 20160, 21840, 22680, 25200, 27720, 30240,
32760, 35280, 36960, 37800, 40320, 42840, 45360, 50400, 55440, 60480, 65520, 75600,
83160, 85680, 90720, 95760, 98280, 100800, 110880, 131040, 138600, 151200, 163800,
166320, 196560, 221760, 262080, 277200, 327600, 332640, 360360, 388080, 393120,
415800, 443520, 471240, 498960, 554400, 655200, 665280, 720720, 831600, 942480,
997920, 1053360, 1081080, 1108800, 1441440, 1663200, 1801800, 1884960, 2162160,
2827440, 2882880, 3603600, 4324320, 5405400, 5765760, 6486480, 7207200, 8648640,
10810800, 12252240, 12972960, 13693680, 14414400, 21621600, 24504480, 27387360,
36756720, 41081040, 43243200, 49008960, 61261200, 73513440, 82162080, 122522400,
136936800, 147026880, 183783600, 205405200, 245044800, 367567200, 410810400,
465585120, 698377680, 735134400, 1102701600, 1163962800, 1396755360, 2327925600,
2793510720, 3491888400, 4655851200, 6983776800, 8454045600, 13967553600,
20951330400, 27935107200, 32125373280, 41902660800, 48886437600, 53542288800,
80313433200, 97772875200, 160626866400, 321253732800, 481880599200,
642507465600, 963761198400, 1124388064800, 1927522396800, 2248776129600,
3373164194400, 4497552259200, 4658179125600, 4979432858400, 6746328388800,
9316358251200, 9958865716800, 13974537376800, 18632716502400, 27949074753600,
32607253879200, 65214507758400, 144403552893600, 195643523275200,
288807105787200, 577614211574400, 866421317361600, 1010824870255200,
2021649740510400, 4043299481020800, 6064949221531200, 12129898443062400,
74801040398884800, 224403121196654400> L_hat[1..10];403806" **n = ", 5040,, 0.0347834895254546 , " n = ", 6300,
,
105
225
127
80
0.466104419770151 , " n = ", 6720,, 0.412387009732545 , " n = ", 7560,
,
35
21

(97)

(98)

403
1922
, 0.412571246633370 , " n = ", 8400,
,
110
525
288
217
0.438089419348736 , " n = ", 9240,, 0.337545919211002 , " n = ", 9360,
,
77
60
39
48
0.551893049840840 , " *n = ", 10080,, 0.0956379758739210 , " n = ", 10920,,
10130.151193721072842 , " n = ", 7920,

0.477998959236476> Lnu(0.582): # computes the elements of Figure 1"nops(L_hat) = ", 161, "nops(L_nu) = ", 16403"k = ", 1, " **n = ", 5040, 2 4 , 3 2 , 5 , 7 , "signsn = ",, "alpha(n) = ",105
0.0347834895254546
39"k = ", 2, " *n = ", 10080, 2 5 , 3 2 , 5 , 7 , "signsn = ",, "alpha(n) = ",10
0.0956379758739210
1612"k = ", 3, " **n = ", 55440, 2 4 , 3 2 , 5 , "...", 11 , "signsn = ",, "alpha(n) = ",385
0.132324767962730
234"k = ", 4, " *n = ", 110880, 2 5 , 3 2 , 5 , "...", 11 , "signsn = ",, "alpha(n) = ",55
0.216922188958564
248"k = ", 5, " **n = ", 720720, 2 4 , 3 2 , 5 , "...", 13 , "signsn = ",, "alpha(n) = ",55
0.257555882476618
252"k = ", 6, " **n = ", 1441440, 2 5 , 3 2 , 5 , "...", 13 , "signsn = ",, "alpha(n) = ",55
0.299035781319322
1984"k = ", 7, " *n = ", 2162160, 2 4 , 3 3 , 5 , "...", 13 , "signsn = ",, "alpha(n) = ",429
0.318975688027496
672"k = ", 8, " **n = ", 4324320, 2 5 , 3 3 , 5 , "...", 13 , "signsn = ",, "alpha(n) = ",143
0.344230467952388
249984"k = ", 9, " **n = ", 6983776800, 2 5 , 3 3 , 5 2 , 7 , "...", 19 , "signsn = ",,46189
"alpha(n) = ", 0.391228244059934
5999616"k = ", 10, " **n = ", 160626866400, 2 5 , 3 3 , 5 2 , 7 , "...", 23 , "signsn = ",,1062347"alpha(n) = ", 0.404423407327998

6047232,
1062347"k = ", 11, " **n = ", 321253732800, 2 6 , 3 3 , 5 2 , 7 , "...", 23 , "signsn = ","alpha(n) = ", 0.416736428677165"k = ", 12, " *n = ", 642507465600, 2 7 , 3 3 , 5 2 , 7 , "...", 23 , "signsn = ",357120,62491"alpha(n) = ", 0.491199055388484"k = ", 13, " **n = ", 9316358251200, 2 6 , 3 3 , 5 2 , 7 , "...", 29 , "signsn = ",181416960, "alpha(n) = ", 0.493964267675243
30808063"k = ", 14, " **n = ", 2021649740510400, 2 6 , 3 3 , 5 2 , 7 2 , 11 , "...", 31 ,
70225920"signsn = ",, "alpha(n) = ", 0.525031437410294
11350339"k = ", 15, " **n = ", 6064949221531200, 2 6 , 3 4 , 5 2 , 7 2 , 11 , "...", 31 ,6437376"signsn = ",, "alpha(n) = ", 0.543691300126343
1031849"k = ", 16, " **n = ", 224403121196654400, 2 6 , 3 4 , 5 2 , 7 2 , 11 , "...", 37 ,244620288"signsn = ",, "alpha(n) = ", 0.570441843864993
38178413> ############################## the end #########################(99)