> ### Computation of the paper "Highly composite numbers and the
Riemann hypothesis ###> read "hcnHR.mpl": # contains the procedures used below> read "hcnHR.m": # this file contains two lists : hcn2HR andlishc10p100># hcn2HR contains the 7739 first shc numbers of type 2 [q,al,
eps,p,(log d(N))/log 2,log N]# computed with "shcntype2(2*10^8,100)"# lishc10p100 contains the 1381 first hc numbers [M,d(M)]# computed with "highlycomp(10^100,0,0)> Digits:=30: # precise computations are carried out with 30
decimals, other ones with 10> lisS:=constrlisteS(10000): # allows the computation of S(t) definedin (1.3)> lisshc:=shc(300,0): # list of shc numbers N with P+(N) <= 300 [q,al,eps,p,d(N),N,xi]> ######> ##############################> ######

Section 1##################> betak(40,0); evalf(op([be[2],be[3],be[4],be[5]]),10);# determines be[j] = beta_j for 1 <= j <= 40, cf. (1.2)
0.5849625007, 0.4150374993, 0.3219280949, 0.2630344058(1)> log2:=log(2.): tautau:=evalf(2+gamma-log(4*Pi)): taup2:=2+tautau:"log2 = ",log2, "tautau = ",tautau,"taup2 = ",evalf(taup2,6); # cf
(1.5)
"log2 = ", 0.693147180559945309417232121458, "tautau = ",(2)
0.04619141793224206762862049581, "taup2 = ", 2.04619> ######> ##############################> ######> #################

Section 2##################

Proof of Lemma 2.1, (2.6) ################> comaxpi(376),comaxpi(2090);"t0=", 376, "comaxpi=", 1.197678720"t0=", 2090, "comaxpi=", 1.159620946> #################

Proof of Lemma 2.2, (2.11) ################(3)>> lempipip1(89693): # computes the champion primes for (p_{i+1}-p_i)/p_{i+1}"x = ", 89693, "ecarmax = ", 0.0006742353"p=", 89689, "pp=", 89753, "diff=", 64, "quo=", 0.0007130681"p=", 79699, "pp=", 79757, "diff=", 58, "quo=", 0.0007272089"p=", 60539, "pp=", 60589, "diff=", 50, "quo=", 0.0008252323"p=", 59281, "pp=", 59333, "diff=", 52, "quo=", 0.0008764094"p=", 58831, "pp=", 58889, "diff=", 58, "quo=", 0.0009849038"p=", 48679, "pp=", 48731, "diff=", 52, "quo=", 0.001067083"p=", 45893, "pp=", 45943, "diff=", 50, "quo=", 0.001088305"p=", 44293, "pp=", 44351, "diff=", 58, "quo=", 0.001307750"p=", 43331, "pp=", 43391, "diff=", 60, "quo=", 0.001382775"p=", 35677, "pp=", 35729, "diff=", 52, "quo=", 0.001455400"p=", 35617, "pp=", 35671, "diff=", 54, "quo=", 0.001513835"p=", 34061, "pp=", 34123, "diff=", 62, "quo=", 0.001816956"p=", 31397, "pp=", 31469, "diff=", 72, "quo=", 0.002287966"p=", 19609, "pp=", 19661, "diff=", 52, "quo=", 0.002644830"p=", 15683, "pp=", 15727, "diff=", 44, "quo=", 0.002797736"p=", 11743, "pp=", 11777, "diff=", 34, "quo=", 0.002886983"p=", 10799, "pp=", 10831, "diff=", 32, "quo=", 0.002954483"p=", 9973, "pp=", 10007, "diff=", 34, "quo=", 0.003397622"p=", 9551, "pp=", 9587, "diff=", 36, "quo=", 0.003755085"p=", 8467, "pp=", 8501, "diff=", 34, "quo=", 0.003999529"p=", 7253, "pp=", 7283, "diff=", 30, "quo=", 0.004119182"p=", 6917, "pp=", 6947, "diff=", 30, "quo=", 0.004318411"p=", 6491, "pp=", 6521, "diff=", 30, "quo=", 0.004600521"p=", 5953, "pp=", 5981, "diff=", 28, "quo=", 0.004681491"p=", 5749, "pp=", 5779, "diff=", 30, "quo=", 0.005191210"p=", 5591, "pp=", 5623, "diff=", 32, "quo=", 0.005690912"p=", 4831, "pp=", 4861, "diff=", 30, "quo=", 0.006171570"p=", 4297, "pp=", 4327, "diff=", 30, "quo=", 0.006933210"p=", 3271, "pp=", 3299, "diff=", 28, "quo=", 0.008487420"p=", 2971, "pp=", 2999, "diff=", 28, "quo=", 0.009336445

"p=", 2477, "pp=", 2503, "diff=", 26, "quo=", 0.01038753"p=", 2179, "pp=", 2203, "diff=", 24, "quo=", 0.01089424"p=", 1951, "pp=", 1973, "diff=", 22, "quo=", 0.01115053"p=", 1669, "pp=", 1693, "diff=", 24, "quo=", 0.01417602"p=", 1327, "pp=", 1361, "diff=", 34, "quo=", 0.02498163"p=", 523, "pp=", 541, "diff=", 18, "quo=", 0.03327172"p=", 317, "pp=", 331, "diff=", 14, "quo=", 0.04229607"p=", 293, "pp=", 307, "diff=", 14, "quo=", 0.04560261"p=", 211, "pp=", 223, "diff=", 12, "quo=", 0.05381166"p=", 199, "pp=", 211, "diff=", 12, "quo=", 0.05687204"p=", 139, "pp=", 149, "diff=", 10, "quo=", 0.06711409"p=", 113, "pp=", 127, "diff=", 14, "quo=", 0.1102362"p=", 47, "pp=", 53, "diff=", 6, "quo=", 0.1132075"p=", 31, "pp=", 37, "diff=", 6, "quo=", 0.1621622"p=", 23, "pp=", 29, "diff=", 6, "quo=", 0.2068966"p=", 13, "pp=", 17, "diff=", 4, "quo=", 0.2352941"p=", 7, "pp=", 11, "diff=", 4, "quo=", 0.3636364"p=", 3, "pp=", 5, "diff=", 2, "quo=", 0.4000000(4)> ############ Section 2.2, logarithmic integral, Table (2.14)################> "Li(0.5) = ",evalf(Li(0.5),10); # computation of the table (2.14)"Li(1.45136) = ", evalf(Li(1.45136),10),"Li(1.45137) = ", evalf(Li(1.45137),10);"Li(1.96904) = ", evalf(Li(1.96904),10),"Li(1.96905) = ", evalf(Li(1.96905),10);"Li(2) = ", evalf(Li(2.),10);"Li(0.5) = ", 0.3786710431"Li(1.45136) = ", 0.00002479135175, "Li(1.45137) = ", 0.000002053962201"Li(1.96904) = ", 0.9999889466, "Li(1.96905) = ", 1.000003706"Li(2) = ", 1.045163780> ############(5)

Section 2.3, function S #########################> sig2:=1-evalf(Pi^2/8)+evalf(2*gamma(1)+gamma^2); # cf. Definition
2.4
sig2
0.046154317295804602757107990375(6)

> ################# Proof of Lemma 2.5, (2.20) ################
assume(x > 1):f:=-int(log(1-1/t^2)/(2*t),t=1..x): series(f,x=infinity,14); f:='f':
1 2
1
1
1
1
1
1
1
O
2
4
6
8
10
12
24
4 x~
16 x~
36 x~
64 x~
100 x~
144 x~
x~14(7)> Snum(10000,0); # computes [S(t),S(t)/sqrt(t)] by using the list
"lisS", cf. (2.19)
0.00839644241796491064816053419078, 0.0000839644241796491064816053419078(8)> psiproc(128,0); # for a_i <= 127 returns [a_i,co_i] where co_i isthe parenthesis of 2.25
2., 0.17595576 , 3., 0.10065818 , 4., 0.13952546 , 5., 0.15401257 , 7., 0.016217800 ,(9)8., 0.048325811 , 9., 0.12129620 , 11., 0.071936988 , 13., 0.085576317 , 16.,
0.098579414 , 17., 0.044331082 , 19., 0.082496117 , 23., 0.030470842 , 25.,
0.042674772 , 27., 0.070810865 , 29., 0.022133708 , 31., 0.054621072 , 32.,
0.069158253 , 37., 0.071899259 , 41., 0.028703965 , 43., 0.031340985 , 47.,
0.0076967229 , 49., 0.031640003 , 53., 0.062823625 , 59., 0.026705691 , 61.,0.0080962657 , 64., 0.042164226 , 67., 0.036905610 , 71., 0.0048988093 , 73.,
0.026166328 , 79., 0.0037322449 , 81., 0.0072177379 , 83., 0.024496878 , 89.,
0.058676039 , 97., 0.050661987 , 101., 0.024288722 , 103., 0.017448653 , 107.,
0.0073928858 , 109., 0.013249339 , 113., 0.014672990 , 121., 0.027020275 ,125., 0.029670043 , 127., 0.00059446945> 0.0672+log(128.)^2/(8*evalf(Pi)*sqrt(128.)); # between (2.25) and(2.26)
0.149994627700364382478980039981(10)> y0:=10.^8: 1/(8*evalf(Pi))+1/(sqrt(y0)*log(y0)^2)*(\log(2*evalf(Pi))+1/2/(y0^2-1));# between (2.28) and (2.29)
0.0397892774059619498693020478960(11)> y0:=10.^8: 0.0924/log(y0)^3; # after (2.30)
0.0000147827401899855530746658564643(12)> ################# Section 2.4 : Four lemmas of calculus###############> #################### Proof of Lemma 2.8 #################> Digits:=10: exp(2/(1/2-be[3])), fsolve(5.12*t^(1/2-be[3])/log(t)^2=
1.52,t=2..10),(13)

>fsolve(5.12*t^(1/2-be[3])/log(t)^2=1.52,t=10^11..10^50);
1.671875165 1010 , 7.373534649, 1.100260199 1040

(13)

> #################### Proof of Lemma 2.10 #################> a[1]:=0.352: a[2]:=0.9132: a[3]:=0.0143:
Phi:=a1*exp((b[3]-b[2]/2)*t)+a2*exp((b[4]-b[2]/2)*t)-a3*t^2;
b

a1 e

1
b t
2 2

3

b

a2 e

4

1
b t
2 2

a3 t2

(14)

> Phi1:=diff(Phi,t);Phi2:=diff(Phi1,t); Phi3:=diff(Phi2,t);
1

2

3

a1 b3

b
1
b2 e 3
2

a1 b3

1
b
2 2

a1 b3

2

b

1
b t
2 2

e

3

1
b
2 2

3

1
b t
2 2

b

e

3

a2 b4

b
1
b2 e 4
2

a2 b4

1
b
2 2

1
b t
2 2

a2 b4

2

b

1
b t
2 2

e

4

1
b
2 2

3

1
b t
2 2

b

e

4

2 a3 t

2 a3

1
b t
2 2

> Phinum:=subs(a1=a[1],a2=a[2],a3=a[3],b[2]=be[2],b[3]=be[3],b[4]=be[4],Phi);Phinum0.352 e0.1225562489 t 0.9132 e0.0294468445 t 0.0143 t2(15)> Phinum1:=subs(a1=a[1],a2=a[2],a3=a[3],b[2]=be[2],b[3]=be[3],b[4]=be[4],Phi1);
Phinum1
0.04313979961 e0.1225562489 t 0.02689085840 e0.0294468445 t 0.0286 t(16)> Phinum2:=subs(a1=a[1],a2=a[2],a3=a[3],b[2]=be[2],b[3]=be[3],b[4]=be[4],Phi2);
Phinum2
0.005287052017 e0.1225562489 t 0.0007918509257 e0.0294468445 t 0.0286(17)> Phinum3:=subs(a1=a[1],a2=a[2],a3=a[3],b[2]=be[2],b[3]=be[3],b[4]=be[4],Phi3); # Phinum3 is positive
Phinum30.0006479612631 e0.1225562489 t 0.00002331751108 e0.0294468445 t(18)> evalf(subs(t=0.,Phinum2)),evalf(subs(t=0.,Phinum1)),evalf(subs(t=0.
,Phinum));t0:=fsolve(subs(Phinum2)=0,t=0...100);evalf(subs(t=t0,Phinum1)),
evalf(subs(t=t0,Phinum));
0.02252109706, 0.07003065801, 1.2652
t0

13.43184612

0.1204470482, 0.602151091(19)> t1:=fsolve(Phinum1=0,t=0..t0);evalf(subs(t=t1,Phinum2)),evalf(subs(t=t1,Phinum));
(20)

t1

3.294863590

0.01981003827, 1.378124554(20)> t2:=fsolve(Phinum1=0,t=t0..100);evalf(subs(t=t2,Phinum2)),evalf(subs(t=t2,Phinum));t2

20.62990647

0.03911357968, 0.001915134(21)> plot(Phinum, t=0..25);

1.2
1.0
0.8
0.6
0.4
0.2

0

5

10

15

20

25

t
> ##################### Proof of Lemma 2.11 ####################> g1:=Li(t): g11:=normal(diff(g1,t)): g12:=normal(diff(g11,t)):"g1 = ",g1," g11 = ",g11, " g12 = ",g12;
11"g1 = ", Li t , " g11 = ",, " g12 = ",ln t
ln t 2 t(22)> g2:=b2*Li(t^b2)-t^b2/log(t): g21:=simplify(diff(g2,t),assume=

(23)

>
positive):g22:=simplify(diff(g21,t),assume=positive): "g2 = ",g2," g21 = ",g21, " g22 = ",g22;
tb2
tb2 1
2
b2 1 ln t tb2 2
b2"g2 = ", b2 Li t
, " g21 = ",, " g22 = ",
(23)
ln t
ln t 2ln t 3> a:='a':g3:=Li(t)/6-a*sqrt(t)/log(t)^2: g31:=normal(diff(g3,t)):g32:=normal(simplify(diff(g31,t),assume=positive)): " g3 = ",g3,"
g31 = ",g31, "g32 = ",g32;
1
a t
1t ln t 2 3 a ln t
12 a
" g3 = ",Li t, " g31 = ",, "g32 = ",
(24)
2
3
6
6
ln t
t ln t1
12

3 ln t 2 a

2
3/2

t

t ln t 2
ln t

72 a

4

> b:='b': g4:=Li(t)/6-b*sqrt(t)/log(t)^3: g41:=normal(diff(g4,t)): "g4 = ",g4," g41 = ",g41;
1b t
1t ln t 3 3 b ln t
18 b
" g4 = ",Li t, " g41 = ",
(25)
3
4
6
6
ln t
t ln t> g5:=c*t^b3/log(t): g51:=normal(diff(g5,t)): "g5 = ",g5,"g51 = ",g51,"exp(1/be[3])= ",evalf(exp(1/be[3]),8);
c tb3
c tb3 ln t b3
"g5 = ",
, "g51 = ",ln t
ln t 2 t

1, "exp(1/be[3])= ", 11.127514(26)> 1/2-0.18/log(2.)-2*tautau/sqrt(2.)/log(2.)^2; ############ end of
Section 2.4
0.104350442813177472877626801997(27)> ########## Section 2.5, Proposition 2.11 : Estimates of A(x)###########> x:=10.^8;-(2/300+0.0009*log(x)^5/sqrt(x)),2/300+log(2.)*log(x)^3/sqrt(x);# proof of (2.51), cf. (2.46) and (2.47)x1.00000000 108
0.1975512646, 0.4399205738> 8-log(x)^3/x^(1/4)/8/evalf(Pi)-0.0009*log(x)^5/sqrt(x); # lower
bound of (2.52), cf. (2.49)
5.322107490

(28)

(29)

> F1til(95),L(1,95)/(95./log(95.)^2), F2til(381),L(2,381)/(381./log(381.)^3); # cf.(2.50)
1.785, 1.784109620, 4.05, 4.040414604(30)

> Q(5,10^8): # upper bound of (2.52) cf. (2.49) with kappa_1 = 5"COMPUTATION of an upper bound of the bracket of (2.50)""ka1 = ", 5, "x = ", 100000000, "log x = ", 18.42068074, "sqrt(x) = ", 10000.0000"T1 = ", 13.28822641"[k,T2[k] = ",3, 4.250187540 , 4, 1.314395987 , 5, 0.6545064041"T2sum = ", 6.219089931"T3 = ", 3.597893553, "T4 = ", 1.468875169, "T5 = ", 0.002770234301, "T6 = ", 0.1908845979"bracket= ", 24.76773989(31)> ######> ###################### Section 3, shc numbers ############> ######> setE(23); # computation of the set E, see (3.1) and (3.5)"set E = ", 1.0000, 2., 1. , 0.63093, 3., 1. , 0.58496, 2., 2. , 0.43068, 5., 1. , 0.41504, 2.,
3. , 0.36907, 3., 2. , 0.35621, 7., 1. , 0.32193, 2., 4. , 0.28906, 11., 1. , 0.27024, 13.,
1. , 0.26303, 2., 5. , 0.26186, 3., 3. , 0.25193, 5., 2. , 0.24465, 17., 1. , 0.23541, 19.,
1. , 0.22239, 2., 6. , 0.22106, 23., 1.
> shc(19,1): # List of shc numbers N with P+(N) <= 19, cf Table(3.20)# the list "lisshc" contains the first 89 shc numbers"i=", 1, 1, 0 , "eps = ", , "xi = ", 1, "N = ", 1, "d(N) = ", 1"i=", 2, 2, 1 , "eps = ", 1.0000, "xi = ", 2, "N = ", 2, 2, 1 , "d(N) = ", 2, 2"i=", 3, 3, 1 , "eps = ", 0.63093, "xi = ", 3, "N = ", 6, 2, 1 , 3, 1 , "d(N) = ", 4, 2 2"i=", 4, 2, 2 , "eps = ", 0.58496, "xi = ", 3.2705, "N = ", 12, 2, 2 , 3, 1 , "d(N) = ", 6,23"i=", 5, 5, 1 , "eps = ", 0.43068, "xi = ", 5, "N = ", 60, 2, 2 , 3, 1 , 5, 1 , "d(N) = ", 12,
2 2 3"i=", 6, 2, 3 , "eps = ", 0.41504, "xi = ", 5.3126, "N = ", 120, 2, 3 , 3, 1 , 5, 1 , "d(N) = ",16, 2 4"i=", 7, 3, 2 , "eps = ", 0.36907, "xi = ", 6.5410, "N = ", 360, 2, 3 , 3, 2 , 5, 1 , "d(N) = ",24, 2 3 3"i=", 8, 7, 1 , "eps = ", 0.35621, "xi = ", 7, "N = ", 2520, 2, 3 , 3, 2 , 5, 1 , 7, 1 ,"d(N) = ", 48, 2 4 3(32)

"i=", 9, 2, 4 , "eps = ", 0.32193, "xi = ", 8.6116, "N = ", 5040, 2, 4 , 3, 2 , 5, 1 , 7, 1 ,"d(N) = ", 60, 2 2 35"i=", 10, 11, 1 , "eps = ", 0.28906, "xi = ", 11, "N = ", 55440, 2, 4 , 3, 2 , 5, 1 , 7, 1 ,11, 1 , "d(N) = ", 120, 2 3 35"i=", 11, 13, 1 , "eps = ", 0.27024, "xi = ", 13, "N = ", 720720, 2, 4 , 3, 2 , 5, ... , 13,"d(N) = ", 240, 2 4 35"i=", 12, 2, 5 , "eps = ", 0.26303, "xi = ", 13.946, "N = ", 1441440, 2, 5 , 3, 2 , 5, ... , 13,"d(N) = ", 288, 2 5 3 2"i=", 13, 3, 3 , "eps = ", 0.26186, "xi = ", 14.112, "N = ", 4324320, 2, 5 , 3, 3 , 5, ... , 13,"d(N) = ", 384, 2 7 3"i=", 14, 5, 2 , "eps = ", 0.25193, "xi = ", 15.664, "N = ", 21621600, 2, 5 , 3, 3 , 5, 2 , 7,... , 13, "d(N) = ", 576, 2 6 3 2"i=", 15, 17, 1 , "eps = ", 0.24465, "xi = ", 17, "N = ", 367567200, 2, 5 , 3, 3 , 5, 2 , 7,... , 17, "d(N) = ", 1152, 2 7 3 2"i=", 16, 19, 1 , "eps = ", 0.23541, "xi = ", 19, "N = ", 6983776800, 2, 5 , 3, 3 , 5, 2 , 7,(33)... , 19, "d(N) = ", 2304, 2 8 3 2> shcntype2(50,1): # compute the shc numbers of type 2 with largest# prime <= 50 (cf. between Lemma 3.12 and Lemma 3.13)# To compute the list of shc numbers N of type 2 with p < 2*
10^8, execute# "hcn2HR:=shcntype2(2*10^8,100):",with Digits = 70, the running
time is about one hour
"kmaxre=", 7.6525953, "kmax=", 7" [q, al, eps, p, (log d(N))/log 2, log N, pi(p), theta(p)] ""i=", 1, "T[i] = ", 2, 2, 0.5849625007, 3, 2.584962501, 2.484906651, 2, 1.791759470 ,"time = ", 0.003"i=", 2, "T[i] = ", 2, 3, 0.4150374989, 5, 4.000000000, 4.787491743, 3, 3.401197382 ,"time = ", 0.003"i=", 3, "T[i] = ", 3, 2, 0.3690702463, 5, 4.584962500, 5.886104032, 3, 3.401197382 ,"time = ", 0.004"i=", 4, "T[i] = ", 2, 4, 0.3219280948, 7, 5.906890595, 8.525161362, 4, 5.347107531 ,"time = ", 0.004"i=", 5, "T[i] = ", 2, 5, 0.2630344058, 13, 8.169925001, 14.18115317, 6, 10.30995216 ,"time = ", 0.004"i=", 6, "T[i] = ", 3, 3, 0.2618595068, 13, 8.584962500, 15.27976546, 6, 10.30995216 ,"time = ", 0.004"i=", 7, "T[i] = ", 5, 2, 0.2519296365, 13, 9.169925001, 16.88920337, 6, 10.30995216 ,"time = ", 0.004"i=", 8, "T[i] = ", 2, 6, 0.2223924217, 19, 11.39231742, 23.36000287, 8, 16.08760448 ,"time = ", 0.004"i=", 9, "T[i] = ", 7, 2, 0.2083678469, 23, 12.97727992, 28.44140724, 9, 19.22309870 ,"time = ", 0.004"i=", 10, "T[i] = ", 3, 4, 0.2031140135, 29, 14.29920802, 32.90731536, 10, 22.59039453 ,"time = ", 0.004"i=", 11, "T[i] = ", 2, 7, 0.1926450780, 31, 15.49185310, 37.03444974, 11, 26.02438173 ,"time = ", 0.004"i=", 12, "T[i] = ", 5, 3, 0.1787469216, 47, 19.90689060, 53.57972535, 15, 40.96021943 ,"time = ", 0.005"total time = ", 0.005(34)> ######### small differences between two epsilons #################> Digits:=30: mineps22(1,7738,hcn2HR):log(1.2)/log(101.)-log(1.5)/log(28669.);# smallest difference between two epsilons of type 2"mineps22=", 5.787955048257834350711 10-10 , "epsmax = ", 5, 101 , "epsmin = ", 2, 28669
5.787955048257834350714 10-10(35)> mineps21(1,7739,hcn2HR);log(1.5)/log(62129.)-log(2.)/log(156383467.
);# smallest difference between an epsilons of type 2 and an
epsilon of type 1"mineps21=", 1.658180660611488482 10-13 , "epsmin = ", 2, 62129 , 1, 156383467
1.658180660611488482 10-13(36)> pmax:=hcn2HR[7739][4]: pf:=evalf(pmax): mineps11:=2*log(2.)/(pf*log(pf)^2);# the difference between two epsilons of type 1 is bigger than(37)

>
mineps11:
mineps11

1.89747174628194616821538836321 10-11> listN0:=calcullogN(10^8+7,1): logN0:=listN0[1]; llN0:=log(logN0);# Computation of xi_k^{(0)}, log(N_0) and log log(N_0) cf.
(3.22), (3.23) and (3.24)"kmaxre=", 37.842407"k=", 1, "beta=", 1.000000000, "xi0=", 1.00000007 108 , "pp = ", 100000007, "np = ",100000037"k=", 2, "beta=", 0.5849625007, "xi0=", 47829.96065, "pp = ", 47819, "np = ", 47837"k=", 3, "beta=", 0.4150374993, "xi0=", 2090.739897, "pp = ", 2089, "np = ", 2099
"k=", 4, "beta=", 0.3219280949, "xi0=", 376.2051781, "pp = ", 373, "np = ", 379"k=", 5, "beta=", 0.2630344058, "xi0=", 127.1379647, "pp = ", 127, "np = ", 131"k=", 6, "beta=", 0.2223924213, "xi0=", 60.13667022, "pp = ", 59, "np = ", 61"k=", 7, "beta=", 0.1926450779, "xi0=", 34.76647259, "pp = ", 31, "np = ", 37"k=", 8, "beta=", 0.1699250014, "xi0=", 22.87704975, "pp = ", 19, "np = ", 23"k=", 9, "beta=", 0.1520030934, "xi0=", 16.44465445, "pp = ", 13, "np = ", 17"k=", 10, "beta=", 0.1375035237, "xi0=", 12.59007143, "pp = ", 11, "np = ", 13"k=", 11, "beta=", 0.1255308821, "xi0=", 10.09827191, "pp = ", 7, "np = ", 11"k=", 12, "beta=", 0.1154772174, "xi0=", 8.391077705, "pp = ", 7, "np = ", 11"k=", 13, "beta=", 0.1069152039, "xi0=", 7.166739760, "pp = ", 7, "np = ", 11"k=", 14, "beta=", 0.09953567355, "xi0=", 6.255836526, "pp = ", 5, "np = ", 7"k=", 15, "beta=", 0.09310940439, "xi0=", 5.557445827, "pp = ", 5, "np = ", 7"k=", 16, "beta=", 0.08746284125, "xi0=", 5.008442967, "pp = ", 5, "np = ", 7"k=", 17, "beta=", 0.08246216019, "xi0=", 4.567696968, "pp = ", 3, "np = ", 5"k=", 18, "beta=", 0.07800251200, "xi0=", 4.207460993, "pp = ", 3, "np = ", 5"k=", 19, "beta=", 0.07400058144, "xi0=", 3.908450840, "pp = ", 3, "np = ", 5"k=", 20, "beta=", 0.07038932789, "xi0=", 3.656912974, "pp = ", 3, "np = ", 5"k=", 21, "beta=", 0.06711419586, "xi0=", 3.442814068, "pp = ", 3, "np = ", 5"k=", 22, "beta=", 0.06413033742, "xi0=", 3.258687399, "pp = ", 3, "np = ", 5"k=", 23, "beta=", 0.06140054466, "xi0=", 3.098877146, "pp = ", 3, "np = ", 5"k=", 24, "beta=", 0.05889368905, "xi0=", 2.959030994, "pp = ", 2, "np = ", 3"k=", 25, "beta=", 0.05658352837, "xi0=", 2.835751880, "pp = ", 2, "np = ", 3"k=", 26, "beta=", 0.05444778402, "xi0=", 2.726354177, "pp = ", 2, "np = ", 3(37)

"k=", 27, "beta=", 0.05246741989, "xi0=", 2.628689926, "pp = ", 2, "np = ", 3"k=", 28, "beta=", 0.05062607307, "xi0=", 2.541022942, "pp = ", 2, "np = ", 3"k=", 29, "beta=", 0.04890960048, "xi0=", 2.461936263, "pp = ", 2, "np = ", 3"k=", 30, "beta=", 0.04730571478, "xi0=", 2.390263169, "pp = ", 2, "np = ", 3"k=", 31, "beta=", 0.04580368961, "xi0=", 2.325035125, "pp = ", 2, "np = ", 3"k=", 32, "beta=", 0.04439411936, "xi0=", 2.265442044, "pp = ", 2, "np = ", 3"k=", 33, "beta=", 0.04306872189, "xi0=", 2.210801632, "pp = ", 2, "np = ", 3"k=", 34, "beta=", 0.04182017569, "xi0=", 2.160535501, "pp = ", 2, "np = ", 3"k=", 35, "beta=", 0.04064198450, "xi0=", 2.114150388, "pp = ", 2, "np = ", 3"k=", 36, "beta=", 0.03952836419, "xi0=", 2.071223260, "pp = ", 2, "np = ", 3"k=", 37, "beta=", 0.03847414781, "xi0=", 2.031389408, "pp = ", 2, "np = ", 3"k=", 38, "beta=", 0.03747470542, "xi0=", 1.994332857, "pp = ", 1, "np = ", 2"xi=", 100000007, "log N=", 1.00037943869466 108 , "log d(N)/log 2=",5.76451574240844 106 , "log d(N) = ", 3.99565783414383 106 , "log log N = ",18.4210601106784"list of exponents of N = ", 2, 37 , 3, 23 , 5, 16 , 7, 13 , 11, 10 , 13, 9 , 19, 8 , 31,
7 , 59, 6 , 127, 5 , 373, 4 , 2089, 3 , 47819, 2 , 100000007, 1
logN0
llN0

1.00037943869465743905928343761 108
18.4210601106783659628373398078(38)> ############### proof of Lemma 3.13 ###################> T3T5(10.^8+7,34): # cf. (3.25)"k0 = ", 34, "T5 = ", 0.1814976520, "T3= ", 1.361436428(39)> ############### Section 3.4 : highly composite numbers##############> highlycomp(1000,0,1); # computes the list [M,d(M)] of hc numbers M
up to 1000
# the list lishc10p100 contains the 1381 hc numbers M < 10^100"p=", 3, "nops(lord) = ", 26, "nops(lis) =", 15"p=", 5, "nops(lord) = ", 26, "nops(lis) =", 14"p=", 7, "nops(lord) = ", 24, "nops(lis) =", 151, 1 , 2, 2 , 4, 3 , 6, 4 , 12, 6 , 24, 8 , 36, 9 , 48, 10 , 60, 12 , 120, 16 , 180,

(40)

(40)
18 , 240, 20 , 360, 24 , 720, 30 , 840, 32> ############### Section 3.5 : benefit ##############> Digits:=10: N:=367567200; eps:=log(2.)/log(17.): print("eps = ",eps);
bensmall(N,eps,0.13,N,2*N,0); # cf. Section 3.5# computes the list of [n,d(n),ben(n)] of numbers n, N <= n <=2*N with ben(n) <= 0.13
N367567200"eps = ", 0.2446505421"nops(liso)=", 286, "nops(lisocourte) = ", 23367567200, 1152, 0. , 410810400, 1152, 0.0272114119 , 428828400, 1080,(41)0.1022515681 , 441080640, 1120, 0.0727759449 , 454053600, 1080, 0.1162354056 ,
465585120, 1152, 0.0578326424 , 479278800, 1080, 0.1294629800 , 490089600,
1152, 0.0703815746 , 492972480, 1120, 0.0999873568 , 497296800, 1152,
0.0739531792 , 514594080, 1152, 0.0823181152 , 537213600, 1152, 0.0928423418 ,
547747200, 1152, 0.0975929865 , 551350800, 1200, 0.0583752641 , 563603040,
1152, 0.1045744097 , 575134560, 1152, 0.1095295271 , 581981400, 1152,
0.1124248330 , 588107520, 1152, 0.1149866429 , 605404800, 1152, 0.1220784593 ,
612612000, 1152, 0.1249737652 , 616215600, 1200, 0.0855866760 , 698377680,
1280, 0.0516693856 , 735134400, 1344, 0.0154281536> ################ Section 3.7, estimates without RH, proof of Lemma
3.18 ############> evalf(1/be[5]); f:=0.8499*t-3.81*log(t);f1:=diff(f,t); t0:=solve(f1=0.,t);evalf(subs(t=11,f));# lower bound of (3.33), cf. (3.39)
3.801784017
f0.8499 t

3.81 ln t
3.81
t

f1

0.8499t0

4.482880339
0.212919010> eta:=7.5*10^(-7):(1+eta)*1.3615; # upper bound of (3.33)
1.361501021

(42)
(43)

> ub334:=1/(1-1/log(10.^8+7)^3); # upper bound of (3.34)
ub334
1.000160012(44)> A:=(1+eta)*(1+(10.^8+7)^(be[2]-1))+1.362*(10.^8+7)^(be[3]-1):
lb334:=1/A; # lower bound of (3.34)
lb334
0.9994927315(45)> lb335:=1+0.00017/llN0: ub335:=1+log(0.99949)/llN0:"lb335 = ",evalf(lb335,10),"ub335 = ",evalf(ub335,10); # proofof (3.35)"lb335 = ", 1.000009229, "ub335 = ", 0.9999723072(46)> numtheory[pi](376),74*be[4],(0.99949)^be[3]/1.00001; # lower boundof (3.36)
74, 23.82267902, 0.9997783015(47)> 1.00017^be[3]/0.99997*(1.15963+1.19768*(10.^8+7)^(be[4]-be[3])+1.25506*0.1815);# upper bound of (3.36)
1.603093648(48)> sqrt(1.00017)*1.00001^2/8/evalf(Pi); # proof of (3.37)
0.03979291349(49)> 1.00017^(be[2]/2)*(1.00001*be[2])^2/8/evalf(Pi); # proof of (3.38)
0.01361590369(50)> ######### Section 3.8, estimates under RH, proof of Lemma 3.20#########> plot(t^be[2]-sqrt(t)*log(t)^2/8/evalf(Pi),t=10^24..10^25); #beginning of the proof of Lemma 3.20

2.

10

12

0
3.
2.

10

12

4.

10

12

6.

10

12

8.

10

12

1.

10

13

1.2

10

13

24

10 5.

24

10 7.

10

24

1.

10

25

t

> 0.9927*0.99949^be[2],(be[2]-1/2)^2*exp(2.)/4,-0.0398+0.9924*
0.013334;# upper bound of (3.43), cf. (3.52)
0.9924038155, 0.01333470909, 0.0265673384(51)> eta:=7.5*10^(-7): 1.00017^be[2]*(1+eta+1.362*xi0[3]/xi0[2]);# lower bound of (3.43), cf. (3.53)
exp(2/(be[2]-1/2)),1.06+(0.0398*log(10.^19/1.00017)^2)/(10.
^19/1.00017)^(be[2]-1/2);
1.059641759
1.671875165 1010 , 2.911556783> 0.0266*be[2]; # upper bound of (3.44)
0.01556000252

(52)
(53)

> h0:=2.92/logN0^(1-be[2]):b:=(1-(1-h0)^be[2])/h0: # lower bound of(3.44), cf. (3.54)" h0 = ",h0, " b = ",b,0.5852*2.92;" h0 = ", 0.001396414908, " b = ", 0.5851321089, 1.708784(54)

> 0.0137+1.71/logN0^(1-3*be[2]/2)/llN0^2; # lower bound of (3.45)
0.01422710446(55)> 0.0137+0.0156/logN0^(1/2-be[2]/2); # upper bound of (3.45), cf.
(3.56)
0.01404114610(56)> 0.0141/be[2]; # upper bound of (3.46)
0.02410410920(57)> 1/1.008, be[2]-log(1.008)/llN0, 0.0143/0.5845; # lower bound of(3.46), cf. (3.57)
0.9920634921, 0.5845299431, 0.02446535500(58)> h0:=2.92/logN0^(1-be[2]):bb:=(1-(1-h0)^(1/2))/h0:" h0 = ",h0, " bb = ",bb,0.5002*2.92; # lower bound of (3.47)" h0 = ", 0.001396414908, " bb = ", 0.5001746945, 1.460584(59)> 0.9629*0.99949^be[3], 0.9134*0.99949^be[4]; # lower bound of (3.48)
0.9626961534, 0.9132500090(60)> 1.362*1.00017^be[3], 4*exp(-2.)/(be[3]-be[2]/2)^2, 1.363+0.0141*
36.05; # upper bound of (3.48)
1.362096093, 36.04127180, 1.871305(61)> 1+1.872/logN0^(be[2]-be[3]), 1/(1-1/logN0^(1-be[2])-1.872/logN0^(1be[3]));
A:=(1-0.0006/llN0)^2/1.0006: 1/A;
0.5858/logN0^(be[3]-2*be[2]+1)/llN0; # lower bound of (3.49), cf.
(3.58)
1.081823459, 1.000517622
1.000665185
0.0003479320029(62)> 0.04/sqrt(0.99949), 4*exp(-2.)/(be[2]-1/2)^2,0.0266*75,2.92*0.041;# proof of (3.50), cf. (3.59)
0.04001020392, 74.99226216, 1.9950, 0.11972(63)> ######> ###################### Section 4, proofs of Theorem 1.1############> ######> ############################################

Proof of Proposition 4.1> 0.994 +0.0242*be[2]*llN0^2/logN0^(be[3]-be[2]/2)+(2.922/llN0+0.12*
llN0+(7.48032)/llN0^2)/logN0^(be[3]+1/2-be[2]); # upper bound of B
1.501930765(64)

> 0.546*llN0/logN0^be[3]+0.12*llN0/logN0^(be[3]-be[2]+1/2); # upperbound of A(xi), cf. (4.6)
0.009866332298(65)> 5.07*be[2]*1.00017^(be[2]/2)/(0.99997*be[2])^2,8.67/llN0/logN0^(be[3]-be[2]/2);# upper bound of be[2]*A(xi_2), cf. (4.7)
8.668173255, 0.04923050544(66)> -1.873-0.0099-0.05+0.9997, -(0.9332+0.0245*be[2]*llN0^2/logN0^(be[3]-be[2]/2));# lower bound of B'
0.9332, 1.441889961(67)> 1.51+llN0/logN0^be[3]; # proof of Corollary 4.2
1.518809398(68)> ###################### Section 4, proof of Theorem 1.1 ############> Digits:=30: N3:=lishc10p100[1162][1]: print("N3 = M1162",impfact(N3),evalf(N3,8)):
thm1_1(fonG1,N3,12,0); # Application of Lemma 3.16 with f = G_1 toend the proof of Lemma 4.3"N3 = M1162", 2, 10 , 3, 6 , 5, 4 , 7, 3 , 11, 2 , ... , 19, 2 , 23, ... , 181,
1.8997706 1086"j=", 1125, "Mj=", 2, 10 , 3, 6 , 5, 4 , 7, 3 , 11, 2 , ... , 19, 2 , 23, ... , 179, "d(Mj)=",1071508441006080"f(logMj)-logdMj/log2 = ", 0.00003409330033"jp1 = ", 1126, "Mjp1/Mj = ",4
78
, "dMjp1/dMj = ",, "f(logMjp1)-logdMjp1/log2 = ",377

0.03696489" N is the smallest shc number= Mj""N/Mj = ", 1, "eps = ", 0.133621675, "logN = ", 193.4655541, "dN = ", 1071508441006080,"logdN = ", 34.60784381"logMj=", 193.465554127103, "f(logMj) = ", 49.9285305451666"logmu =", 193.465730457522, "f(logmu)=", 49.928564638466888437"logdMj/log2 = ", 49.928564638466888437, "logdMjp1/log2 = ", 49.9471803166342"logMjp1=", 193.753236199555, "f(logMjp1) = ", 49.9841452138125"log2*f(logMj) = ", 34.607820, "benmaxn=", 0.000047193241"nops(liso)=", 34, "nops(lisocourte) = ", 4"list of [n/Mj, d(n)/d(Mj), ben_eps(n)] with Mj
benmaxn

0.01 = "1, 1, 0. ,181
, 1, 0.0014847005 ,179=n= Mjp1 and ben_eps(n)181, 1, 0.0060404262 ,
173

=

191
, 1, 0.0086703927
179"n/Mj = ", 1, "f(log n)-(log d(n))/log 2 = ", 0.00003409330033"n/Mj = ",181
, "f(log n)-(log d(n))/log 2 = ", 0.002114238120
179"n/Mj = ",181
, "f(log n)-(log d(n))/log 2 = ", 0.008706124391
173"n/Mj = ",191
, "f(log n)-(log d(n))/log 2 = ", 0.01251143548
179"j=", 1125, "nmaxi/Mj=", 1, "d(nmaxi)/dMj=", 1, "bennmaxi=", 0.,(69)"f(nmaxi)-logdnmaxi/log2 = ", 0.00003409330033> M1125:=lishc10p100[1125][1]: print("N2=M1125 = ",impfact(M1125),
evalf(M1125,8)):
thm1_1(fonH0,M1125,12,0);# Application of Lemma 3.16 with f=H_0 to end the proof of(1.8), Section 4.2 cf. (4.9)"N2=M1125 = ", 2, 10 , 3, 6 , 5, 4 , 7, 3 , 11, 2 , ... , 19, 2 , 23, ... , 179,
1.0495970 1084
"j=", 60, "Mj=", 2, 4 , 3, 3 , 5, 2 , 7, ... , 17, "d(Mj)=", 960"f(logMj)-logdMj/log2 = ", 0.02487234930"jp1 = ", 61, "Mjp1/Mj = ",4
21
, "dMjp1/dMj = ",, "f(logMjp1)-logdMjp1/log2 = ",3
20

0.026598307" N is the smallest shc number= Mj""N/Mj = ", 2, "eps = ", 0.244650542, "logN = ", 19.72241672, "dN = ", 1152, "logdN = ",7.049254841"logMj=", 19.0292695364837, "f(logMj) = ", 9.88201824630488"logmu =", 19.0876861823733, "f(logmu)=", 9.9068905956085185293"logdMj/log2 = ", 9.9068905956085185293, "logdMjp1/log2 = ", 9.97727992349992"logMjp1=", 19.3169516089355, "f(logMjp1) = ", 10.0038782280108"log2*f(logMj) = ", 6.8496931, "benmaxn=", 0.044274586"nops(liso)=", 40, "nops(lisocourte) = ", 3"list of [n/Mj, d(n)/d(Mj), ben_eps(n)] with Mj=n= Mjp1 and ben_eps(n)=benmaxn

0.01 = "1, 1, 0.0127427233 ,19, 1, 0.0399541352 ,
17

4 21
,
, 0.0343341337
3 20"n/Mj = ", 1, "f(log n)-(log d(n))/log 2 = ", 0.02487234930"n/Mj = ",19
, "f(log n)-(log d(n))/log 2 = ", 0.02242833543
17"n/Mj = ",4
, "f(log n)-(log d(n))/log 2 = ", 0.02659830451
3"j=", 60, "nmaxi/Mj=", 1, "d(nmaxi)/dMj=", 1, "bennmaxi=", 0.012742723,(70)"f(nmaxi)-logdnmaxi/log2 = ", 0.02487234930> M60:=183783600: print("M60 = ",M60,impfact(M60)); thm1_1(fonG2,M60,
12,0);# Application of Lemma 3.16 with f=G_2 to end the proof of(1.10), Section 4.3.1, cf. (4.10)"M60 = ", 183783600, 2, 4 , 3, 3 , 5, 2 , 7, ... , 17"j=", 58, "Mj=", 2, 5 , 3, 2 , 5, 2 , 7, ... , 17, "d(Mj)=", 864"f(logMj)-logdMj/log2 = ", 0.01235488515"jp1 = ", 59, "Mjp1/Mj = ",628
, "dMjp1/dMj = ",, "f(logMjp1)-logdMjp1/log2 = ",527

0.011226569" N is the smallest shc number= Mj""N/Mj = ", 3, "eps = ", 0.244650542, "logN = ", 19.72241672, "dN = ", 1152, "logdN = ",7.049254841"logMj=", 18.6238044283756, "f(logMj) = ", 9.74253261701293"logmu =", 18.6533583495846, "f(logmu)=", 9.7548875021634685444"logdMj/log2 = ", 9.7548875021634685444, "logdMjp1/log2 = ", 9.80735492205760"logMjp1=", 18.8061259851695, "f(logMjp1) = ", 9.81858149152794"log2*f(logMj) = ", 6.7530090, "benmaxn=", 0.034700117"nops(liso)=", 30, "nops(lisocourte) = ", 2"list of [n/Mj, d(n)/d(Mj), ben_eps(n)] with Mj
benmaxn=n= Mjp1 and ben_eps(n)=0.01 = "1, 1, 0.0189059801 ,6 28,, 0.0271434041
5 27"n/Mj = ", 1, "f(log n)-(log d(n))/log 2 = ", 0.01235488515

(71)

6
, "f(log n)-(log d(n))/log 2 = ", 0.01122656947
5"n/Mj = ","j=", 58, "nmaxi/Mj=", 1, "d(nmaxi)/dMj=", 1, "bennmaxi=", 0.018905980,(71)"f(nmaxi)-logdnmaxi/log2 = ", 0.01235488515> evalf(1.52^(1/(be[2]-be[3])),10);
M60:=183783600: print("M60 = ",M60,impfact(M60)); thm1_1(fonG3,
lishc10p100[60][1],1,0);# Application of Lemma 3.16 with f=G_3 to end the proof of(1.11), Section 4.3.2
11.75277106"M60 = ", 183783600, 2, 4 , 3, 3 , 5, 2 , 7, ... , 17"j=", 6, "Mj=", 2, 3 , 3, 1 , "d(Mj)=", 8"f(logMj)-logdMj/log2 = ", 0.09552541021"jp1 = ", 7, "Mjp1/Mj = ",3
9
, "dMjp1/dMj = ", , "f(logMjp1)-logdMjp1/log2 = ",2
80.185536359" N is the smallest shc number"N/Mj = ",

= Mj"

5
, "eps = ", 0.430676558, "logN = ", 4.094344562, "dN = ", 12, "logdN = ",2

2.484906650"logMj=", 3.17805383034795, "f(logMj) = ", 2.90447458979213"logmu =", 3.26016009742919, "f(logmu)=", 3.0000000000000000000"logdMj/log2 = ", 3.0000000000000000000, "logdMjp1/log2 = ", 3.16992500144231"logMjp1=", 3.58351893845611, "f(logMjp1) = ", 3.35546135993709"log2*f(logMj) = ", 2.0132284, "benmaxn=", 0.11241458"nops(liso)=", 12, "nops(lisocourte) = ", 3"list of [n/Mj, d(n)/d(Mj), ben_eps(n)] with Mj
benmaxn=n= Mjp1 and ben_eps(n)=0.01 = "1, 1, 0.0108401697 ,5, 1, 0.1069428661 ,4

3 9, , 0.0676814514
2 8"n/Mj = ", 1, "f(log n)-(log d(n))/log 2 = ", 0.09552541021"n/Mj = ",5
, "f(log n)-(log d(n))/log 2 = ", 0.1588672278
4"n/Mj = ",3
, "f(log n)-(log d(n))/log 2 = ", 0.1855363585
2(72)

"j=", 6, "nmaxi/Mj=", 1, "d(nmaxi)/dMj=", 1, "bennmaxi=", 0.010840170,(72)"f(nmaxi)-logdnmaxi/log2 = ", 0.09552541021> lisG3n:=[1]: # list of exceptions <= 35 for (1.11), cf. Section
4.3.2
for n from 2 to 35 do
logn:=log(evalf(n)): dn:=numtheory[tau](n): ldnsl2:=log(evalf(dn))/log2;
dif:=fonG3(logn)-ldnsl2:
if dif < 0 then lisG3n:=[op(lisG3n),n]: fi:
od: lisG3n;1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 18, 24(73)> evalf(exp(2/(1/2-be[3])),10); # beginning of the proof of (1.12),
Section 4.3.3
evalf(subs(t=1.56*10^17,t^(1/2-be[3])/log(t)*(2-tautau+5.12/log(t))),10);
1.671875165 1010
1.520262334(74)> M1125:=lishc10p100[1125][1]: print("N2=M1125 = ",impfact(M1125),
evalf(M1125,8)):
thm1_1(fonF,M1125,12,0);# Application of Lemma 3.16 with f=F to end the proof of (1.12),
Section 4.3.3"N2=M1125 = ", 2, 10 , 3, 6 , 5, 4 , 7, 3 , 11, 2 , ... , 19, 2 , 23, ... , 179,
1.0495970 1084
"j=", 44, "Mj=", 2, 5 , 3, 3 , 5, ... , 13, "d(Mj)=", 384"f(logMj)-logdMj/log2 = ", 0.03479614862"jp1 = ", 45, "Mjp1/Mj = ",325
, "dMjp1/dMj = ",, "f(logMjp1)-logdMjp1/log2 = ",2
24

0.071636484" N is the smallest shc number= Mj""N/Mj = ", 1, "eps = ", 0.261859507, "logN = ", 15.27976546, "dN = ", 384, "logdN = ",5.950642553"logMj=", 15.2797654605534, "f(logMj) = ", 8.55016635210030"logmu =", 15.3647087540624, "f(logmu)=", 8.5849625007211561815"logdMj/log2 = ", 8.5849625007211561815, "logdMjp1/log2 = ", 8.64385618977472"logMjp1=", 15.6852305686615, "f(logMjp1) = ", 8.71549267821705"log2*f(logMj) = ", 5.9265237, "benmaxn=", 0.046362061

"nops(liso)=", 28, "nops(lisocourte) = ", 1"list of [n/Mj, d(n)/d(Mj), ben_eps(n)] with Mj
benmaxn=n= Mjp1 and ben_eps(n)=0.01 = "1, 1, 0.
"n/Mj = ", 1, "f(log n)-(log d(n))/log 2 = ", 0.03479614862"j=", 44, "nmaxi/Mj=", 1, "d(nmaxi)/dMj=", 1, "bennmaxi=", 0., "f(nmaxi)-logdnmaxi/log2 = ",0.03479614862> M1125:=lishc10p100[1125][1]: print("N2=M1125 = ",impfact(M1125),
evalf(M1125,8)):
thm1_1(fonFmR,M1125,12,0);# Application of Lemma 3.16 with f=F-R to end the proof of(1.13), Section 4.3.4"N2=M1125 = ", 2, 10 , 3, 6 , 5, 4 , 7, 3 , 11, 2 , ... , 19, 2 , 23, ... , 179,
1.0495970 1084
"j=", 975, "Mj=", 2, 10 , 3, 6 , 5, 4 , 7, 3 , 11, 2 , ... , 19, 2 , 23, ... , 157, "d(Mj)=",66969277562880"f(logMj)-logdMj/log2 = ", 0.0005356266061
478, "dMjp1/dMj = ",, "f(logMjp1)-logdMjp1/log2 = ",3
77

"jp1 = ", 976, "Mjp1/Mj = ",0.03788281" N is the smallest shc number= Mj""N/Mj = ", 1, "eps = ", 0.137087319, "logN = ", 172.9131327, "dN = ", 66969277562880,"logdN = ", 31.83525509"logMj=", 172.913132713541, "f(logMj) = ", 45.9280290118607"logmu =", 172.915844250895, "f(logmu)=", 45.928564638466888437"logdMj/log2 = ", 45.928564638466888437, "logdMjp1/log2 = ", 45.9471803166342"logMjp1=", 173.200814785993, "f(logMjp1) = ", 45.9850631281111"log2*f(logMj) = ", 31.834884, "benmaxn=", 0.00074298546"nops(liso)=", 108, "nops(lisocourte) = ", 4"list of [n/Mj, d(n)/d(Mj), ben_eps(n)] with Mj
benmaxn

0.01 = "1, 1, 0. ,163
, 1, 0.0051413771 ,157=n= Mjp1 and ben_eps(n)167, 1, 0.0084648687 ,
157

=

163
, 1, 0.0104831172
151"n/Mj = ", 1, "f(log n)-(log d(n))/log 2 = ", 0.0005356266061(75)

"n/Mj = ",163
, "f(log n)-(log d(n))/log 2 = ", 0.006872551797
157"n/Mj = ",167
, "f(log n)-(log d(n))/log 2 = ", 0.01166101228
157"n/Mj = ",163
, "f(log n)-(log d(n))/log 2 = ", 0.01456875904
151"j=", 975, "nmaxi/Mj=", 1, "d(nmaxi)/dMj=", 1, "bennmaxi=", 0., "f(nmaxi)-logdnmaxi/log2 = ",0.0005356266061>##################### THE END ####################(76)