> read "gdenHR.mpl":> read "TabT2.m": # [q,alpha,p,n,log g(n),rho,excadd,excmul]> read "superhden.m": # [p,n=pi_1(p),log(h(n)=theta(p)]> x0:=10.^10+19: logx0:=log(x0):sqrtx0:=sqrt(x0): c:=
0.0461176444215090238270908: # cf. (1.11)"x0=",x0,"logx0=",logx0,"sqrtx0=",sqrtx0,"c=",c;"x0=", 1.0000000019 1010 , "logx0=", 23.025850931840456838, "sqrtx0=",(1)1.0000000009499999995 105 , "c=", 0.0461176444215090238270908> "pi_1(x0) = ",pi1(x0);"pi_1(x0) = ", 2220822442581729257(2)> nu0:=2220832950051364840: nu0f:=evalf(nu0): L0:=log(nu0f):lambda0:=log(L0):"nu0 =",nu0,"nu0f =",nu0f,"L0 =",L0,"lambda0 =",lambda0;"nu0 =", 2220832950051364840, "nu0f =", 2.220832950051364840 1018 , "L0 =",(3)42.244414002131909364, "lambda0 =", 3.7434721320949859580> f746(48757,70877,0.0746); # Proof of (2.4)"for t satisfying pmin t = pmax with pmin = ", 48757, "pmax = ", 70877,(4)0.0746"thetaminus(t) is", t 1, " holds"
ln t> f746(48000,70877,0.0746); # Proof of (2.4)"for t satisfying pt=pwith p = ", 48751, "p= ", 48757, "theta(t) is", t 1(5)0.0746, " fails"
ln t> calculW(10^6): # Proof of Lemma 2.2, (2.8)"list of primes = y = ", 1000000, " s.t. W(p)p :""p = ", 3, "W(p) = ", 3.034212794, "W(p)/p = ", 1.011404265"p = ", 5, "W(p) = ", 5.046010185, "W(p)/p = ", 1.009202037"p = ", 7, "W(p) = ", 7.316238692, "W(p)/p = ", 1.045176956"y=", 1000000, "W(y) = ", 9.98497413976641 105 , "theta(y) = ", 9.98484175025634 105 ,(6)" a = ", 12.24046583, " b = ", 1.000013990> a:=0.045: K:=2: alpha:=1/2: b:=(1-a/logx0^(K+1))*(a-2*alpha/logx0a*alpha/logx0^(K+1)):"a=",a,"b=",b,"sqrt(x0)/logx0^3=",sqrtx0/logx0^3,"8.19 b=",8.19*b;
a:='a':K:='K':alpha:='alpha': b:='b':a,b,K,alpha: # Proof ofCorollary 2.4, (2.12)"a=", 0.045, "b=", 0.0015687029879308942561, "sqrt(x0)/logx0^3=",(7)
8.1913019292091958460, "8.19 b=", 0.012847677471154023957> Lim1Newtonx0(1,0,6); # Computation of li^(-1)(x). To avoid thedetails, execute Lim1Newtonx0(1,0,3)# cf. Section 2.2 andSection 8.1 where the result is used"computation of Li^(-1)(x), x=", 1

"x=", 1., "yanc=", 1.4, "ynou=", 1.785257684292964378141217, "dif=",0.3852576842929643781412167"x=", 1., "yanc=", 1.785257684292964378141217, "ynou=", 1.954849863902451855418587,"dif=", 0.1695921796094874772773698"x=", 1., "yanc=", 1.954849863902451855418587, "ynou=", 1.968971308048332347947542,"dif=", 0.01412144414588049252895480"x=", 1., "yanc=", 1.968971308048332347947542, "ynou=", 1.969047487049610247978378,"dif=", 0.00007617900127790003083557618"x=", 1., "yanc=", 1.969047487049610247978378, "ynou=", 1.969047489224750848374546,"dif=", 2.175140600396167970986952 10-9"x=", 1., "yanc=", 1.969047489224750848374546, "ynou=", 1.969047489224750850147703,"dif=", 1.773156943069393933440876 10-18"x=", 1., "yanc=", 1.969047489224750850147703, "ynou=", 1.969047489224750850147703,"dif=", 0.
1.9690474892247508501> #################### Study of pi_r(x), Section 2.3########################> lis:=[227325003281036, 13501147086873627946348,
906608880604777977466210365146,
65064408117405555953424106561636940128,
4868694384660993648813605308147109183464084346]:# lis has been computed by lis:=sommepr(5,89967803); in about 450
seconds> calculC0(5,lis); # The values of C_0 and C_0h (defined by (2.15)
and (2.18)) are used in the proofsof Corollary 2.7
and Corollary 2.8. A[0] is the sum of positive terms, A[1] negative"r=", 1, "A[0]=", 8.2082626525957611659 1014 , "A[1]=", 8.2083250195076881395 1014 ,"C0=", 6.23669119269736 109"r=", 1, "Ah[0]=", 8.4986765008763478598 1014 , "Ah[1]=", 8.4979737672803381019 1014 ,"C0h=", 7.027335960097579 1010
"r=", 2, "A[0]=", 1.2921973093720948207 1023 , "A[1]=", 1.2922077098998901986 1023 ,"C0=", 1.04005277953779 1018"r=", 2, "Ah[0]=", 1.3154396844999538478 1023 , "Ah[1]=", 1.3153594634478156376 1023 ,"C0h=", 8.02210521382102 1018"r=", 3, "A[0]=", 2.3829697485026415367 1031 , "A[1]=", 2.3829814063566907065 1031 ,"C0=", 1.16578540491698 1026"r=", 3, "Ah[0]=", 2.4026013375556204250 1031 , "Ah[1]=", 2.4025222390607564728 1031 ,"C0h=", 7.90984948639522 1026(8)

"r=", 4, "A[0]=", 4.1157065632902816301 1039 , "A[1]=", 4.1157182783765692068 1039 ,"C0=", 1.17150862875767 1034
"r=", 4, "Ah[0]=", 4.1326834879070137524 1039 , "Ah[1]=", 4.1326087219068440232 1039 ,"C0h=", 7.47660001697292 1034
"r=", 5, "A[0]=", 6.4750249362423601240 1047 , "A[1]=", 6.4750361698286499611 1047 ,"C0=", 1.12335862898371 1042
"r=", 5, "Ah[0]=", 6.4898896736972046997 1047 , "Ah[1]=", 6.4898202592338292804 1047 ,(9)"C0h=", 6.94144633754193 1042
> verifpi2max(60169,70000,0.001); # To prove (2.21) executeverifpi2max(60169,89967803,0.001)"p=", 60169, "pi2(p)=", 6.838284919887 1012 , "est=", 6.8369963635222830269 1012 , "S/est=",1.000189"p=", 60223, "pi2(p)=", 6.849162940386 1012 , "est=", 6.8548372071821490682 1012 , "S/est=",0.9991723"p=", 60259, "pi2(p)=", 6.860055176517 1012 , "est=", 6.8667480752922893415 1012 , "S/est=",0.9990253
60172.903253337664416> verifpi2maxbis(60293,315011,0.0001); # Proof of (2.22)"p=", 60293, "pi2(p)=", 6.870957779328 1012 , "c=", 6.8697770137142241757 1012 , "S/c=",1.000172(10)"p=", 60353, "pi2(p)=", 6.889160065205 1012 , "c=", 6.8896629823335629583 1012 , "S/c=",
0.9999270"p=", 96017, "pi2(p)=", 2.6582175089330 1013 , "c=", 2.6583289513724537426 1013 , "S/c=",
0.9999582
60296.565793200319808> verifpi2min(1091200,1100000,0.00001): # To prove (2.23) executeverifp2min(32300,89967803,0.0001)
t3
t3
2 t3
1069 t3"F=",, "F1=",
3 ln t9 ln t 2
27 ln t 3
648 ln t 4
t2 648 ln t 43351 ln t

648 ln t

4276

5"FF1=", 648 T4 3351 T 4276, "FF2=diff of FF1=", 2592 T3 3351
"zero of FF2=(3351/2592)^(1/3)=", 1.089381, "the minimum of FF1=", 1538.113, "is positive""p=", 1091219, "p =", 1091221, "pi2(p)=", 3.1879086471058513 1016 , "F(p )=",
3.1879429840931769633 1016 , "S/est=", 0.9999892291"p=", 1091221, "p =", 1091239, "pi2(p)=", 3.1880277234329354 1016 , "F(p )=",

(11)

3.1880968855097752579 1016 , "S/est=", 0.9999783059"p=", 1091239, "p =", 1091243, "pi2(p)=", 3.1881468036884475 1016 , "F(p )=",
3.1881310864894094542 1016 , "S/est=", 1.000004930"p=", 1091243, "p =", 1091257, "pi2(p)=", 3.1882658848169524 1016 , "F(p )=",

(12)

3.1882507918219931002 1016 , "S/est=", 1.000004734> verifpi2minbis(32300,120000,0.0001): # To prove (2.24) executeverifp2minbis(32300,1100000,0.0001)
t3
t3
2 t3
1069 t3
t3
31 t3
"F=",
, "f=",
3 ln t3 ln t9 ln t 2
27 ln t 3
648 ln t 4
375 ln t 2"F-f=",

t3 2304 ln t 26000 ln t

133625, "trinom=", 2304 T2

6000 T 133625
81000 ln t
"zero of trinom=", 9.028167905, 6.424001239, "exp(6.425)=", 617.0808
4"p=", 32303, "p =", 32309, "p =", 32309, "pi2(p)=", 1.108260898891 1012 , "f(p )=",
1.1085961771097900465 1012 , "S/est=", 0.9996975653"p=", 32309, "p =", 32321, "p =", 32321, "pi2(p)=", 1.109304770372 1012 , "f(p )=",
1.1097912620024024453 1012 , "S/est=", 0.9995616365"p=", 69959, "p =", 69991, "p =", 69991, "pi2(p)=", 1.0472883706424 1013 , "f(p )=",
1.0472266558975163065 1013 , "S/est=", 1.000058932> verifpi3max(600,70000,0.01);# To prove (2.25) execute verifpi3max(600,89967803,0.01)"r=3, coef=", 0.25380853990085065148"p=", 619, "pi3(p)=", 6.171715615 109 , "est=", 6.1894050708312291332 109 , "S3/est=",0.9971421"p=", 661, "pi3(p)=", 8.076458694 109 , "est=", 7.9667154778525718127 109 , "S3/est=",1.013775"p=", 677, "pi3(p)=", 8.691568644 109 , "est=", 8.7343685487645216716 109 , "S3/est=",0.9950998"p=", 683, "pi3(p)=", 9.010180631 109 , "est=", 9.0359143427976750482 109 , "S3/est=",0.9971521"p=", 1129, "pi3(p)=", 6.2106256796 1010 , "est=", 6.2639185948171564573 1010 , "S3/est=",0.9914921"p=", 1307, "pi3(p)=", 1.09476567065 1011 , "est=", 1.1020976973386898407 1011 , "S3/est=",0.9933472"p=", 1321, "pi3(p)=", 1.14076510985 1011 , "est=", 1.1483773743956371603 1011 , "S3/est=",0.9933715(13)

"p=", 1327, "pi3(p)=", 1.16413263768 1011 , "est=", 1.1686467608217762700 1011 , "S3/est=",0.9961374"p=", 1621, "pi3(p)=", 2.52427126454 1011 , "est=", 2.5316868568936880752 1011 , "S3/est=",0.9970707"p=", 1627, "pi3(p)=", 2.56734005337 1011 , "est=", 2.5680950525219821688 1011 , "S3/est=",0.9997060"p=", 1637, "pi3(p)=", 2.61120787190 1011 , "est=", 2.6296374104779834565 1011 , "S3/est=",0.9929918"solution of pi_3(t) log(t) = 0.271 t^4 is", 663.3555(14)> verifpi4max(150,300000,0.01); # To prove (2.26) execute verifpi4max(150,89967803,0.01)"r=4, coef=", 0.20254144050183350040"p=", 199, "pi4(p)=", 1.4185215405 1010 , "est=", 1.3972908775495022172 1010 , "S4/est=",1.015194"p=", 283, "pi4(p)=", 7.5911179980 1010 , "est=", 7.6204777072473669256 1010 , "S4/est=",0.9961472"solution of pi_4(t) log(t) = 0.237 t^5 is", 199.6247(15)> verifpi5max(31,100000,0.01); # To prove (2.27) execute verifpi3max(31,89967803,0.01)"r=5, coef=", 0.16852606262213529074"p=", 31, "pi5(p)=", 6.0025150 107 , "c=", 5.8409021339400157698 107 , "S5/c=", 1.027669"p=", 43, "pi5(p)=", 3.92233751 108 , "c=", 3.7983303337492375910 108 , "S5/c=", 1.032648"p=", 109, "pi5(p)=", 8.0528815750 1010 , "c=", 8.0792265797600247385 1010 , "S5/c=",
0.9967392"p=", 113, "pi5(p)=", 9.8953167543 1010 , "c=", 9.9530885582769023493 1010 , "S5/c=",
0.9941956"solution of pi_5(t) log(t)=0.226 t^6 is", 43.24158(16)> verifpi5maxbis(5,43);# Compute the max of pi_5(p)/(p^6/log(p)) for
pmin <= p <= pmax
(proof of (2.28))
0.35021, 0.33422, 0.24534, 0.29362, 0.23152, 0.27842, 0.23055, 0.17773, 0.23225, 0.18207,
0.19171, 0.23338"pmaxi=", 5, "maxi=", 0.35021369, "log(3.)/3=", 0.36620410(17)>> ################# Definition of Superchampions, Section 3.1################>> n:=nu0: SupChn(n);# determines rho, xi, n', n'', N', N'' (as defined in Definition
3.6) and g(n)/N'

" rho = pquo/log(pquo) = ", 4.342944827 108 , " xi = pquo = ", 10000000019" n = ", 2220832950051364840, " n' =ell(N') = ", 2220832950051364840, " n'' = ell(N'') = ",2220832960051364859" N' = ", 2, 29 , 3, 18 , 5, 12 , 7, 10 , 11, 8 , 13, 8 , 17, 7 , 19, 7 , 23, 6 , ... , 31,
6 , 37, 5 , ... , 71, 5 , 73, 4 , ... , 211, 4 , 223, 3 , ... , 1459, 3 , 1471, 2 , ... ,69557, 2 , 69593, ... , 9999999967" N''/N' = pquo = ", 10000000019, " g(n)/N' = ", 1, " = ", 1., " time = ", 170.371(18)> calculdexij(10^10+19); # from xi_1, this proc determines xi[j] as
described in Definition 3.6,
p1[j] and p2[j] are the primes surrounding xi[j] cf. also (3.19).
"expo" is the exponent of p in N_rho = N' (cf. (3.6) and (3.18)). J
is defined in (3.9)"j=", 2, "p1=", 69557, "xi[j]=", 69588.85912, "p2=", 69593 , "j=", 3, "p1=", 1459, "xi[j]=",1468.859920, "p2=", 1471 , "j=", 4, "p1=", 211, "xi[j]=", 220.2592091, "p2=", 223 ,"j=", 5, "p1=", 71, "xi[j]=", 71.59540467, "p2=", 73 , "j=", 6, "p1=", 31, "xi[j]=",34.12559548, "p2=", 37 , "j=", 7, "p1=", 19, "xi[j]=", 20.20220124, "p2=", 23 , "j=", 8,"p1=", 13, "xi[j]=", 13.67872852, "p2=", 17 , "j=", 9, "p1=", 7, "xi[j]=", 10.12216225,"p2=", 11 , "j=", 10, "p1=", 7, "xi[j]=", 7.967318437, "p2=", 11 , "j=", 11, "p1=", 5,"xi[j]=", 6.557245414, "p2=", 7 , "j=", 12, "p1=", 5, "xi[j]=", 5.579189860, "p2=", 7 ,"j=", 13, "p1=", 3, "xi[j]=", 4.869318441, "p2=", 5 , "j=", 14, "p1=", 3, "xi[j]=",4.335130355, "p2=", 5 , "j=", 15, "p1=", 3, "xi[j]=", 3.921159360, "p2=", 5 , "j=", 16,"p1=", 3, "xi[j]=", 3.592464449, "p2=", 5 , "j=", 17, "p1=", 3, "xi[j]=", 3.326110823,"p2=", 5 , "j=", 18, "p1=", 3, "xi[j]=", 3.106512206, "p2=", 5 , "j=", 19, "p1=", 2,"xi[j]=", 2.922756826, "p2=", 3 , "j=", 20, "p1=", 2, "xi[j]=", 2.767006733, "p2=", 3 ,"j=", 21, "p1=", 2, "xi[j]=", 2.633504368, "p2=", 3 , "j=", 22, "p1=", 2, "xi[j]=",2.517936236, "p2=", 3 , "j=", 23, "p1=", 2, "xi[j]=", 2.417013857, "p2=", 3 , "j=", 24,"p1=", 2, "xi[j]=", 2.328190911, "p2=", 3 , "j=", 25, "p1=", 2, "xi[j]=", 2.249468069,"p2=", 3 , "j=", 26, "p1=", 2, "xi[j]=", 2.179255654, "p2=", 3 , "j=", 27, "p1=", 2,"xi[j]=", 2.116275247, "p2=", 3 , "j=", 28, "p1=", 2, "xi[j]=", 2.059488059, "p2=", 3 ,"j=", 29, "p1=", 2, "xi[j]=", 2.008042006, "p2=", 3 , "j=", 30, "p1=", 1, "xi[j]=",1.961232078, "p2=", 2"expo=", 0, 10000000019, "... ,infinity" , "expo=", 1, 69593, "...", 9999999967 ,"expo=", 2, 1471, "...", 69557 , "expo=", 3, 223, "...", 1459 , "expo=", 4, 73, "...",
211 , "expo=", 5, 37, "...", 71 , "expo=", 6, 23, "...", 31 , "expo=", 7, 17, "...",
19 , "expo=", 8, 11, "...", 13"p=", 7, "expo=", 10 , "p=", 5, "expo=", 12 , "p=", 3, "expo=", 18 , "p=", 2, "expo=",29"J=", 29.165332010963717010(19)

> calculch(1,6): # Computation of table of Fig. 1 and enumeration ofsuperchampion numbers"i= ", "ell(N)= ", "N=", " rho= ", " xi= "1, " ", 7, " ", 2, 2 , 3, " ", 3.1067, " ", 5
1, " ", 12, " ", 2, 2 , 3, 5, " ", 3.5973, " ", 7
1, " ", 19, " ", 2, 2 , 3, ... , 7, " ", 4.5874, " ", 11
1, " ", 30, " ", 2, 2 , 3, ... , 11, " ", 5.0683, " ", 13
1, " ", 43, " ", 2, 2 , 3, ... , 13, " ", 5.4614, " ", 14.667
2, " ", 49, " ", 2, 2 , 3, 2 , 5, ... , 13, " ", 5.7708, " ", 16.000
3, " ", 53, " ", 2, 3 , 3, 2 , 5, ... , 13, " ", 6.0003, " ", 173, " ", 70, " ", 2, 3 , 3, 2 , 5, ... , 17, " ", 6.4528, " ", 19
3, " ", 89, " ", 2, 3 , 3, 2 , 5, ... , 19, " ", 7.3354, " ", 23
3, " ", 112, " ", 2, 3 , 3, 2 , 5, ... , 23, " ", 8.6123, " ", 29
3, " ", 141, " ", 2, 3 , 3, 2 , 5, ... , 29, " ", 9.0274, " ", 31
3, " ", 172, " ", 2, 3 , 3, 2 , 5, ... , 31, " ", 10.247, " ", 37
3, " ", 209, " ", 2, 3 , 3, 2 , 5, ... , 37, " ", 11.041, " ", 41
3, " ", 250, " ", 2, 3 , 3, 2 , 5, ... , 41, " ", 11.433, " ", 43
3, " ", 293, " ", 2, 3 , 3, 2 , 5, ... , 43, " ", 11.542, " ", 43.559
4, " ", 301, " ", 2, 4 , 3, 2 , 5, ... , 43, " ", 12.207, " ", 474, " ", 348, " ", 2, 4 , 3, 2 , 5, ... , 47, " ", 12.427, " ", 48.143
5, " ", 368, " ", 2, 4 , 3, 2 , 5, 2 , 7, ... , 47, " ", 13.349, " ", 53
5, " ", 421, " ", 2, 4 , 3, 2 , 5, 2 , 7, ... , 53, " ", 14.470, " ", 59
5, " ", 480, " ", 2, 4 , 3, 2 , 5, 2 , 7, ... , 59, " ", 14.839, " ", 615, " ", 541, " ", 2, 4 , 3, 2 , 5, 2 , 7, ... , 61, " ", 15.935, " ", 67
5, " ", 608, " ", 2, 4 , 3, 2 , 5, 2 , 7, ... , 67, " ", 16.384, " ", 69.488
6, " ", 626, " ", 2, 4 , 3, 3 , 5, 2 , 7, ... , 67, " ", 16.656, " ", 71
6, " ", 697, " ", 2, 4 , 3, 3 , 5, 2 , 7, ... , 71, " ", 17.014, " ", 73
6, " ", 770, " ", 2, 4 , 3, 3 , 5, 2 , 7, ... , 73, " ", 18.080, " ", 79
6, " ", 849, " ", 2, 4 , 3, 3 , 5, 2 , 7, ... , 79, " ", 18.783, " ", 83
6, " ", 932, " ", 2, 4 , 3, 3 , 5, 2 , 7, ... , 83, " ", 19.828, " ", 89
6, " ", 1021, " ", 2, 4 , 3, 3 , 5, 2 , 7, ... , 89, " ", 21.204, " ", 97
6, " ", 1118, " ", 2, 4 , 3, 3 , 5, 2 , 7, ... , 97, " ", 21.584, " ", 99.230(20)> lemma3_8ii(9); # Proof of Lemma 3.8 (ii) and computation of lambda[j]"j=", 2, "e^j=", 7.3890561, "lambda[j]=", 79.877781309766419843, 80"j=", 3, "e^j=", 20.085537, "lambda[j]=", 585.45826414894707700, 586"j=", 4, "e^j=", 54.598150, "lambda[j]=", 6380.4599469708695867, 6381
"j=", 5, "e^j=", 148.41316, "lambda[j]=", 89016.683602195247707, 89017(21)

"j=", 6, "e^j=", 403.42879, "lambda[j]=", 1.4997496877890318902 106 , 1499750"j=", 7, "e^j=", 1096.6332, "lambda[j]=", 2.9511243343943428453 107 , 29511244"j=", 8, "e^j=", 2980.9580, "lambda[j]=", 6.6318407453999995851 108 , 663184075"j=", 9, "e^j=", 8103.0839, "lambda[j]=", 1.6743263688753398443 1010 , 16743263689> lemma3_11(t); # Convexity : Proof of Lemma 3.11, cf. (3.25)
1 ln ln t
1u ln ln t 2"Phi =", t ln t
12
ln t
ln t 2
1"Phi_1 = d Phi/dt=",
4

2

2L

2 L3

u

L2

2

6
L2

L2

u

L

8

u

(21)

3L

tL

"Phi_1 as written =",
2

1 2L
4

L2

u

L L

L2

3

2u

6u

1

L2 t L"Phi1(L,lambda) - Phi1 as written =", 0"Phi_2 = d Phi1/dt=",
2 L2

1
8

2

2 L4

u

L3

L3

L
1
1
L2
8 L2 t L 3 / 2"Phi_2 as written =",
L4

11 L

2

30
2

tL
2

2

2 L2

u

3L

64

u

11 L

L3

1

L 2L

16 u

3/2

L2

u

3

2

22 u

2 u 1521
8"Phi2(L,lambda) - Phi2 as written =", 0(22)> subs(a=log(2),devx2(3,0)); # Computation of (3.30) with t = log(xi)
1
1
1
1
1
ln 2
ln 2 2ln 2
ln 2 2ln 2 3
1 ln 2
2
8
2
2
16
1
(23)
2
t
t2
t3
> prop3_14(); # Proof of Proposition 3.14 : effective estimate of
xi_2, a satisfies O < a <= 0.4
1a"phi = ",
2 x 1
2
ln x
"Wx = (phi^2-phi)*log(x)/x-log(phi) = ",
1
x 1
2

a
ln x

2

1
2

2

x

a
ln x

ln x

2

1
2

1

1
2

ln

x

2

a
ln x
3
"Wt = ",
ln 2
2

ln t

ln 2 t

a

1 a
4 t

a

2 a
et

t

2
,

x

1

"change of variable x = e^(2t), t = log(x)/2"
1 a2
4 t

"W as written = ", a1
4 t2

"W1 = dW/dt = ",4e t

2 t2 a3ln 2
ln t
ln 2 t2"W as written - Wt = ", 0

1
2t

a

8e t

2 t3

8

2 a

8

"UU1 = dU/dt = ",

8

2 t4

t2

4

2 a2

2 t4

"U=", 82e ta3

2 a2 t28e t

2 a2 t

4ta

t3

2

2 a2

8

2 a

40

2

8

2 a t e t
2

2 e t

2

a

1
e t
2

2

2 a t38e t

2 t4

2 a t2 e t

4

2t

2 a2

a

4a t

3

a

24

2

"U1 as writte n =", 2 a a
2a a

2

t3

2

2 a2

4a

4 t4

4 a

2 a2

28

2 a

5 t3

a2

14 a

2 a2

52

2 a

12 t2

2 t
"U1 as written - UU1 = ", 0

"U2 = dU1/dt = ", 8
144

2

t2

2 t4
8

8
2 a2

64

"V = U2* exp(t)/(2*sqrt(2)) = ", 4 t4
32 a

24 t

2 a2

72

2

t3

2 a

48

2

4a

36 t3

2 a

2
t
a2

4

2 a2
26 a

8

2 a e t

72 t2

4 a2

4a

"******** UPPER BOUNDa = log(2)/2 = ", 0.34657359, " **********""real roots of Va = ", 0.051671859, 0.51498620, 2.3922609, 6.3876546

15

10

5

0

2

4

6

8

10

t

5

10
"U1a = subs(a = log(2)/2,U1), t = -0.3..10)""list of extrema of U1a = ", 1.5146439, 3.3570388, 16.098644, 10.036034"roots of U1a = ", 0.96460820, 4.2370082"lim U1a_{t -- -inf} = ",, "U1a(log 2) = ", 2.7975979, "lim U1a_{t -inf} = ",1.6265209

20

10

0

2

4

6

8

10

"Ua = subs(a = log(2)/2,U), t = -1..10)""list of extrema of Ua = ", 2.1898697, 29.691640"roots of Ua = ", 0.026388103, 1.4925111, 8.0010041-inf} = ", , "Ua(log 2) = ", 1.7641814, "lim Ua_{t--inf} = ",t

10"lim Ua_{t--

0.2

0.1

0

1

2

3

4

5

6

7

8

9

10

t

0.1

0.2

"Wa = subs(a = log(2)/2,W), t = 0.5..10 (Wa is defined for t a/2)""list of extrema of Wa = ", 0.27587993, 0.021939313"roots of Wa: w1 = ", 0.65426442, "w2 = ", 5.1811243, "exp(2*w1)=", 3.7007252,"exp(2*w2)=", 31642.265"lim Wa_{t -- a/2,right} = ", , "Wa(log 2) = ", 0.03659308, "lim Wa_{t -inf} = ", 0"*************** LOWER BOUND a = aa = 0.366 ******************""real roots of Va = ", 0.054243563, 0.52068533, 2.3978548, 6.3932163

15

10

5

0

2

4

6

8

10

t

5

10
"U1a = subs(a = log(2)/2,U1), t = -0.3..10)""list of extrema of U1a = ", 1.6071816, 3.4814326, 15.910864, 10.097391"roots of U1a = ", 0.97973211, 4.2315297"lim U1a_{t -- -inf} = ",, "U1a(log 2) = ", 2.9572804, "lim U1a_{t -inf} = ",1.7319120

20

10

0

2

4

6

8

10

"Ua = subs(a = log(2)/2,U), t = -1..10)""list of extrema of Ua = ", 2.3080321, 29.049736"roots of Ua = ", 0.029272690, 1.5208774, 7.8921186-inf} = ", , "Ua(log 2) = ", 1.8306254, "lim Ua_{t--inf} = ",t

10"lim Ua_{t--

0.2

0.1

0

2

4

6

8

10

12

14

t

0.1

0.2

"Wa = subs(a = 0.366,W), t = 0.5..15 (Wa is defined for t a/2)""list of extrema of Wa = ", 0.28265348, 0.0042037371"root of Wa: w1 = ", 0.66717791, " w2 = ", 6.3701275, "w3 = ", 11.088031"exp(2*w1) = ", 3.7975488, "exp(2*w2) = ", 3.4121055 105 , "exp(2*w3) = ", 4.2750516 109"lim Wa_{t -- a/2,right} = ", , "Wa(log 2) = ", 0.02529747, "lim Wa_{t -inf} = ",0.01942641"*************** REMARK

a = 0.370612465... ******************""roots of {U=0,W=0} = ", a = 0.37061246510421873686, t = 7.8668240753167659431

0.2

0.1

0

2

4

6

8

10

12

14

t

0.1

0.2

"subs(a = 0.3706..,W), t = 0.5..15""lim Wa_{t -- aa/2,right} = ", , "Wa(log 2) = ", 0.02252275, "lim Wa_{t -inf} = ",(24)
0.02403888> -log(2.)/2+log(1-0.366/logx0),2/(1-0.726/logx0),0.726/(1-0.726/logx0), 1+0.75/logx0;# Proof of Corollary 3.16, (3.36)
0.36259645149756300309, 2.0651125428792345374, 0.74963585306516213708,(25)
1.0325720861400561704> 0.0746*2.07,0.000079*2.07,log(2.)/2-0.000164; # Proof of Corollary
3.17, (3.37)
0.154422, 0.00016353, 0.34640959027997265471(26)> 0.13-0.496*0.366/logx0,log(2.)/2-sqrt(2.)*(1+2/sqrtx0)*
logx0^2/sqrtx0,2.07*0.385,0.797-0.339;# Proof ofCorollary 3.18, (3.38)
0.12211599169397168407, 0.33907542411137549851, 0.79695, 0.458(27)> addexc(x0); # Computation of the additive excess, Section 3.5.1,(3.40)
10517469635602(28)> 0.122-0.057*3*sqrt(2.)*logx0^2/sqrt(x0); # Proof of Proposition
(29)

>

3.19, (3.42), the lower bound0.12071783922838529422> majexcadd(8,3); # Proof of Proposition 3.19, (3.42), the upperbound"j=", 3, "tot=", 1.9173213501338885334, "be=", 3., "ga=", 0.19732, "de=", 0.4152847, "et=",1.459321, "xi0=", 1468.859920"j=", 4, "tot=", 1.0917301233710986024, "be=", 4., "ga=", 0.17832, "de=", 0.05508744, "et=",0.2184455, "xi0=", 220.2592091"j=", 5, "tot=", 0.99917712229214889068, "be=", 5., "ga=", 0.17611, "de=", 0.01720423, "et=",0.07080502, "xi0=", 71.59540467"j=", 6, "tot=", 0.97921454582973396434, "be=", 6., "ga=", 0.32045, "de=", 0.01453040, "et=",0.03363820, "xi0=", 34.12559548"j=", 7, "tot=", 0.97995402905538685099, "be=", 7., "ga=", 0.32066, "de=", 0.008403627,"et=", 0.01984729, "xi0=", 20.20220124"j=", 8, "tot=", 0.98190626528105069885, "be=", 8., "ga=", 0.32253, "de=", 0.005602969,"et=", 0.01339590, "xi0=", 13.67872852"total minimum =", 0.97921454582973396434, "j0 =", 6> remark3_20(x0); # Remark 3.20
"x = ", 1.0000000019 1010 , "additive excess = ", 10517469635602, "coe = ",0.63219636253854929280> mulexc(x0); # Computation of the multiplicative excess, Section
3.5.2, (3.49)
70954.464472086132180> remark3_22(x0); # Remark 3.22
"x = ", 1.0000000019 1010 , "multiplicative excess = ", 70954.464472086132180, "coe = ",0.079385275754404565538> majexcmul(29,3); # Proof of Proposition 3.21 the upper bound,(3.55)"j0=", 3, "total[j0] = ", 15.812988, "be = ", 3., "S1 = ", 0., "S2 = ", 16.15898, "xi0 = ",1468.8599"j0=", 4, "total[j0] = ", 2.5592690, "be = ", 4., "S1 = ", 0.4864336, "S2 = ", 2.418833, "xi0 = ",220.25921"j0=", 5, "total[j0] = ", 0.99726837, "be = ", 5., "S1 = ", 0.5592478, "S2 = ", 0.7840196,"xi0 = ", 71.595405"j0=", 6, "total[j0] = ", 0.60932361, "be = ", 6., "S1 = ", 0.5828491, "S2 = ", 0.3724738,"xi0 = ", 34.125595"j0=", 7, "total[j0] = ", 0.46783010, "be = ", 7., "S1 = ", 0.5940617, "S2 = ", 0.2197678,"xi0 = ", 20.202201"j0=", 8, "total[j0] = ", 0.40300986, "be = ", 8., "S1 = ", 0.6006774, "S2 = ", 0.1483319,"xi0 = ", 13.678729

(29)

(30)
(31)

(32)
(33)

"j0=", 9, "total[j0] = ", 0.36733275, "be = ", 9.9866, "S1 = ", 0.6051426, "S2 = ", 0.1081896,"xi0 = ", 10.122162"j0=", 10, "total[j0] = ", 0.34718051, "be = ", 11.435, "S1 = ", 0.6083994, "S2 = ", 0.08478056,"xi0 = ", 7.9673184"j0=", 11, "total[j0] = ", 0.33445562, "be = ", 12.979, "S1 = ", 0.6109516, "S2 = ", 0.06950353,"xi0 = ", 6.5572454"j0=", 12, "total[j0] = ", 0.32597944, "be = ", 14.621, "S1 = ", 0.6130438, "S2 = ", 0.05893509,"xi0 = ", 5.5791899"j0=", 13, "total[j0] = ", 0.32010274, "be = ", 16.360, "S1 = ", 0.6148180, "S2 = ", 0.05128427,"xi0 = ", 4.8693184"j0=", 14, "total[j0] = ", 0.31590357, "be = ", 18.198, "S1 = ", 0.6163618, "S2 = ", 0.04554130,"xi0 = ", 4.3351304"j0=", 15, "total[j0] = ", 0.31283469, "be = ", 20.135, "S1 = ", 0.6177327, "S2 = ", 0.04110149,"xi0 = ", 3.9211594"j0=", 16, "total[j0] = ", 0.31055495, "be = ", 22.172, "S1 = ", 0.6189700, "S2 = ", 0.03758447,"xi0 = ", 3.5924644"j0=", 17, "total[j0] = ", 0.30884279, "be = ", 24.308, "S1 = ", 0.6201014, "S2 = ", 0.03474091,"xi0 = ", 3.3261108"j0=", 18, "total[j0] = ", 0.30754928, "be = ", 26.545, "S1 = ", 0.6211472, "S2 = ", 0.03240159,"xi0 = ", 3.1065122"j0=", 19, "total[j0] = ", 0.30657129, "be = ", 28.882, "S1 = ", 0.6221226, "S2 = ", 0.03044822,"xi0 = ", 2.9227568"j0=", 20, "total[j0] = ", 0.30583556, "be = ", 31.319, "S1 = ", 0.6230392, "S2 = ", 0.02879591,"xi0 = ", 2.7670067"j0=", 21, "total[j0] = ", 0.30528891, "be = ", 33.856, "S1 = ", 0.6239060, "S2 = ", 0.02738241,"xi0 = ", 2.6335044"j0=", 22, "total[j0] = ", 0.30489193, "be = ", 36.493, "S1 = ", 0.6247303, "S2 = ", 0.02616114,"xi0 = ", 2.5179362"j0=", 23, "total[j0] = ", 0.30461493, "be = ", 39.231, "S1 = ", 0.6255178, "S2 = ", 0.02509661,"xi0 = ", 2.4170139"j0=", 24, "total[j0] = ", 0.30443521, "be = ", 42.070, "S1 = ", 0.6262733, "S2 = ", 0.02416141,"xi0 = ", 2.3281909"j0=", 25, "total[j0] = ", 0.30433513, "be = ", 45.009, "S1 = ", 0.6270006, "S2 = ", 0.02333400,"xi0 = ", 2.2494681"j0=", 26, "total[j0] = ", 0.30430085, "be = ", 48.048, "S1 = ", 0.6277031, "S2 = ", 0.02259730,"xi0 = ", 2.1792557"j0=", 27, "total[j0] = ", 0.30432139, "be = ", 51.188, "S1 = ", 0.6283833, "S2 = ", 0.02193759,"xi0 = ", 2.1162752

"j0=", 28, "total[j0] = ", 0.30438791, "be = ", 54.428, "S1 = ", 0.6290437, "S2 = ", 0.02134372,"xi0 = ", 2.0594881"j0=", 29, "total[j0] = ", 0.30449327, "be = ", 57.769, "S1 = ", 0.6296862, "S2 = ", 0.02080657,"xi0 = ", 2.0080420"total minimum =", 0.30430085464403772470, "j0min =", 26(34)> 0.98+3*sqrt(2.)*logx0^2/sqrt(x0),1.01+3*sqrt(2.)*logx0^2/sqrt(x0),1/3/sqrt(2.)/logx0*(1+1.033/logx0); #(3.60)--(3.62)
1.0024940486248193996, 1.0324940486248193996, 0.010695651685592331819(35)> 0.045/logx0,0.12-0.002,0.118-0.002*0.12/logx0,0.1179-3*sqrt(2.)*
logx0^2/sqrt(x0);# (3.63)--(3.64)
0.0019543251684033702237, 0.118, 0.11798957693243518203, 0.095405951375180600404(36)> 0.002/logx0,(1.0331+0.0001033/logx0)/3, 0.095-3*sqrt(2.)*
logx0^2/sqrt(x0);# (3.65)--(3.67)
0.000086858896373483121052, 0.34436816208733256347, 0.072505951375180600404(37)> ############### Section 5 : Proof of Theorem 1.7###############> logx0^4/x0+logx0^3/3/sqrt(2*x0)*(1+0.98/logx0),107/160+0.0301; #(5.7)--(5.8)
0.030027486686917837222, 0.69885000000000000000(38)> Digits:=8: P:=expand((t/2+t^2/4+t^3/4+7*t^4/10)*(2/t-1-0.584*t));fsolve(P-1=0);1/logx0; Digits:=20: # between (5.8) and (5.9)
P
1 0.04200000 t2 1.0040000 t3 0.84600000 t4 0.40880000 t5
2.9220110, 0., 0., 0.043457433, 0.80908191
0.043429448(39)> 1+1.168/L0; # (5.11)
1.0276486259210757577(40)> (lambda0-1)/2/L0; # between (5.14) and (5.15)
0.032471418966263015433(41)> 2*(1+1/2/L0),1/lambda0,2.03*lambda0/L0,(-2+0.27+0.18)/8; # (5.17)
2.0236717687680443131, 0.26713168008555813164, 0.17988765160215779833,(42)0.19375000000000000000> (0.584+0.48*1.03)/lambda0; # (5.18)
0.28807480380426588916(43)> 3/20-logx0^4/x0; # between (5.18) and (5.19)
0.14997188987647023669(44)> Digits:=8: Q:=expand((t/2+t^2/4+t^3/4+0.149*t^4)*(2/t-1-0.492*t));fsolve(Q-1=0);1/logx0; Digits:=20: # between (5.19) and ((5.20)
Q
1 0.00400000 t2 0.07500000 t3 0.27200000 t4 0.073308 t5
3.4052224, 0.35082465, 0., 0., 0.045674419
0.043429448(45)> 2-log(2*logx0)/logx0,1.8336*0.492; # before (5.20)
1.8336754315737963672, 0.9021312(46)

> 1+(lambda0-1)/2/L0,normal(expand((1+(\lambda-1)/(2*L))^2-(1+(lambda-1-b/L)/L+(lambda-1)/L^2))),evalf([fsolve(0.989*lambda^2-6*
lambda+8.608)],8),0.011/(4*2.066); # between (5.21)and (5.22)
2
1
4b 6
5
1.0324714189662630154,, 2.3279775, 3.7387566 ,
(47)
2
4
L
0.0013310745401742497580> y:=1+(t^2/4-0.76*t)*exp(-t): y1:=normal(diff(y,t)):y,evalf(y1,3);
A:=[solve(exp(t)*y1,t)]:A,evalf(subs(t=A[1],y));# Proof ofCorollary 5.3
1 2
1t
0.76 t e t , 1.26 e 1. t t 0.76 e 1. t 0.250 e 1. t t2
44.3394504664870654291, 0.70054953351293457094 , 1.0183880248221891576(48)> np0:=nu0-addexc(x0): # np0 is equal to pi_1(x0)
Lp0:=log(evalf(np0)):lambdap0:=log(Lp0):np0,Lp0,lambdap0;
1/2+logx0^4/x0,8*0.500029,0.38-0.026*(1+lambdap0/2/Lp0+0.38*
lambdap0/Lp0^2);# Section 5.2, (5.24)--(5.25)
2220822432581729238, 42.244409266298653707, 3.7434720199894299222
0.50002811012352976331, 4.000232, 0.35282728508916522658(49)> minslice1_7(380000000,400000000,1): # Proof of Theorem 1.7, (1.13)
cf. Section 5.3# To prove (1.13), execute minslice1_7(400000000,
2220822442581729257,100):"i=", 74, "wmin = ", 0.0066219950, "ppmin = ", 88813, "nppmin = ", 361092676, "temps = ",0.142
"Conclusion""smallest difference = ", 2.67300676360905 10-8 , " for the interval = ", 361181493,
361270312 , "time = ", 0.142"For ", 398898277, " = n = ", 411137805, "log h(n)Phi_{1/8}(n) holds""For p = ", 93337, "pp = ", 93371, "np = ", 398804906, "npp =", 398898277, "k1 = pi(pp) = ",9018, " log h(np) = ", 93084.9779186447, "Phi_{1/8}(npp) = ", 93084.9901194923,"holds", " w = ", 0.000018031046> okrec1_7(2,398898277,500); # completes the proof of (1.13), see
Section 5.3"n1=", 2, "n2=", 398898277, "test log h(n)phi1s8(n)""comp=", 500, 392202900, 392227247 , "log h(n1)=", 92267.2008144, "phi1s8(n2)=",92265.6222929, "time=", 19.958"comp=", 1000, 386651825, 386663998 , "log h(n1)=", 91577.3325886, "phi1s8(n2)=",91577.2084028, "time=", 40.350"comp=", 1500, 382147664, 382245051 , "log h(n1)=", 91016.7948888, "phi1s8(n2)=",91027.0126491, "time=", 59.176(50)

"comp=", 2000, 378690415, 378714762 , "log h(n1)=", 90583.2159550, "phi1s8(n2)=",90585.2735537, "time=", 78.558"comp=", 2500, 376067045, 376073132 , "log h(n1)=", 90253.0781314, "phi1s8(n2)=",90253.4417908, "time=", 95.753"comp=", 3000, 374161908, 374356683 , "log h(n1)=", 90012.8133791, "phi1s8(n2)=",90037.2302934, "time=", 112.710"comp=", 3500, 373623233, 373624755 , "log h(n1)=", 89944.6953125, "phi1s8(n2)=",89944.8893040, "time=", 129.366"Conclusion""number of intervals=", 3577, "mini=", 0.000010316671, "lismini=", 373714534, 373714724 ,"time=", 132.184"For n s.t. ", 373623863, " = n = ", 398898277, " log h(n)phi1s8(n) holds""For n =", 373623862, "log h(n) =", 89944.7747662060, "phi1s8(n) =", 89944.7765891595,"log h(n) = phi1s8(n) is false"(51)

Digits20(52)> 0.72+logx0/sqrt(x0),0.73*(1+(lambda0-1)/4/L0),0.75*L0^(7/4)/evalf(n0)^(1/4)/lambda0^2;#(5.28)--(5.29)
0.72023025850909965898, 0.74185206792268600063, 0.00097046379802116620280(53)> nu1:=10.^36:L1:=log(nu1):lambda1:=log(L1):"L1 = ",L1,"lambda1 = ",lambda1;
evalf(1+(lambda1-1)/2/L1,8),evalf(1.2*1.021,5),evalf(1.23/L1/lambda1^2,8),
evalf(0.75*L1^(7/4)/nu1^(1/4)/lambda1^2,8);# (5.28)--(5.29)"L1 = ", 82.893063347785644625, "lambda1 = ", 4.4175513837040658014
1.0206142, 1.2252, 0.00076036781, 8.7519510 10-8> calculzn(19); # Proof of (5.32) for n <= 19 [Computation of z_n
defined by (3.30)]"n=", 2, "g(n)=", 2., "zn=", 2.054536"n=", 3, "g(n)=", 3., "zn=", 2.383142"n=", 4, "g(n)=", 4., "zn=", 3.033933"n=", 5, "g(n)=", 6., "zn=", 2.351073"n=", 6, "g(n)=", 6., "zn=", 3.183046"n=", 7, "g(n)=", 12., "zn=", 2.057540"n=", 8, "g(n)=", 15., "zn=", 2.191439"n=", 9, "g(n)=", 20., "zn=", 2.165139"n=", 10, "g(n)=", 30., "zn=", 1.944815"n=", 11, "g(n)=", 30., "zn=", 2.342347"n=", 12, "g(n)=", 60., "zn=", 1.730138"n=", 13, "g(n)=", 60., "zn=", 2.073518(54)

"n=", 14, "g(n)=", 84., "zn=", 1.963177"n=", 15, "g(n)=", 105., "zn=", 1.988366"n=", 16, "g(n)=", 140., "zn=", 1.933928"n=", 17, "g(n)=", 210., "zn=", 1.752946"n=", 18, "g(n)=", 210., "zn=", 1.997081"n=", 19, "g(n)=", 420., "zn=", 1.531456(55)> Digits:=30; majslice1_7(6466*10^12,6482*10^12,1,3): Digits:=20:# To prove Theorem 1.7 (1.14) execute majslice1_7(19,nu0,50,3), cf.
Section 5.5
Digits30"i=", 1999, "minz=", 0.00545504803632000559309654456114, "nminz=", 6473549497145122," pminz = ", 502748593, "time = ", 12.436"Conclusion""mini = ", 3.05746569703860669 10-15 , "nmini = ", 6472496283558071, " pmini = ",502706629, "time = ", 12.436"For ", 6465984410320754, " = n = ", 6482105188612314"minimum of z_n = ", 0.00545504803632000559309654456114, " for n = ",6473549497145122, " and p = ", 502748593(56)> ##################### Section 6 : Study of g(n)/h(n) for large n's##################> (lambda0-1.019)^2/8/L0,-11.433-10.392*(lambda0-1.041)/4/L0,3/2*(0.305-0.0724/3),2.3704+3.3704*(lambda0-1)/4/L0; # Proof of
Proposition 6.1, (6.5)--(6.6)
0.021963697964266949319, 11.599200023483068084, 0.42130000000000000000,
2.4251208352419464336(57)> ######### Section 7.1 : Proof of Theorem 1.5(i), Asymptotic
expansion of log(g(n)/h(n)) ########> devx2(4,0); # Asymptotic expansion of xi_2/sqrt(xi/2) with t=log(xi), a=log 2 (cf. (3.31))
1
1
1 2
1
1 2
1 3
a
a
a
a
a
a
2
2
8
2
2
16
1
t
t2
t3
1
a
2

9 2
a
8

23 3
a
48

(58)

5
a4
128

t4
> devx2log(4,0); # Asymptotic expansion of log(xi_2)/(log(xi)/2))
with t=log(xi), a=log 2
(59)

a
t

1

1 2
a
2

a

a

3 2
a
2

a

1 3
a
3

(59)
t
t3
t4
> loggmoinsloghx(4,0): # log(g(n)/h(n)) = sqrt(2 xi)/3*F(1/log xi),a=log 2 cf. (7.1)--(7.2)
1
1
4
2
1 2
8
4
5 2
1 3
a
a
a
a
a
a
3
2
9
3
8
9
3
8
16
"F(t) =", 1
(60)
t
t2
t3
64
27

32
a
9

2

55 2
a
24

7 3
a
12

5
a4
128

t4
> Limoinsunasy(4,logn,0); # Expansion of Li^(-1)(n) with logn=log(n),L=log(logn))# cf. Salvy [30, p. 235, B2]
11
1 2
131
1 3
7 2
L
3L
L
L
14 L
L 1
2 L
2
2
6
3
2
1
(61)
logn
logn2
logn3
logn4
> loggdenasy(4,t,0); # Asymptotic expansion of sqrt(Li^(-1)(n))/sqrt(n*log(n))# with logn=log(n), L=log(logn) : cf. (1.7) and Salvy [30, p.
235, B3]
1
1
9
3
1 2
11 2
39
53
1 3
L
L
L
L
L
L
2
2
8
4
8
16
16
16
16
1
(62)
logn
logn2
logn3
5071
384

61 3
L
96

239 2
L
64

343
L
32

5
L4
128

logn4
> Loggmoinsloghn(3,0); # log(g(n)/h(n)) with logn=log(n), L=log(logn)) cf. Thm. 1.5 (i)
11
1
635
29
1 2
15
3
3 2
a
L
a
a
a L
L
12
4
288
12
2
16
4
32
1
(63)
logn
logn2
1
3

logn
117
128

1119
128

333
a
32

21
a L2
32

7
L3
128

39 2
a
8

1 3
a
2

5525
1152

239
a
48

7 2
a L
8

> #################### The method of B. Salvy : cf. [30]###########################>> E:= convert([seq(i!*n^i, i=0..10)],`+`); G:=expand(subs(a=log(2),t=
1/2/n,loggmoinsloghx(5,0)));E3628800 n10 362880 n9 40320 n8 5040 n7 720 n6 120 n5 24 n4 6 n3

(64)

2 n2

n

1

8 2
1 2
64 3
32 3
n ln 2n ln 2 2nn ln 2
(64)
3
2
9
3
1 3
1024 4512 4110 4
5 n3 ln 2 2
n ln 2 3nn ln 2n ln 2 2
2
27
9
3
28 4
5 4
20480 510240 52452 5n ln 2 3n ln 2 4nn ln 2n ln 2 2
3
8
81
27
9
938 5
211 5
7 5
n ln 2 3n ln 2 4n ln 2 59128> sqrt(2)/3*theorem2_part2(1,1/4,0,E,subs(t=1/(2*n),G),n,3); # Salvy
method for Thm. 1.5 (i)111ln ln n
ln 2
1
4
12
1
3
1/4
1/4
2 n
ln n
1
ln ln n 2
(65)
2
3
ln n
32
ln n
G

2
n
3

1

1
ln ln n
4
1
ln n

3

16 2
n
9

n ln 23
4

7ln ln n1281
ln ln n43
41
ln 2 2 ln ln n2O

ln ln n

11
12
3

11
12

ln 2

5
ln ln n
16
ln 2

1119
128

635
288

ln ln n2ln ln n
333
ln 2
32

21
32

29
ln 2
12
11
12

7
4

635
288

39
ln 2 28ln 2

1ln 2 22ln ln n

2

29
ln 2
12
1
ln 2 32

4

ln n 4> thm2(1,1/4,0,E,subs(t=1/(2*n),G),n,6): # C is defined by Salvy [30,p.229, (8)]
5 2
47 3
379 4
9337 5109139 6"C=", n
n
n
n
n
n
O n7
2
6
12
60
120
11
635
29
"generating series of constant coeff = ", 1
ln 2 n
ln 2
(66)
12
288
12
1
1119
333
39
1
ln 2 2 n2ln 2
ln 2 2ln 2 3 n3
2
128
32
8
2
2553311
67297
6935
221
5
ln 2
ln 2 2ln 2 3ln 2 4 n4
55296
1152
192
24
8
199949215
22023041
637417
59381
1673
ln 2
ln 2 2ln 2 3ln 2 4663552
55296
2304
576
96
7
8635843619
37818307
3170699
5
5
2
ln 2
n
ln 2
ln 2
8
26542080
22118436864

18913ln 2 32304

1741
ln 2 4768

7ln 2 5 n632

O n7> ################ Section 7.2 : Proof of Theorem 1.5 (ii)#################> majslice1_5_2(22*10^12,24*10^12,2.43,1,5):# To prove Theorem 1.5 (ii) execute majslice1_5_2(22*10^12,nu0,
2.43,50,3):"i=", 603, "betamaj a=", 2.43, 2.4300064058024262524, "grnp=", 21962830729180,"grnpp=", 21962857728827"et2 fin i=", 603, 22255802707198, 22255815714040 , "pg=", 27183829, "temps=", 10.611"i=", 604, "betamaj a=", 2.43, 2.4300064058024262524, "grnp=", 21962830729180,"grnpp=", 21962857728827"et2 fin i=", 604, 22397215968791, 22397229018947 , "pg=", 27274099, "temps=", 15.074"i=", 605, "betamaj a=", 2.43, 2.4300140812407573615, "grnp=", 22488775657598,"grnpp=", 22488802989495"et2 fin i=", 605, 22493094303400, 22493107382472 , "pg=", 27334357, "temps=", 18.340"i=", 606, "betamaj a=", 2.43, 2.4300140812407573615, "grnp=", 22488775657598,"grnpp=", 22488802989495"et2 fin i=", 606, 22637672158340, 22637685280846 , "pg=", 27424871, "temps=", 23.013"i=", 607, "betamaj a=", 2.43, 2.4300140812407573615, "grnp=", 22488775657598,"grnpp=", 22488802989495"et2 fin i=", 607, 22832033620345, 22832046800875 , "pg=", 27545821, "temps=", 29.126"i=", 608, "betamaj a=", 2.43, 2.4300140812407573615, "grnp=", 22488775657598,"grnpp=", 22488802989495"et2 fin i=", 608, 22980073934412, 22980087158544 , "pg=", 27636683, "temps=", 33.965"i=", 609, "betamaj a=", 2.43, 2.4300143514977784168, "grnp=", 22998745945409,"grnpp=", 22998773594040"et2 fin i=", 609, 23125585762674, 23125599030480 , "pg=", 27727709, "temps=", 38.536"i=", 610, "betamaj a=", 2.43, 2.4300080809917574916, "grnp=", 23270400847898,"grnpp=", 23270428664755"et2 fin i=", 610, 23522231323698, 23522244708320 , "pg=", 27971147, "temps=", 51.235"i=", 611, "betamaj a=", 2.43, 2.4300133958176945950, "grnp=", 23542024585503,"grnpp=", 23542052569006"et2 fin i=", 611, 23821956516556, 23821969989126 , "pg=", 28154449, "temps=", 60.541"i=", 612, "betamaj a=", 2.43, 2.4300133958176945950, "grnp=", 23542024585503,"grnpp=", 23542052569006"et2 fin i=", 612, 23871407733729, 23871421220985 , "pg=", 28185019, "temps=", 62.062"i=", 613, "betamaja=", 2.43, 2.4300133958176945950, "grnp=", 23542024585503,

"grnpp=", 23542052569006"et2 fin i=", 613, 23973111551325, 23973125067977 , "pg=", 28246289, "temps=", 65.184"i=", 614, "betamaj a=", 2.43, 2.4300133958176945950, "grnp=", 23542024585503,"grnpp=", 23542052569006"et2 fin i=", 614, 24328169348333, 24328182968123 , "pg=", 28461241, "temps=", 76.342"Conclusion"-8"mini = ", 2.35154099257 10 , "lismini = ", 22436166605748, 22436193904301 , " time = ",76.342"For ", 23542052569006, " = n = ", 24328182968123, "beta_n a = ", 2.43, "holds""For [n1,n2] = ", 23542024585503, 23542052569006 , "betamaj(n1,n2) = ",(67)
2.4300133958176945950, "a fails"> okrec1_5_2(3997000000000,3997100000000,2.43,50); # cf. Section 7.2# To complete the proof of Theorem 1.5(ii) execute okrec1_5_2(3997*
10 ^9,24*10^12,2.43,10000);"n1=", 3997000000000, "n2=", 3997100000000, "a=", 2.43, "test beta_n = beta(n,n)
a""comp=", 50, 3997087500000, 3997088281250 , "beta(n1,n2) = ", 2.39351707394114,"time=", 23.478"comp=", 100, 3997078027343, 3997078076171 , "beta(n1,n2) = ", 2.39727296570309,"time=", 48.813"comp=", 150, 3997057031250, 3997057812500 , "beta(n1,n2) = ", 1.61529486876097,"time=", 75.550"comp=", 200, 3997043750000, 3997045312500 , "beta(n1,n2) = ", 2.53868961373140,"time=", 99.137"comp=", 250, 3997032812500, 3997033593750 , "beta(n1,n2) = ", 2.46854345234483,"time=", 125.551"comp=", 300, 3997022460937, 3997022656250 , "beta(n1,n2) = ", 2.43154734638921,"time=", 147.300"conclusion""number of intervals=", 336, "mini=", 5.2129424 10-7 , "lismini=", 3997089332579,
3997089332960 , "time=", 163.924"For n s.t. ", 3997022083663, " = n = ", 3997100000000, " beta_n = beta(n,n)", 2.43," holds""For n =", 3997022083662, " beta_n = ", 2.43000186560077, "a = ", 2.43, " is wrong"> ################ Section 7.2 Proof of Theorem 1.5(iii)###########################

(68)

>
> minslice1_5_3(10^6,2*10^6,-11.6,1,5):# To prove Theorem 1.5(iii) execute minslice1_5_3(10^6,nu0,-11.6,50,3): cf. Section 7.2"i=", 27, "betamij a=", 11.6, 14.259827401957466163, "grnp=", 1013843, "grnpp=",1017810"et2 fin i=", 27, 1017810, 1019616 , "pg=", 3967, "betamin = ", 14.259827401957466163,"temps=", 0.039"i=", 28, "betamij a=", 11.6, 14.259827401957466163, "grnp=", 1013843, "grnpp=",1017810"et2 fin i=", 28, 1129985, 1131195 , "pg=", 4211, "betamin = ", 9.6175507206549377901,"temps=", 0.061"i=", 29, "betamij a=", 11.6, 14.259827401957466163, "grnp=", 1013843, "grnpp=",1017810"et2 fin i=", 29, 1417703, 1419865 , "pg=", 4751, "betamin = ", 8.9517383512064067588,"temps=", 0.120"i=", 30, "betamij a=", 11.6, 14.259827401957466163, "grnp=", 1013843, "grnpp=",1017810"et2 fin i=", 30, 2218331, 2221087 , "pg=", 6043, "betamin = ", 9.5553174164055259285,"temps=", 0.209"Conclusion""mini = ", 0.0062114090788275400, "lismini = ", 1448575, 1453376 , " time = ", 0.209"For ", 1017810, " = n = ", 2221087, "beta_n a = ", 11.6, "holds""For [n1,n2] = ", 1013843, 1017810 , "betamij(n1,n2) = ", 14.259827401957466163,"
a fails"(69)> okrec1_5_3(2,1017810,-11.6,200); # completes the proof of Theorem
1.5(iii), cf. Section 7.2"n1=", 2, "n2=", 1017810, "a=", 11.6, "test beta_na""comp=", 200, 747454, 763358 , "beta116(n1,n2) = ", 69.4051239908777, "time=", 6.641"comp=", 400, 548663, 550651 , "beta116(n1,n2) = ", 8.51372480376293, "time=", 12.959"comp=", 600, 373728, 381680 , "beta116(n1,n2) = ", 50.7659876545890, "time=", 18.759"comp=", 800, 239544, 240538 , "beta116(n1,n2) = ", 9.77690435726574, "time=", 24.383"comp=", 1000, 139155, 141143 , "beta116(n1,n2) = ", 20.3678436990832, "time=", 29.918"comp=", 1200, 73554, 75542 , "beta116(n1,n2) = ", 38.4158042644535, "time=", 34.133"comp=", 1400, 36777, 37771 , "beta116(n1,n2) = ", 34.4180583442533, "time=", 36.478"comp=", 1600, 17395, 17892 , "beta116(n1,n2) = ", 26.7926306456169, "time=", 38.485"comp=", 1800, 7673, 7704 , "beta116(n1,n2) = ", 12.7975973642547, "time=", 39.792"conclusion"(70)

"number of intervals=", 1948, "mini=", 0.0092603867, "lismini=", 616252, 618240 , "time=",40.807"For n s.t. ", 4230, " = n = ", 1017810, " beta_n", 11.6, " holds""For n =", 4229, " beta_n = ", 12.3466353525204, "a = ", 11.6, " is wrong"(70)> ################## Section 7.3 : Proof of Theorem 1.5(iv)#################> proc4230(4230): # cf. Section 7.3"list of n's s.t. g(n)=h(n)", 1, 2, 3, 5, 6, 8, 10, 11, 15, 17, 18, 28, 41, 58, 77(71)> ################## Section 7.4 : Proof of Theorem 1.5(v)#################> majslice1_5_5(4*10^7,7*10^7,0.62,1,5): # see Section 7.4# To prove Theorem 1.5 (v), execute majslices1_5_5(7,4*10^12,0.62,50,3)"i=", 53, "Mn1n2 = ", 0.62285689, "a = ", 0.62, "for n1 = ", 47242215, "n2 = ", 47272712"fin i=", 53, 47272712, 47274760 , "pg=", 30497, "max = ", 0.62685834, "temps=", 0.150"i=", 54, "Mn1n2 = ", 0.62002071, "a = ", 0.62, "for n1 = ", 49435860, "n2 = ", 49467083"fin i=", 54, 60123385, 60139387 , "pg=", 34513, "max = ", 0.62499533, "temps=", 0.344"i=", 55, "Mn1n2 = ", 0.62002071, "a = ", 0.62, "for n1 = ", 49435860, "n2 = ", 49467083"fin i=", 55, 67443934, 67460964 , "pg=", 36713, "max = ", 0.61358790, "temps=", 0.455"i=", 56, "Mn1n2 = ", 0.62002071, "a = ", 0.62, "for n1 = ", 49435860, "n2 = ", 49467083"fin i=", 56, 76655164, 76666802 , "pg=", 39251, "max = ", 0.61469502, "temps=", 0.748"Conclusion""For ", 49467083, " = n = ", 76666802, "log(g(n)/h(n))/(n*log n)^(1/4)", 0.62, "holds""For [n1,n2] = ", 49435860, 49467083 , "M(n1,n2) = ", 0.62002070729155826870> okrec1_5_5(2,5*10^7,0.62059,3000); # see Section 7.4"n1=", 2, "n2=", 50000000, "a=", 0.62059, "test log g(n)/h(n)a(n log n)^(1/4)""comp=", 3000, 19604493, 19628907 , "log(g(n2)/h(n1)) = ", 83.061683,"a(n log n)^(1/4) = ", 83.591970, "time=", 201.974"comp=", 6000, 5957032, 5963135 , "log(g(n2)/h(n1)) = ", 60.874494, "a(n log n)^(1/4) = ",60.931797, "time=", 363.801"comp=", 9000, 1149749, 1150512 , "log(g(n2)/h(n1)) = ", 32.997979, "a(n log n)^(1/4) = ",39.277010, "time=", 488.763"comp=", 12000, 128175, 128938 , "log(g(n2)/h(n1)) = ", 21.285434, "a(n log n)^(1/4) = ",21.745433, "time=", 577.182(72)"Conclusion""number of intervals=", 14289, "mini=", 0.000032758901, "lismini=", 2244 , "time=", 619.307(73)

"For n s.t. ", 2244, " = n = ", 50000000, " log(g(n)/h(n)) = ", 0.62059," (n log n)^(1/4) holds""For n =", 2243, "log g(n)/h(n) = ", 0.62059, " (n log n)^(1/4) is false"(73)> proc2243(120000,0.62); # computes the n's up to 120000 with d_n >0.62, cf. (7.12)"list of n's s.t. log(g(n)/h(n))/(n*log(n))^(1/4)", 0.62, 2243, 2244, 118869"n=", 2243, "log(g[n]/h[n])/(nf*log(nf))^(1/4)=", 0.62066526568375620112"n=", 2244, "log(g[n]/h[n])/(nf*log(nf))^(1/4)=", 0.62058714421610282114"n=", 118869, "log(g[n]/h[n])/(nf*log(nf))^(1/4)=", 0.62014609683999469464
2243, 2244, 118869(74)> ##################### Section 8.2 : Proof of Theorem 1.1(ii)######################> 0.23+sqrt(2.)/12+2.43*sqrt(2.)/lambda0/12; # Section 8.2
0.42435183229335061604(75)> Digits:=30: minslice1_1_2(6466*10^12,6482*10^12,1,3): Digits:=20: #cf. Lemma 8.1# To prove Lemma 8.1, execute minslice1_1_2(43,nu0,50,3), cf.
Section 8.2"i=", 1999, "minan=", 0.193938608602741816619480762052, "nmin=", 6473580667603736,"pmin=", 502750063, "time=", 229.446"Conclusion""mini = ", 2.327448795937405289 10-12 , "nmini = ", 6473581170353835, " pmini = ",502750099, " time = ", 229.446"For ", 6465984410320754, " = n = ", 6482105188612314"minimum of a_n = ", 0.193938608602741816619480762052, " for n = ", 6473580667603736," and p = ", 502750063> functLandau(6473580667603736,2): # computation of g
(6473580667603736) cf. Lemma 8.1, Section 8.2
N=, 2, 25 , 3, 15 , 5, 11 , 7, 9 , 11, 7 , 13, 7 , 17, 6 , 19, 6 , 23, 5 , ... , 37, 5 ,41, 4 , ... , 103, 4 , 107, 3 , ... , 523, 3 , 541, 2 , ... , 15559, 2 , 15569, ... ,502750063
n=, 6473580667603736, ell(N)=, 6473580667603736, n-ell(N)=, 0
rho=, 2.5092835 107 , prime number corresponding to rho=, 502750099, "xi = ", 5.0275010 108
n=, 6473580667603736, g(n)/N=, 1, n-ell(g(n))=, 0, total time=, 0.909

(76)

(77)

> ##################### Section 8.3 : Proof of Theorem 1.1(iii)######################> 0.77-sqrt(2.)/12+11.6*sqrt(2.)/lambda0/12; # Section 8.3
1.0173374065548656454(78)

> majslice1_1_3(5000000,7000000,1.02,1);# To prove Theorem 1.1(iii), execute majslice1_1_3(5000000,nu0,
1.02,50) cf. Section 8.3"et2 fin i=", 35, 5389332, 5393754 , "Rmax = ", 1.0815115344417261072, "temps=", 0.960"et2 fin i=", 36, 5451806, 5453864 , "Rmax = ", 1.0242496353855823670, "temps=", 0.992"et2 fin i=", 37, 6673601, 6678571 , "Rmax = ", 1.0164861795871539357, "temps=", 1.557"et2 fin i=", 38, 7402436, 7407692 , "Rmax = ", 0.85062122654949382314, "temps=", 1.954" Conclusion""mini=", 0.0000029920292, "lismini=", 4116861, 4125250 , "time=", 1.954"For n in ", 5432420, 7000000 , "a_n (2-sqrt(2))/3 c ", 1.02, "log(log(n))/log n holds""R( ", 5422741, 5432420, ") is equal to", 1.02036341295143, "", 1.0235, 1.0815115344417261072, 4442781 , 36, 1.0242496353855823670, 5393754 , 37,
1.0164861795871539357, 5453864 , 38, 0.85062122654949382314, 6678571> okrec1_1_3(2,5500000,1.02,1000);# proves the second step of the proof of Theorem 1.1(iii), cf.
Section 8.3"n1=", 2, "n2=", 5500000, "a=", 1.02,"test sqrt(Li^(-1)(n)(n1*log n1)^(1/4)(((2-sqrt(2))/3 c) a*log(log n)/log n)""comp=", 1000, 2814453, 2825195 , " sqrt(Li^(-1)(n2) = ", 6846.8354, " fa102(n1,a) = ",6840.4235, "time=", 28.933"comp=", 2000, 1326660, 1329346 , " sqrt(Li^(-1)(n2) = ", 4580.8164, " fa102(n1,a) = ",4582.6575, "time=", 54.184"comp=", 3000, 520997, 523682 , " sqrt(Li^(-1)(n2) = ", 2782.2470, " fa102(n1,a) = ",2779.2417, "time=", 75.068"comp=", 4000, 162475, 163818 , " sqrt(Li^(-1)(n2) = ", 1488.2739, " fa102(n1,a) = ",1484.7226, "time=", 93.122"comp=", 5000, 43221, 43305 , " sqrt(Li^(-1)(n2) = ", 722.72295, " fa102(n1,a) = ",723.06874, "time=", 108.046"Conclusion""fa102(n,a) = log g(n)(n*log n)^(1/4)(deux3pc a*log(log n)/log n)", " with a = ", 1.02
"number of intervals=", 5993, "mini=", 0.000025672391, "lismini=", 43473, 43641 , "time=",120.586"For n s.t. ", 19425, " = n = ", 5500000, "and a= ", 1.02,"sqrt(Li^(-1)(n)fa102(n,a) holds""For n =", 19424, " sqrt(Li^(-1)(n) = ", 465.831202938283, "fa102(n,a) = ",465.809244112473, " fails"

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> ##################### Section 8.4 : Proof of Theorem 1.1(iv)######################> "c=",c, (2-sqrt(2.))/3+c+1.02*log(log(19425.))/log(19425.); # cf.
Section 8.4"c=", 0.0461176444215090238270908, 0.47792637844227154606(82)> test19424(19424,0.477927): # upper bound of a_n : cf. Section 8.4"com=", 1, " For n = ", 19425, " a_n", 0.47793"com=", 2, " For n = ", 18962, " a_n", 0.47898"com=", 3, " For n = ", 18501, " a_n", 0.48205"com=", 4, " For n = ", 18500, " a_n", 0.48293"com=", 5, " For n = ", 18043, " a_n", 0.48314"com=", 6, " For n = ", 17595, " a_n", 0.48584"com=", 7, " For n = ", 17152, " a_n", 0.48710"com=", 8, " For n = ", 7517, " a_n", 0.48771"com=", 9, " For n = ", 7499, " a_n", 0.48855"com=", 10, " For n = ", 7228, " a_n", 0.49027"com=", 11, " For n = ", 6959, " a_n", 0.49656"com=", 12, " For n = ", 6696, " a_n", 0.49944"com=", 13, " For n = ", 6436, " a_n", 0.50280"com=", 14, " For n = ", 5662, " a_n", 0.50292"com=", 15, " For n = ", 5429, " a_n", 0.50654"com=", 16, " For n = ", 5200, " a_n", 0.51198"com=", 17, " For n = ", 4086, " a_n", 0.51480"com=", 18, " For n = ", 3896, " a_n", 0.51542"com=", 19, " For n = ", 3705, " a_n", 0.52049"com=", 20, " For n = ", 3521, " a_n", 0.52315"com=", 21, " For n = ", 3340, " a_n", 0.52904"com=", 22, " For n = ", 2970, " a_n", 0.53044"com=", 23, " For n = ", 2786, " a_n", 0.53046"com=", 24, " For n = ", 2629, " a_n", 0.53196"com=", 25, " For n = ", 2477, " a_n", 0.53559"com=", 26, " For n = ", 1409, " a_n", 0.53934"com=", 27, " For n = ", 1304, " a_n", 0.54145"com=", 28, " For n = ", 1302, " a_n", 0.54286"com=", 29, " For n = ", 1199, " a_n", 0.54621"com=", 30, " For n = ", 906, " a_n", 0.54840"com=", 31, " For n = ", 883, " a_n", 0.55563"com=", 32, " For n"com=", 33, " For n= ", 823, " a_n= ", 810, " a_n", 0.55740", 0.56014

"com=", 34, " For n"com=", 35, " For n"com=", 36, " For n"com=", 37, " For n"com=", 38, " For n"com=", 39, " For n"com=", 40, " For n"com=", 41, " For n"com=", 42, " For n"com=", 43, " For n"com=", 44, " For n> ######################= ", 731, " a_n= ", 659, " a_n= ", 287, " a_n= ", 200, " a_n= ", 198, " a_n= ", 166, " a_n= ", 161, " a_n= ", 23, " a_n= ", 7, " a_n= ", 3, " a_n= ", 2, " a_n

The END", 0.57408", 0.59835", 0.61324", 0.62046", 0.62710", 0.62923", 0.64447", 0.65227", 0.68836", 0.72355", 0.91024#################################(83)