/* Copyright (C) 2019 Alessandro Languasco */ /**************** A. LANGUASCO ******************** ************* COMPUTATION OF THE max_chi |L'/L| MOD q(PRIME) *******/ /************************* Building the T approach ****************/ \\ Global variables: global(Dilcher_matrix_T); global(psi_matrix); global(tab); \\global variable used to initialise sumnum global(phir); global(pir); global(defaultprecision); /************* COMPUTATION OF THE max_chi |L'/L| MOD q(PRIME) ******* ************* FOR r1<=q<=r2 ************* ************* Using the T-function ********************/ {maxT(r1,r2,defaultprecision)=local(minutes, millisec, seconds, rows, columns, elaptimefinalcomp, elaptimeprecomp, M, derlog, evenresult,oddresult, phiq,qminusonehalfminusone, logtwoPi, EulerpluslogtwoPi, minuslogq, qminusonehalf,twoPi, twoPiI, i, qminusone, qminustwo, correctionderlog, u, numer, denom, v, chi ); \\ minutes,millisec,seconds: used just to compute the elapsed computation time; local variables \\ defaultprecision: used to fix the precision used in the computations; global variable print("************ A. LANGUASCO *************"); print("********* COMPUTATION OF THE MAX |L'/L|**********"); print("******* FOR q IN ONE INTERVAL ********"); print("********* with the T-function **********"); \\ precision setting if(defaultprecision >90 , print("Internal constant are defined with 100 decimal digits; you cannot set the default precision larger than 90 decimal digits"); return); default(realprecision,defaultprecision); r1=nextprime(r1); r2=precprime(r2); if (r1>r2, print("error: no odd primes in this interval. END PROGRAM");return); if (r1 < 3, r1=3); print("Interval=[",r1,",",r2,"]"); phir=r2-1; pir=primepi(r2)-primepi(r1)+1; tab = sumnuminit(); \\ needed for the sumnum functions gettime(); print("Performing precomputations of T(a/q), psi(a/q) for this interval"); precompT(r1,r2); elaptimeprecomp=gettime(); twoPi=2*Pi; logtwoPi = log(twoPi); EulerpluslogtwoPi = Euler + logtwoPi; twoPiI=twoPi*I; i=0; gettime(); print("Starting computation of max_q"); forprime(q=r1,r2, i+=1; phiq=q-1; evenresult=0; oddresult=0; qminusone=phiq; qminustwo=q-2; minuslogq=-log(q); qminusonehalf=qminusone/2; qminusonehalfminusone=qminusonehalf-1; correctionderlog= minuslogq ; u=twoPiI/qminusone; M=0;derlog=0; \\ even characters for (m=1, qminusonehalfminusone, numer=0; denom=0; v=exp(u*2*m); \\ j=2m+1; j-1 even (j odd) means an even character chi=1; for (k=0,qminustwo, \\chi=v^(k) numer += chi*Dilcher_matrix_T[i,k+1]; \\Dilcher_T(g^k%q/q) denom += chi*psi_matrix[i,k+1]; \\psi(g^k%q/q); chi=chi*v; \\print(numer); print(denom); ); derlog=abs(correctionderlog+numer/denom); if (derlog>M, M=derlog); ); evenresult= M; print("max_{chi even}|L'/L(1,chi)|(",q") = ", evenresult); M=0;derlog=0; \\ odd characters for(m=1, qminusonehalf, \\\ odd characters numer=0; denom=0; v=exp(u*(2*m-1)); \\ j=2m; j-1 odd (j even) means an odd character chi=1; for (k=0,qminustwo, \\chi=v^(k) numer += chi*Dilcher_matrix_T[i,k+1]; \\Dilcher_T(g^k%q/q) denom += chi*psi_matrix[i,k+1]; \\psi(g^k%q/q); chi=chi*v; \\print(numer); print(denom); ); derlog=abs(correctionderlog+numer/denom); if (derlog>M, M=derlog); ); oddresult=M; print("max_{chi odd}|L'/L(1,chi)|(",q") = ", oddresult); print("max_{chi}|L'/L(1,chi)|(",q") = ", max(oddresult,evenresult)); ); elaptimefinalcomp=gettime(); seconds=floor(elaptimeprecomp/1000)%60; minutes=floor(elaptimeprecomp/60000); millisec=elaptimeprecomp- minutes*60000 - seconds*1000; \\print(elaptimeprecomp); print("Precomputation time: ", minutes, " min, ", seconds, " sec, ", millisec, " millisec"); \\print(elaptimefinalcomp); seconds=floor(elaptimefinalcomp/1000)%60; minutes=floor(elaptimefinalcomp/60000); millisec=elaptimefinalcomp- minutes*60000 - seconds*1000; print("Final step max computation time (output time included): ", minutes, " min, ", seconds, " sec, ", millisec, " millisec"); seconds=floor((elaptimefinalcomp+elaptimeprecomp)/1000)%60; minutes=floor((elaptimefinalcomp+elaptimeprecomp)/60000); millisec=(elaptimefinalcomp+elaptimeprecomp)- minutes*60000 - seconds*1000; print("Total elapsed time: ", minutes, " min, ", seconds, " sec, ", millisec, " millisec"); print("****** END PROGRAM ********"); } /***************** T-function ****************/ {DilcherT(x)= local(T,m,gamma1,ord,eps); \\ computes -T(x) \\tab is a global variable gamma1=-0.072815845483676724860586375874901319137736338334337952599006559741401433571511484878086928244844014604; eps=1/100.; ord=x; \\rate of decay for the integral formula of S-DIF if (ord < eps, T=sumnum(m=0, (log(x+m)/(x+m))-(log(1+m)/(1+m)),tab), T = - gamma1 +intnum(t=0,[+oo,ord], (1-exp(-t)-t*exp((1-x)*t))*(Euler+log(t))/(t*(exp(t)-1))) ); return(T); } {DilcherTtab(x,tabstandalone)= local(T,m,gamma1,ord,eps); \\ for the standalone precomputations \\ computes -T(x) gamma1=-0.072815845483676724860586375874901319137736338334337952599006559741401433571511484878086928244844014604; eps=1/100.; ord=x; \\rate of decay for the integral formula of S-DIF if (ord < eps, T=sumnum(m=0, (log(x+m)/(x+m))-(log(1+m)/(1+m)),tabstandalone), T = - gamma1 +intnum(t=0,[+oo,ord], (1-exp(-t)-t*exp((1-x)*t))*(Euler+log(t))/(t*(exp(t)-1))) ); return(T); } /***************** Precomputations ****************/ {precompT(r1,r2)= local(rows, columns, i, aoverq, m, g1, g, a); Dilcher_matrix_T=matrix(pir,phir+1,rows,columns,0); psi_matrix=matrix(pir,phir+1,rows,columns,0); i=0; forprime(q=r1,r2, print("Precomputation for q = ",q); g1=znprimroot(q); \\ needed here to sort the sequences T(a,q) psi(a/q) using a=a=g^k%q g=lift(g1); i+=1; Dilcher_matrix_T[i,phir+1]=q; psi_matrix[i,phir+1]=q; a=1; for(k=0, q-2, aoverq = a/q; Dilcher_matrix_T[i,k+1]=DilcherT(aoverq); \\ already sorted for the final summation psi_matrix[i,k+1] = psi(aoverq); a=(a*g)%q \\a=g^k%q; ); ); } /************************************ **** RESULTS **** gp2.11.2 and gp2c0.0.11 compiled by myself, see below ******************* gp 2.11.2 on macmini languasc-macmini:gp_scripts languasc$ gp2c-run -pmy_ -g -W Max-T-final.gp GP/PARI CALCULATOR Version 2.11.2 (released) i386 running darwin (x86-64/GMP-6.1.2 kernel) 64-bit version compiled: Aug 3 2019, Apple LLVM version 10.0.1 (clang-1001.0.46.4) threading engine: single (readline v8.0 enabled, extended help enabled) Copyright (C) 2000-2018 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?17 for how to get moral (and possibly technical) support. parisizemax = 400003072, primelimit = 500000 *************** with the T-function ************ ? init_Max_T_final();maxT(3,300,30) ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[3,293] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 3 Precomputation for q = 5 Precomputation for q = 7 Precomputation for q = 11 Precomputation for q = 13 Precomputation for q = 17 Precomputation for q = 19 Precomputation for q = 23 Precomputation for q = 29 Precomputation for q = 31 Precomputation for q = 37 Precomputation for q = 41 Precomputation for q = 43 Precomputation for q = 47 Precomputation for q = 53 Precomputation for q = 59 Precomputation for q = 61 Precomputation for q = 67 Precomputation for q = 71 Precomputation for q = 73 Precomputation for q = 79 Precomputation for q = 83 Precomputation for q = 89 Precomputation for q = 97 Precomputation for q = 101 Precomputation for q = 103 Precomputation for q = 107 Precomputation for q = 109 Precomputation for q = 113 Precomputation for q = 127 Precomputation for q = 131 Precomputation for q = 137 Precomputation for q = 139 Precomputation for q = 149 Precomputation for q = 151 Precomputation for q = 157 Precomputation for q = 163 Precomputation for q = 167 Precomputation for q = 173 Precomputation for q = 179 Precomputation for q = 181 Precomputation for q = 191 Precomputation for q = 193 Precomputation for q = 197 Precomputation for q = 199 Precomputation for q = 211 Precomputation for q = 223 Precomputation for q = 227 Precomputation for q = 229 Precomputation for q = 233 Precomputation for q = 239 Precomputation for q = 241 Precomputation for q = 251 Precomputation for q = 257 Precomputation for q = 263 Precomputation for q = 269 Precomputation for q = 271 Precomputation for q = 277 Precomputation for q = 281 Precomputation for q = 283 Precomputation for q = 293 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(3) = 0 max_{chi odd}|L'/L(1,chi)|(3) = 0.368281615970147842633237904076 max_{chi}|L'/L(1,chi)|(3) = 0.368281615970147842633237904076 max_{chi even}|L'/L(1,chi)|(5) = 0.827679476715504887991046989677 max_{chi odd}|L'/L(1,chi)|(5) = 0.180899098585657908884214228728 max_{chi}|L'/L(1,chi)|(5) = 0.827679476715504887991046989677 max_{chi even}|L'/L(1,chi)|(7) = 0.693743252999179022242316373933 max_{chi odd}|L'/L(1,chi)|(7) = 0.251084483958898199841792124425 max_{chi}|L'/L(1,chi)|(7) = 0.693743252999179022242316373933 max_{chi even}|L'/L(1,chi)|(11) = 0.566852343328803197958852026226 max_{chi odd}|L'/L(1,chi)|(11) = 0.649609999423979953636904530776 max_{chi}|L'/L(1,chi)|(11) = 0.649609999423979953636904530776 max_{chi even}|L'/L(1,chi)|(13) = 0.565582635433404890770098107244 max_{chi odd}|L'/L(1,chi)|(13) = 0.696309862992037155840892183520 max_{chi}|L'/L(1,chi)|(13) = 0.696309862992037155840892183520 max_{chi even}|L'/L(1,chi)|(17) = 0.510627102336359538761550412731 max_{chi odd}|L'/L(1,chi)|(17) = 1.36293176857311326439833395890 max_{chi}|L'/L(1,chi)|(17) = 1.36293176857311326439833395890 max_{chi even}|L'/L(1,chi)|(19) = 0.644329999573610570346026468705 max_{chi odd}|L'/L(1,chi)|(19) = 1.56821936415476775304938942269 max_{chi}|L'/L(1,chi)|(19) = 1.56821936415476775304938942269 max_{chi even}|L'/L(1,chi)|(23) = 0.617943642128736392603203757765 max_{chi odd}|L'/L(1,chi)|(23) = 1.07370241439895666993863022504 max_{chi}|L'/L(1,chi)|(23) = 1.07370241439895666993863022504 max_{chi even}|L'/L(1,chi)|(29) = 0.908711562024443188256150455943 max_{chi odd}|L'/L(1,chi)|(29) = 1.37173438584080190328583030799 max_{chi}|L'/L(1,chi)|(29) = 1.37173438584080190328583030799 max_{chi even}|L'/L(1,chi)|(31) = 0.791265179022304116114210355804 max_{chi odd}|L'/L(1,chi)|(31) = 1.41315141911004437078399808370 max_{chi}|L'/L(1,chi)|(31) = 1.41315141911004437078399808370 max_{chi even}|L'/L(1,chi)|(37) = 0.733183107188977342750203978087 max_{chi odd}|L'/L(1,chi)|(37) = 1.29518958101078356915278401821 max_{chi}|L'/L(1,chi)|(37) = 1.29518958101078356915278401821 max_{chi even}|L'/L(1,chi)|(41) = 0.916079374735210610265214032861 max_{chi odd}|L'/L(1,chi)|(41) = 1.29673609198958173353796568380 max_{chi}|L'/L(1,chi)|(41) = 1.29673609198958173353796568380 max_{chi even}|L'/L(1,chi)|(43) = 1.39288786688690318884121465229 max_{chi odd}|L'/L(1,chi)|(43) = 1.41176882240051173489451389181 max_{chi}|L'/L(1,chi)|(43) = 1.41176882240051173489451389181 max_{chi even}|L'/L(1,chi)|(47) = 1.35156929889543656315617537620 max_{chi odd}|L'/L(1,chi)|(47) = 1.39567565425273602292102717603 max_{chi}|L'/L(1,chi)|(47) = 1.39567565425273602292102717603 max_{chi even}|L'/L(1,chi)|(53) = 1.00112944067413655687127416377 max_{chi odd}|L'/L(1,chi)|(53) = 1.30627572903790815149667975264 max_{chi}|L'/L(1,chi)|(53) = 1.30627572903790815149667975264 max_{chi even}|L'/L(1,chi)|(59) = 1.81899383678937843989348366929 max_{chi odd}|L'/L(1,chi)|(59) = 1.32353200689013556740700834632 max_{chi}|L'/L(1,chi)|(59) = 1.81899383678937843989348366929 max_{chi even}|L'/L(1,chi)|(61) = 1.41809980889441627035459190983 max_{chi odd}|L'/L(1,chi)|(61) = 1.29707099312546569340773776449 max_{chi}|L'/L(1,chi)|(61) = 1.41809980889441627035459190983 max_{chi even}|L'/L(1,chi)|(67) = 1.67019193303154369921782607634 max_{chi odd}|L'/L(1,chi)|(67) = 1.40107639865766134900254448893 max_{chi}|L'/L(1,chi)|(67) = 1.67019193303154369921782607634 max_{chi even}|L'/L(1,chi)|(71) = 1.47455511100236771011015896767 max_{chi odd}|L'/L(1,chi)|(71) = 1.23112352788523565627982472042 max_{chi}|L'/L(1,chi)|(71) = 1.47455511100236771011015896767 max_{chi even}|L'/L(1,chi)|(73) = 1.78248970799598673447282517891 max_{chi odd}|L'/L(1,chi)|(73) = 1.48272839696663020051708640270 max_{chi}|L'/L(1,chi)|(73) = 1.78248970799598673447282517891 max_{chi even}|L'/L(1,chi)|(79) = 1.27095088466385163871005360770 max_{chi odd}|L'/L(1,chi)|(79) = 1.34616837027813468918588610688 max_{chi}|L'/L(1,chi)|(79) = 1.34616837027813468918588610688 max_{chi even}|L'/L(1,chi)|(83) = 1.16333003167253880184592380510 max_{chi odd}|L'/L(1,chi)|(83) = 1.34527786237910789501875868023 max_{chi}|L'/L(1,chi)|(83) = 1.34527786237910789501875868023 max_{chi even}|L'/L(1,chi)|(89) = 1.61654649274126300156782088673 max_{chi odd}|L'/L(1,chi)|(89) = 1.27381998100989573937802515309 max_{chi}|L'/L(1,chi)|(89) = 1.61654649274126300156782088673 max_{chi even}|L'/L(1,chi)|(97) = 1.60286118570076458480362218799 max_{chi odd}|L'/L(1,chi)|(97) = 1.47070211417034258063739679739 max_{chi}|L'/L(1,chi)|(97) = 1.60286118570076458480362218799 max_{chi even}|L'/L(1,chi)|(101) = 1.51871979857079618912367283335 max_{chi odd}|L'/L(1,chi)|(101) = 1.27994704538217415996880600332 max_{chi}|L'/L(1,chi)|(101) = 1.51871979857079618912367283335 max_{chi even}|L'/L(1,chi)|(103) = 1.56072764165486011343921965820 max_{chi odd}|L'/L(1,chi)|(103) = 1.33555360325105672672035007620 max_{chi}|L'/L(1,chi)|(103) = 1.56072764165486011343921965820 max_{chi even}|L'/L(1,chi)|(107) = 1.51176315288720940464356960246 max_{chi odd}|L'/L(1,chi)|(107) = 1.55529418086936504978552066530 max_{chi}|L'/L(1,chi)|(107) = 1.55529418086936504978552066530 max_{chi even}|L'/L(1,chi)|(109) = 1.64357129023277185742674081202 max_{chi odd}|L'/L(1,chi)|(109) = 1.65357828827908326582841136643 max_{chi}|L'/L(1,chi)|(109) = 1.65357828827908326582841136643 max_{chi even}|L'/L(1,chi)|(113) = 1.51486982889352164427060492878 max_{chi odd}|L'/L(1,chi)|(113) = 1.24748116143699351889256349382 max_{chi}|L'/L(1,chi)|(113) = 1.51486982889352164427060492878 max_{chi even}|L'/L(1,chi)|(127) = 1.55590143040596443193792941854 max_{chi odd}|L'/L(1,chi)|(127) = 1.48849928643174274996361403061 max_{chi}|L'/L(1,chi)|(127) = 1.55590143040596443193792941854 max_{chi even}|L'/L(1,chi)|(131) = 1.43797882292531602089564238879 max_{chi odd}|L'/L(1,chi)|(131) = 1.27572700301367543915138664345 max_{chi}|L'/L(1,chi)|(131) = 1.43797882292531602089564238879 max_{chi even}|L'/L(1,chi)|(137) = 1.53929870904867707257469538680 max_{chi odd}|L'/L(1,chi)|(137) = 1.36270081382481387074802450457 max_{chi}|L'/L(1,chi)|(137) = 1.53929870904867707257469538680 max_{chi even}|L'/L(1,chi)|(139) = 1.58828875478913218915240825692 max_{chi odd}|L'/L(1,chi)|(139) = 1.46843197510913207331559965477 max_{chi}|L'/L(1,chi)|(139) = 1.58828875478913218915240825692 max_{chi even}|L'/L(1,chi)|(149) = 1.55933423387754689170927007457 max_{chi odd}|L'/L(1,chi)|(149) = 1.49066249516909038748573912245 max_{chi}|L'/L(1,chi)|(149) = 1.55933423387754689170927007457 max_{chi even}|L'/L(1,chi)|(151) = 1.48171078244888795642226012230 max_{chi odd}|L'/L(1,chi)|(151) = 1.21630093914275672755098193481 max_{chi}|L'/L(1,chi)|(151) = 1.48171078244888795642226012230 max_{chi even}|L'/L(1,chi)|(157) = 1.52915091159611605159149879696 max_{chi odd}|L'/L(1,chi)|(157) = 1.47930592186725803914452237782 max_{chi}|L'/L(1,chi)|(157) = 1.52915091159611605159149879696 max_{chi even}|L'/L(1,chi)|(163) = 1.57513694273693708178977104660 max_{chi odd}|L'/L(1,chi)|(163) = 2.16832712928352380386400324642 max_{chi}|L'/L(1,chi)|(163) = 2.16832712928352380386400324642 max_{chi even}|L'/L(1,chi)|(167) = 1.56607236656750344030293511154 max_{chi odd}|L'/L(1,chi)|(167) = 1.35615244416151765582440171259 max_{chi}|L'/L(1,chi)|(167) = 1.56607236656750344030293511154 max_{chi even}|L'/L(1,chi)|(173) = 1.54242401828716131644723995819 max_{chi odd}|L'/L(1,chi)|(173) = 1.43963149697994500743617408030 max_{chi}|L'/L(1,chi)|(173) = 1.54242401828716131644723995819 max_{chi even}|L'/L(1,chi)|(179) = 1.60085064594072009293300914735 max_{chi odd}|L'/L(1,chi)|(179) = 1.39782757591912800423472060801 max_{chi}|L'/L(1,chi)|(179) = 1.60085064594072009293300914735 max_{chi even}|L'/L(1,chi)|(181) = 1.50955704471891625386577768178 max_{chi odd}|L'/L(1,chi)|(181) = 1.65656567095010010041093792977 max_{chi}|L'/L(1,chi)|(181) = 1.65656567095010010041093792977 max_{chi even}|L'/L(1,chi)|(191) = 1.69400806335478035992195123369 max_{chi odd}|L'/L(1,chi)|(191) = 1.51238257796640956592393446369 max_{chi}|L'/L(1,chi)|(191) = 1.69400806335478035992195123369 max_{chi even}|L'/L(1,chi)|(193) = 1.72106839151430000218016220949 max_{chi odd}|L'/L(1,chi)|(193) = 1.42940396032541735361413928822 max_{chi}|L'/L(1,chi)|(193) = 1.72106839151430000218016220949 max_{chi even}|L'/L(1,chi)|(197) = 1.58142886014710250392544294033 max_{chi odd}|L'/L(1,chi)|(197) = 1.58425224704856913591906318269 max_{chi}|L'/L(1,chi)|(197) = 1.58425224704856913591906318269 max_{chi even}|L'/L(1,chi)|(199) = 1.43355422198296835867339511842 max_{chi odd}|L'/L(1,chi)|(199) = 1.52055512030192431037107983792 max_{chi}|L'/L(1,chi)|(199) = 1.52055512030192431037107983792 max_{chi even}|L'/L(1,chi)|(211) = 1.42636194836883457299103667906 max_{chi odd}|L'/L(1,chi)|(211) = 1.58887689723521687477342354947 max_{chi}|L'/L(1,chi)|(211) = 1.58887689723521687477342354947 max_{chi even}|L'/L(1,chi)|(223) = 1.57809439787964273689310796956 max_{chi odd}|L'/L(1,chi)|(223) = 1.55303288836237605943021818591 max_{chi}|L'/L(1,chi)|(223) = 1.57809439787964273689310796956 max_{chi even}|L'/L(1,chi)|(227) = 1.61440476278289514090073256762 max_{chi odd}|L'/L(1,chi)|(227) = 1.50389687420786831791141229315 max_{chi}|L'/L(1,chi)|(227) = 1.61440476278289514090073256762 max_{chi even}|L'/L(1,chi)|(229) = 1.64147596839111917230029367252 max_{chi odd}|L'/L(1,chi)|(229) = 1.64391627222705529854073112016 max_{chi}|L'/L(1,chi)|(229) = 1.64391627222705529854073112016 max_{chi even}|L'/L(1,chi)|(233) = 1.34601818744675837966996822954 max_{chi odd}|L'/L(1,chi)|(233) = 1.56534808865669695863593307680 max_{chi}|L'/L(1,chi)|(233) = 1.56534808865669695863593307680 max_{chi even}|L'/L(1,chi)|(239) = 1.62409294223527640292514612201 max_{chi odd}|L'/L(1,chi)|(239) = 1.83593237895342242137799671838 max_{chi}|L'/L(1,chi)|(239) = 1.83593237895342242137799671838 max_{chi even}|L'/L(1,chi)|(241) = 1.52053007064284698420889751143 max_{chi odd}|L'/L(1,chi)|(241) = 1.74483502309356231328685290592 max_{chi}|L'/L(1,chi)|(241) = 1.74483502309356231328685290592 max_{chi even}|L'/L(1,chi)|(251) = 1.48698867799867919284125336441 max_{chi odd}|L'/L(1,chi)|(251) = 1.60634233356394595761434310531 max_{chi}|L'/L(1,chi)|(251) = 1.60634233356394595761434310531 max_{chi even}|L'/L(1,chi)|(257) = 1.52986363395322517571321794433 max_{chi odd}|L'/L(1,chi)|(257) = 1.44048311367778018043569879961 max_{chi}|L'/L(1,chi)|(257) = 1.52986363395322517571321794433 max_{chi even}|L'/L(1,chi)|(263) = 1.47652133868945448396166271321 max_{chi odd}|L'/L(1,chi)|(263) = 1.61873689910065712561008039262 max_{chi}|L'/L(1,chi)|(263) = 1.61873689910065712561008039262 max_{chi even}|L'/L(1,chi)|(269) = 1.58662353583078976012953348699 max_{chi odd}|L'/L(1,chi)|(269) = 1.50439156819694711118224374474 max_{chi}|L'/L(1,chi)|(269) = 1.58662353583078976012953348699 max_{chi even}|L'/L(1,chi)|(271) = 1.46458989230894491096657135270 max_{chi odd}|L'/L(1,chi)|(271) = 1.51145118046000075647340279932 max_{chi}|L'/L(1,chi)|(271) = 1.51145118046000075647340279932 max_{chi even}|L'/L(1,chi)|(277) = 1.60164832656696765575321238671 max_{chi odd}|L'/L(1,chi)|(277) = 1.72974155675277125427451583060 max_{chi}|L'/L(1,chi)|(277) = 1.72974155675277125427451583060 max_{chi even}|L'/L(1,chi)|(281) = 1.49191529286515462825136060300 max_{chi odd}|L'/L(1,chi)|(281) = 1.60536366070704717918242357661 max_{chi}|L'/L(1,chi)|(281) = 1.60536366070704717918242357661 max_{chi even}|L'/L(1,chi)|(283) = 1.55609186296142373233316514603 max_{chi odd}|L'/L(1,chi)|(283) = 1.49992248252889901008064735773 max_{chi}|L'/L(1,chi)|(283) = 1.55609186296142373233316514603 max_{chi even}|L'/L(1,chi)|(293) = 1.41713157874130820443673390762 max_{chi odd}|L'/L(1,chi)|(293) = 1.58515317244284064528356780036 max_{chi}|L'/L(1,chi)|(293) = 1.58515317244284064528356780036 Precomputation time: 0 min, 27 sec, 120 millisec Final step max computation time (output time included): 0 min, 0 sec, 694 millisec Total elapsed time: 0 min, 27 sec, 814 millisec ****** END PROGRAM ******** ? for(n=1,10,m=nextprime(n*10^3); init_Max_T_final();maxT(m,m,30)) ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[1009,1009] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 1009 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(1009) = 1.66163071579662920308405512837 max_{chi odd}|L'/L(1,chi)|(1009) = 1.91905070366722993140044134955 max_{chi}|L'/L(1,chi)|(1009) = 1.91905070366722993140044134955 Precomputation time: 0 min, 3 sec, 313 millisec Final step max computation time (output time included): 0 min, 0 sec, 417 millisec Total elapsed time: 0 min, 3 sec, 730 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[2003,2003] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 2003 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(2003) = 1.87590052118752805867444411545 max_{chi odd}|L'/L(1,chi)|(2003) = 1.93559000482102812476860401386 max_{chi}|L'/L(1,chi)|(2003) = 1.93559000482102812476860401386 Precomputation time: 0 min, 6 sec, 576 millisec Final step max computation time (output time included): 0 min, 1 sec, 648 millisec Total elapsed time: 0 min, 8 sec, 224 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[3001,3001] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 3001 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(3001) = 1.85170875270322364850217009333 max_{chi odd}|L'/L(1,chi)|(3001) = 2.02047653064549796546951465975 max_{chi}|L'/L(1,chi)|(3001) = 2.02047653064549796546951465975 Precomputation time: 0 min, 9 sec, 861 millisec Final step max computation time (output time included): 0 min, 3 sec, 669 millisec Total elapsed time: 0 min, 13 sec, 530 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[4001,4001] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 4001 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(4001) = 1.92493112516329229196301756391 max_{chi odd}|L'/L(1,chi)|(4001) = 1.98111048196548026386404591492 max_{chi}|L'/L(1,chi)|(4001) = 1.98111048196548026386404591492 Precomputation time: 0 min, 13 sec, 197 millisec Final step max computation time (output time included): 0 min, 6 sec, 590 millisec Total elapsed time: 0 min, 19 sec, 787 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[5003,5003] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 5003 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(5003) = 2.16914336482541724701723239482 max_{chi odd}|L'/L(1,chi)|(5003) = 2.09085309222589149130565282238 max_{chi}|L'/L(1,chi)|(5003) = 2.16914336482541724701723239482 Precomputation time: 0 min, 16 sec, 520 millisec Final step max computation time (output time included): 0 min, 10 sec, 285 millisec Total elapsed time: 0 min, 26 sec, 805 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[6007,6007] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 6007 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(6007) = 2.19761938716235219714332977972 max_{chi odd}|L'/L(1,chi)|(6007) = 2.27649543589754496157793572989 max_{chi}|L'/L(1,chi)|(6007) = 2.27649543589754496157793572989 Precomputation time: 0 min, 20 sec, 4 millisec Final step max computation time (output time included): 0 min, 14 sec, 786 millisec Total elapsed time: 0 min, 34 sec, 790 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[7001,7001] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 7001 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(7001) = 2.16381618373619426178026988823 max_{chi odd}|L'/L(1,chi)|(7001) = 2.08777770385597106873908214147 max_{chi}|L'/L(1,chi)|(7001) = 2.16381618373619426178026988823 Precomputation time: 0 min, 23 sec, 64 millisec Final step max computation time (output time included): 0 min, 20 sec, 146 millisec Total elapsed time: 0 min, 43 sec, 210 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[8009,8009] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 8009 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(8009) = 2.08775398337120103585896611130 max_{chi odd}|L'/L(1,chi)|(8009) = 2.15077860002341223830865273302 max_{chi}|L'/L(1,chi)|(8009) = 2.15077860002341223830865273302 Precomputation time: 0 min, 26 sec, 425 millisec Final step max computation time (output time included): 0 min, 26 sec, 280 millisec Total elapsed time: 0 min, 52 sec, 705 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[9001,9001] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 9001 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(9001) = 2.16225139707852719114127441722 max_{chi odd}|L'/L(1,chi)|(9001) = 1.99923255049969643889532283355 max_{chi}|L'/L(1,chi)|(9001) = 2.16225139707852719114127441722 Precomputation time: 0 min, 29 sec, 715 millisec Final step max computation time (output time included): 0 min, 33 sec, 761 millisec Total elapsed time: 1 min, 3 sec, 476 millisec ****** END PROGRAM ******** ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[10007,10007] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 10007 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(10007) = 2.02551416039253687835130081402 max_{chi odd}|L'/L(1,chi)|(10007) = 2.17148657398748131199086566182 max_{chi}|L'/L(1,chi)|(10007) = 2.17148657398748131199086566182 Precomputation time: 0 min, 33 sec, 509 millisec Final step max computation time (output time included): 0 min, 41 sec, 83 millisec Total elapsed time: 1 min, 14 sec, 592 millisec ****** END PROGRAM ******** ****************** larger q's ********************** ? init_Max_T_final();maxT(20011,20011,30) ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[20011,20011] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 20011 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(20011) = 2.28807406721414248609296533589 max_{chi odd}|L'/L(1,chi)|(20011) = 2.34670358729048423962840680872 max_{chi}|L'/L(1,chi)|(20011) = 2.34670358729048423962840680872 Precomputation time: 1 min, 6 sec, 492 millisec Final step max computation time (output time included): 2 min, 45 sec, 229 millisec Total elapsed time: 3 min, 51 sec, 721 millisec ****** END PROGRAM ******** ? init_Max_T_final();maxT(30011,30011,30) ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[30011,30011] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 30011 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(30011) = 2.18119895632481240565959034653 max_{chi odd}|L'/L(1,chi)|(30011) = 2.28268986824865518173486616741 max_{chi}|L'/L(1,chi)|(30011) = 2.28268986824865518173486616741 Precomputation time: 1 min, 39 sec, 76 millisec Final step max computation time (output time included): 6 min, 8 sec, 214 millisec Total elapsed time: 7 min, 47 sec, 290 millisec ****** END PROGRAM ******** ? init_Max_T_final();maxT(83089,83089,10) ************ A. LANGUASCO ************* ********* COMPUTATION OF THE MAX |L'/L|********** ******* FOR q IN ONE INTERVAL ******** ********* with the T-function ********** Interval=[83089,83089] Performing precomputations of T(a/q), psi(a/q) for this interval Precomputation for q = 83089 Starting computation of max_q max_{chi even}|L'/L(1,chi)|(83089) = 2.378136547 max_{chi odd}|L'/L(1,chi)|(83089) = 2.378137472 max_{chi}|L'/L(1,chi)|(83089) = 2.378137472 Precomputation time: 2 min, 10 sec, 212 millisec Final step max computation time (output time included): 36 min, 40 sec, 257 millisec Total elapsed time: 38 min, 50 sec, 469 millisec ****** END PROGRAM ******** *******/