/*
Copyright (C) 2021 Alessandro Languasco
*/

/****************   A. LANGUASCO  ***************************/  

\\ Global variables:

  
global(tab);   \\global variable used to initialise sumnum 
global(defaultprecision);

 /************* COMPUTATION OF gamma2    *********/

  
{gamma2(Pbound,defaultprecision)=local(minutes, millisec, seconds, 
elaptimefinalcomp, s1, S, errs1,   Pbound1
);

 
 
\\ 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 gamma for q=2 **********"); 
\\ precision setting

default(realprecision,defaultprecision);   
default(format,f);   
default(primelimit, Pbound);  

Pbound1=Pbound+1; 

tab = sumnuminit(); \\ needed for the sumnum functions in the errors

gettime(); 
 
 
gettime();
print("Starting computation of gamma(1,2)");
print("internal sums over primes p up to ", Pbound);
print("---------");

errs1=0;  
s1=0; 
S=0;  
 
forprime(p=3, Pbound, s1 += log(p)/(p^2-1);  
);
 
S = 2*s1;

\\ error computation
errs1 = sumnum(n=Pbound1, log(n)/(n^2-1), tab);
errS = 2*abs(errs1)   ;  

print("sum over primes = ", s1 );  
 
print("gamma(1,2) = ", -log(2) - S );  
print("gammaprime(1,2) = ", - S );  
print("errS",r,"(",q,") = ", errS); 
if (abs(S) < 10*errS,  print("*********  recompute  !!!")); 
 
 
 
elaptimefinalcomp=gettime(); 

seconds=floor(elaptimefinalcomp/1000)%60;
minutes=floor(elaptimefinalcomp/60000);
millisec=elaptimefinalcomp- minutes*60000 - seconds*1000;
 
print("gammas(2) computation time (output time included): ", minutes, " min, ", seconds, " sec, ", millisec, " millisec");

print("******  END PROGRAM   ********"); 
}
 


 
 
/**** RESULTS ****

 gp2.13.1 and gp2c0.0.12 compiled by myself, see below 
 

languasc@languasc-macmini$ gp2c-run -pmy_ -g -W  gamma(q=2).gp
 
GP/PARI CALCULATOR Version 2.13.1 (released)
i386 running darwin (x86-64/GMP-6.2.1 kernel) 64-bit version
compiled: Jan 25 2021, Apple clang version 12.0.0 (clang-1200.0.32.28)
threading engine: single
(readline v8.0 enabled, extended help enabled)
Copyright (C) 2000-2020 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.

parisize = 8000000, primelimit = 500000

-------
  
? gamma2(10^7,19)
************ A. LANGUASCO *************
********* COMPUTATION of gamma for q=2 **********
Starting computation of gamma(1,2)
internal sums over primes p up to 10000000
---------
sum over primes = 0.3389118329051014684
gamma(1,2) = -1.370970846370148246
gammaprime(1,2) = -0.6778236658102029368
errgammas(2) = 0.000003423618969010723622
gammas(2) computation time (output time included): 0 min, 1 sec, 560 millisec
******  END PROGRAM   ********

? gamma2(10^8,19)
************ A. LANGUASCO *************
********* COMPUTATION of gamma for q=2 **********
Starting computation of gamma(1,2)
internal sums over primes p up to 100000000
---------
sum over primes = 0.3389119229079501492
gamma(1,2) = -1.370971026375845608
gammaprime(1,2) = -0.6778238458159002983
errgammas(2) = 0.0000003884136130369792535
gammas(2) computation time (output time included): 0 min, 13 sec, 828 millisec
******  END PROGRAM   ********


? gamma2(10^9,19)
************ A. LANGUASCO *************
********* COMPUTATION of gamma for q=2 **********
Starting computation of gamma(1,2)
internal sums over primes p up to 1000000000
---------
sum over primes = 0.3389119319072067970
gamma(1,2) = -1.370971044374358903
gammaprime(1,2) = -0.6778238638144135939
errgammas(2) = 0.00000004344653165316955650
gammas(2) computation time (output time included): 2 min, 1 sec, 888 millisec
******  END PROGRAM   ********
  
  


? gamma2(10^10,19)
************ A. LANGUASCO *************
********* COMPUTATION of gamma for q=2 **********

Starting computation of gamma(1,2)
internal sums over primes p up to 10000000000
---------
sum over primes = 0.3389119328016886185
gamma(1,2) = -1.370971046163322547
gammaprime(1,2) = -0.6778238656033772371
errSr(q) = 0.000000004805170185757832859
Sr(q) computation time (output time included): 18 min, 8 sec, 490 millisec
******  END PROGRAM   ********

---------
*******/