/*
Copyright (C) 2024 Alessandro Languasco
*/

/**********
****** COMPUTATION OF OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
******  and gamma_S in Moree EK-Artin paper (two different ways) ******
******  and gammaK7, gammaN12, gammaSerre in L.-Moree Euler's constants in AP ******
**********/

\\ Global variables
global(CHI_VECTOR, NUM_PRIME_FACTORS_D,P_PRIMES, VALCHI_CONJ_A_MATRIX, VALCHI_P_MATRIX);
global(DFACTORS, TWOPII, S5VECTOR, S2VECTOR, GAMMA1VECTOR, LOG_DFACTORS, LOG_P, LOG_P_OVER_P_TO_S, LOG_DER_AT_1, SJCHI_VECTOR);
global(SQUARE_FREE_UP_TO_J, NUMPRIMES_UP_TO_P, ONE_OVER_P_MINUS_ONE);
global(ONE_OVER_P_D_MINUS_ONE, ORD_CHI_J_MATRIX, CHI_J_MATRIX, ISPRIMEDIVISOR_OF_D, NUMPRIMES_UP_TO_D, PRIMES_UP_TO_D);

\\ Routine to compute the value of L'/L(s,chi) when chi is a Dirichlet character (also imprimitive)

{L_derlog(L, chi, G, d, s) = local(conductor, Gstar, chistar, corr_chi_imprim, val_chistar_d1,
ud1, L_value, L_prime_value, L_derlog_value, d1);

L_value = lfun(L, s); \\ value of L at s for the attached primitive character		
L_prime_value = lfun(L, s, 1); \\ value of L'at s for the attached primitive character		
L_derlog_value = L_prime_value / L_value; \\ value of logarithmic derivatie at s for the attached primitive character	

conductor = zncharconductor(G, chi);
				
if (conductor < d, \\ chi is imprimitive
			\\ computing the correction sum for the logarithmic derivative
			[Gstar,chistar] = znchartoprimitive(G, chi); 
			\\ chistar is the primitive character such that chi is induced by chistar
	 											  
			corr_chi_imprim = 0.;
			for(i=1, NUM_PRIME_FACTORS_D, 
						 d1 = DFACTORS[i];
						 \\print("d1 = ", d1);	
	 					 if(conductor % d1 <> 0, \\ d1 not divides both d and conductor of chi and conductor <> 1
				 				ud1 = chareval(Gstar, chistar, d1); 
	 							val_chistar_d1 = exp(TWOPII * ud1); \\ value of chistar(d1)								
		 						corr_chi_imprim += val_chistar_d1 * LOG_DFACTORS[i] / (d1^s - val_chistar_d1) 
		 						\\corr_chi_imprim += val_chistar_d1 * log(d1) / (d1^s - val_chistar_d1 ) ;
	 							);
	 						);

						L_derlog_value += corr_chi_imprim ;
				);

return(L_derlog_value);
}

\\ Routine to compute the sum p <= P of chi(p) log(p) / (p^s - chi(p))); to get the truncated log derivative L(s, chi)
{chi_sum_fin(P, chi, G, d, s) = local(fin_sum_chi,  val_chi_p, up, p);
 
fin_sum_chi = 0.; \\ computation of the truncated sum over chi(p) log(p) / (p^s - chi(p)


\\ working on (p,d) == 1
\\i = 1 ;
\\forprime (p = 2, d, 
for (i = 1, NUMPRIMES_UP_TO_D,
			\\if ( d % p <> 0,	\\ p does not divide d	
			if ( ISPRIMEDIVISOR_OF_D[i] == 0,
						p = PRIMES_UP_TO_D[i];
						up = chareval(G, chi , p ); 
						val_chi_p = exp(TWOPII * up);
						fin_sum_chi += val_chi_p * LOG_P[i] /(p^s - val_chi_p);	

						);
			\\i+=1;								
		); 		
\\i = NUMPRIMES_UP_TO_D + 1 ;
\\ working on (p,d) == 1
\\forprime (p = d+1, P, 
for (i = NUMPRIMES_UP_TO_D + 1, NUMPRIMES_UP_TO_P,
					p = P_PRIMES[i];
					up = chareval(G, chi , p ); 
					val_chi_p = exp(TWOPII * up);
					fin_sum_chi += val_chi_p * LOG_P[i] /(p^s- val_chi_p);					
		); 		

return(fin_sum_chi);
}


{sum_primes_derlog_AP(d, u, s, prec=19)  = local(dminusone, A, K, P, err1, G, moebius_vector, 
Smallprimes_sum, chi, dcoprimes, toterr, t, real_s, resfile,
L2, elaptimecomp, seconds, minutes, millisec, ua, Sjchi,num_square_free,
Sjkchi, err2, factorp, i, accuracy, phid, J, point,  jtimess, jtimesu,
aux, sum_chars, numdcoprimes, p, chi_to_j, j,  real_u, kjtimess,
L2_derlog_value, L2trunc_derlog, fin_sum_chi_j, results);

if (d > 100, error("d must be <= 100"));
if (d <= 2, error("d must be >= 3"));

if (floor(prec)<> prec, prec = floor(prec);
						print("prec forced to be an integer");
						);
if (prec < 5, error("prec must be an integer >= 5"));

default(realprecision, prec+10);        \\ set working precision;  

dminusone = d-1;
phid = eulerphi(d);


resfile = fileopen("sum_primes_derlog_AP-results.csv", "a"); 
\\filewrite(resfile,"d; a; u; s ; sum_primes_derlog_AP; accuracy"); 



print("****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******");
print("****** WITH A PRECISION OF AT LEAST ", prec," DECIMAL DIGITS *******");
print("d = ", d);
print("u = ", u);
print("s = ", s);
\\ settings of the main parameters for a precision of about 100 decimal digits 

gettime();  

real_s = real(s);
real_u = real(u);
if (real_s <= real_u + 1, error ("Re(s) must be > Re(u) + 1"));

t = imag(s);

A = 3200;  


P = A*d; \\ is P in the file
if (P > 9600, P = 9600);

accuracy = 10^(-prec-10);

K = 2;
err1 =  P^(1+real_u)*log(P) / real_s * 1.0/(P^(K*real_s)*(P^real_s-1.0)) 	;
until (err1 < accuracy/2, 
		K += 1;
		err1 =  P^(1+real_u)*log(P) / real_s * 1.0/(P^(K*real_s)*(P^real_s-1.0)) 	;
		);

\\J = 26;
J = 2; 
err2 = P^(2+real_u)/real_s * 1.0/(P^(real_s*(J+1)) -1.0); 
until (err2 < accuracy/2, 
		J += 1;
		err2 = P^(2+real_u)/real_s * 1.0/(P^(real_s*(J+1)) -1.0); 
		);	


print("-------------------------- ");
default(format,e); 
print ("Accuracy parameters: ");
if (prec == 19 , print("default precision is with prec = 19"));
print("prec = ", prec, "; A = ", A, "; P = ", P, "; K = ", K, "; J = ", J);
print("Accuracy is 10^(-",prec+10,")");
print("err1 = ", err1); 
print("err2 = ", err2);  
toterr = err1+err2;
print("toterr = ", toterr); 

print("-------------------------- ");
default(format,f); 

\\ setting the group
G = znstar(d, 1); 
TWOPII = 2*Pi*I;

\\ PRECOMPUTATIONS 

moebius_vector = vector(J);

moebius_vector[1] = 1;
for (j = 2, J, moebius_vector[j] = moebius(j) * 1.0);

num_square_free = hammingweight(moebius_vector);			
SQUARE_FREE_UP_TO_J = vector(num_square_free);
SQUARE_FREE_UP_TO_J[1] = 1;
i=2;
for (j = 2, J, 
			if (moebius_vector[j] <> 0,
					SQUARE_FREE_UP_TO_J[i] = j;
					i+=1
					);
	);

NUMPRIMES_UP_TO_P = primepi(P);
LOG_P = vector(NUMPRIMES_UP_TO_P);
LOG_P_OVER_P_TO_S = vector(NUMPRIMES_UP_TO_P);
ONE_OVER_P_MINUS_ONE = vector(NUMPRIMES_UP_TO_P);
P_PRIMES = vector(NUMPRIMES_UP_TO_P);

i=1;
forprime ( p=2, P, 
			factorp = p;
			P_PRIMES[i] = factorp;
			LOG_P[i] = log(factorp);
			LOG_P_OVER_P_TO_S[i] = LOG_P[i]/factorp^s;
			ONE_OVER_P_MINUS_ONE[i] = 1.0/(factorp - 1.0);
			i += 1					
	);

\\ generating the list of integers <= d-1 coprime with d

dcoprimes = vector(phid);
i=1;
dcoprimes[1]=1;
for(m=2, dminusone, 
	if (gcd(m,d) == 1, 
		i+=1;
		dcoprimes[i]=m
		)
	);
numdcoprimes = #dcoprimes;	

\\ generating the non-principal Dirichlet characters mod d
\\ and the orders of his powers
CHI_VECTOR = vector(phid);
CHI_J_MATRIX = matrix(phid, J);
i=1;
foreach(dcoprimes, m,
	\\ m = 1 corresponds to the principal character mod d	
		CHI_VECTOR[i]=znconreychar(G,m);
		CHI_J_MATRIX[i,1] = CHI_VECTOR[i];
		\\for ( j = 2, J, 
		foreach(SQUARE_FREE_UP_TO_J,j,
			CHI_J_MATRIX[i,j] =  charpow(G, CHI_J_MATRIX[i,1], j);
			);
		i+=1;
	);

\\ DFACTORS contains the list of prime factors of d
\\ it is needed to handle the L functions associated to imprimitive characters

DFACTORS=factor(d)[,1]; 
NUM_PRIME_FACTORS_D = #DFACTORS;
\\print(DFACTORS);
\\print(NUM_PRIME_FACTORS_D);


LOG_DFACTORS = vector(NUM_PRIME_FACTORS_D);
ONE_OVER_P_D_MINUS_ONE = vector(NUM_PRIME_FACTORS_D);

for(i=1, NUM_PRIME_FACTORS_D, 
			p = DFACTORS[i];
			factorp = p*1.0;
			ONE_OVER_P_D_MINUS_ONE[i] = 1.0/(factorp - 1.0);
			LOG_DFACTORS[i] = log(factorp);
	);
	
\\ ISPRIMEDIVISOR_OF_D contains 1 or 0 depending if p<=d divides d or not	
NUMPRIMES_UP_TO_D = primepi(d);
ISPRIMEDIVISOR_OF_D = vector(NUMPRIMES_UP_TO_D);
PRIMES_UP_TO_D = vector(NUMPRIMES_UP_TO_D);
i = 1 ;
forprime (p = 2, d, 
			PRIMES_UP_TO_D[i] = p;
			if ( d % p == 0,
					ISPRIMEDIVISOR_OF_D[i] = 1 
					);
			i+=1
			);

\\ S5VECTOR stores the value of the sum over each non trivial character
S5VECTOR = vector(numdcoprimes); \\ initialised with zeros
SJCHI_VECTOR = vector(phid);

for(m=1, phid, \\ we run over the characters
		chi = CHI_VECTOR[m];
		Sjchi = 0;    
		\\for (j=1, J ,
		foreach(SQUARE_FREE_UP_TO_J,j,
				chi_to_j = CHI_J_MATRIX[m,j];
				L2 = lfuncreate([G, chi_to_j]);
				Sjkchi = 0;
				jtimess = j*s; 
				jtimesu = j*u;
				kjtimess = jtimess;
				for(k =  1 , K,	
						point =  kjtimess - jtimesu;
						
						fin_sum_chi_j = chi_sum_fin(P, chi_to_j, G, d, point);
						L2_derlog_value = L_derlog(L2, chi_to_j, G, d, point);
						L2trunc_derlog = L2_derlog_value + fin_sum_chi_j;	
						
						Sjkchi += L2trunc_derlog;
						kjtimess += jtimess;
					
					);
			 Sjchi += moebius_vector[j]* Sjkchi;
			);
		SJCHI_VECTOR[m] = Sjchi;
	);
	
VALCHI_CONJ_A_MATRIX = matrix(numdcoprimes,phid);

i=0;
foreach(dcoprimes, a,
	i += 1 ;
	sum_chars = 0; 
	for(m=1, phid, \\ we run over the characters
		chi = CHI_VECTOR[m];
		ua = chareval(G, chi , a ); 
		
		VALCHI_CONJ_A_MATRIX[i,m] = exp(-TWOPII * ua); \\ conjugate character
		sum_chars += VALCHI_CONJ_A_MATRIX[i,m]* SJCHI_VECTOR[m];
	);
	S5VECTOR[i] = -sum_chars/phid;
);	


S2VECTOR = vector(numdcoprimes); 

i=0;
foreach(dcoprimes, a,
	i += 1 ;
	Smallprimes_sum = 0;
	j = 0;
	forprimestep(p = 2, P, Mod( a, d),
			j+=1;
			Smallprimes_sum += p^u * log(p)/( p^s-1) ;	
		    );
	S2VECTOR[i] = Smallprimes_sum;
	);

	    

print("          RESULTS          ");
elaptimecomp=gettime();

print("-------------------------- ");
i=0;
results = matrix(numdcoprimes,2);
if (u == 0, print("d;a;s"), print("d;a;u;s"));
foreach(dcoprimes, a,
	i += 1 ;
	aux = S5VECTOR[i] + S2VECTOR[i] ;
	if (t == 0, aux = real(aux));
	results[i,1] = aux;
	results[i,2] = toterr;

	print("sum_primes_AP (", d , ";" , a , ";" , u , ";" , s ,") = ", aux);
	filewrite1(resfile,d); 
	filewrite1(resfile,";");
	filewrite1(resfile,a); 
	filewrite1(resfile,";");	
	filewrite1(resfile,s); 
	filewrite1(resfile,";");	
	filewrite1(resfile,aux); 
	filewrite1(resfile,";");	
	default(format,e); 
	default(realprecision,5);  
	filewrite(resfile,toterr); 
	default(format,f); 
	default(realprecision, prec+10);   
	);
	


print("-------------------------- ");

\\fileclose(resfile);


print("-------------------------- ");
print("******  COMPUTATION TIME   ********");
seconds=floor(elaptimecomp/1000)%60;
minutes=floor(elaptimecomp/60000);
millisec=elaptimecomp - minutes*60000 - seconds*1000; 
print("Computation time: ", minutes, " min, ", seconds, " sec, ", millisec, " millisec");

print("******  END PROGRAM   ********");
return(results);
}
		
 
		
{gammaK7() = local(ret_val, toterr, prec, class2, class3, class4, class5, 
results, results_pow3, errors_pow3, err_class2, err_class3, err_class4, 
err_class5, gamma71, gamma76);

prec = 100;
default(realprecision, prec+10);        \\ set working precision;  

results = matrix(6,2);
results_pow3 = vector(6);
errors_pow3 = vector(6); 
results = sum_primes_derlog_AP(7,0,3,prec); \\ second parameter = 0 : is (-1)*(log der at 3)
results_pow3 = results[,1];
errors_pow3 = results[,2]; 

class2 = results_pow3[2];
class3 = results_pow3[3];
class4 = results_pow3[4];
class5 = results_pow3[5]; 
err_class2 = errors_pow3[2];
err_class3 = errors_pow3[3];
err_class4 = errors_pow3[4];
err_class5 = errors_pow3[5]; 

gamma71 = 0.52481504069249735626252234114953328990619486140490967834864672236535655116974342670993724978972138602916219370;
gamma76 = 0.39336339395050524247328579386987665754303292490810610600748306027135634682241512916893364947514410015989585045;

ret_val = -log(7)/6 + 3*gamma71 + 3*gamma76 - 3*(class2+class3+class4+class5);
toterr = 3*(err_class2 + err_class3 + err_class4 + err_class5);

print(" ***************************");
print("gammaK7 = ", ret_val);
default(format,e); 
print("toterr = ", toterr);
default(format,f); 
\\ verification
d = 7;
G = znstar(d, 1); 
phid = eulerphi(7);
dminusone = d-1;
dcoprimes = vector(phid);
i=1;
dcoprimes[1]=1;
for(m=2, dminusone, 
	if (gcd(m,d) == 1, 
		i+=1;
		dcoprimes[i]=m
		)
	);

S = 0;
foreach(dcoprimes, m,
	\\ m = 1 corresponds to the principal character mod d	
		if ( m > 1,
		chi = znconreychar(G,m);
		if (zncharisodd(G, chi) == 0,
			L = lfuncreate([G, chi]);
			S+= lfun(L,1,1)/lfun(L,1);
			);
 		);
	);

verif = Euler + S;
print("verification K7 = ", verif);
}

{gammaS_ver() = local(class1,class2,class3,class4,corr_class1,gamma54, gammaS, prec,
results, results_pow2, results_pow3, results_pow4, results_pow5, toterr,
errors_pow2, errors_pow3, errors_pow4, errors_pow5,
err_class1,err_class2,err_class3,err_class4);

\\ using gamma(5,4) from Euler constants for primes in AP
prec = 100;
default(realprecision, prec+10); 
 
results= matrix(4,2);


results_pow4 = vector(4);
errors_pow4 = vector(4);
results = sum_primes_derlog_AP(5,0,4,prec);
results_pow4 = results[,1];
errors_pow4 = results[,2]; 

 
results_pow5 = vector(4);
errors_pow5 = vector(4);
results =  sum_primes_derlog_AP(5,0,5,prec);
results_pow5 = results[,1];
errors_pow5 = results[,2];

results_pow3 = vector(4);
errors_pow3 = vector(4);
results =  sum_primes_derlog_AP(5,0,3,prec);
results_pow3 = results[,1];
errors_pow3 = results[,2];

results_pow2 = vector(4);
errors_pow2 = vector(4);
results = sum_primes_derlog_AP(5,0,2,prec);
results_pow2 = results[,1];
errors_pow2 = results[,2];

class1 = 4*results_pow4[1] - 5*results_pow5[1];
err_class1 = 4*errors_pow4[1] + 5*errors_pow5[1];

corr_class1 = -(4*log(11)/(11^4-1) -5*log(11)/(11^5-1));

class2 = 3*results_pow3[2]-4*results_pow4[2];
err_class2 = 3*errors_pow3[2] + 4*errors_pow4[2];

class3 = 3*results_pow3[3]-4*results_pow4[3];
err_class3 = 3*errors_pow3[3] + 4*errors_pow4[3];

B1 = class1 + corr_class1 + class2 + class3;

class4 = -2*results_pow2[4];
err_class4 = 2*errors_pow2[4];

B2  = - class4; 


gamma54 = 0.75093255582908325836434166257848050997541691176698326316531897994046526250473123739850379507613130955547404848;

toterr = err_class1+ err_class2+ err_class3 + err_class4; 
gammaS = Euler - gamma54 + class1 + corr_class1 + class2 + class3 +class4;
\\ gammaS = Euler - gamma54 + B1 - B2;

print(" ***************************");
print("gammaS verification using gamma(5,4) =", gammaS);
print("B1 = ", B1);
print("B2 = ", B2);
print("B = ", B1-B2);
\\print("gammaS verification using gamma(5,4) =", Euler - gamma54 + B1 - B2);
default(format,e); 
print("toterr = ", toterr);
}



{gammaS_ver_alt() = local(class1,class2,class3,class4,corr_class1,prec,
gammaK1, gammaK2, several_gammas, corr_summand, gammaS, Aplusminusone, Aplusminustwo,
results, results_pow2, results_pow3, results_pow4, results_pow5, toterr,
errors_pow2, errors_pow3, errors_pow4, errors_pow5, err_Aplusminusone, err_Aplusminustwo,
err_class1,err_class2,err_class3,err_class4);
\\ using EK for cyclotomic zeta_5 and Q(sqrt(5))

prec = 100;
default(realprecision, prec+10); 

\\ gammaK_2(5) = 1.404895141617037748597559079759776077796023288614892553451082518014620888343781526587951824425696490;
\\ EK(5) = 1.720624212513404761695728788649107298352195374910267779571623964233579926129987185729663334641543216;

\\---------
\\ output of  gamma_K_div(5,5,110)
\\ into gammaK-divisors.gp for CLM
\\ EK(5) = 1.7206242125134047616957287886491072983521953749102677795716239642335799261299871857296633346415432163620997662;
\\ gammaK_2(5) = 1.4048951416170377485975590797597760777960232886148925534510825180146208883437815265879518244256964898942433246;
\\ ------------------------------------

gammaK2 = 1.4048951416170377485975590797597760777960232886148925534510825180146208883437815265879518244256964898942433246;
gammaK1 = 1.7206242125134047616957287886491072983521953749102677795716239642335799261299871857296633346415432163620997662;


results= matrix(4,2);

results_pow4 = vector(4);
errors_pow4 = vector(4);
results = sum_primes_derlog_AP(5,0,4,prec);
results_pow4 = results[,1];
errors_pow4 = results[,2]; 

results_pow5 = vector(4);
errors_pow5 = vector(4);
results = sum_primes_derlog_AP(5,0,5,prec);
results_pow5 = results[,1];
errors_pow5 = results[,2];

results_pow2 = vector(4);
errors_pow2 = vector(4);
results = sum_primes_derlog_AP(5,0,2,prec);
results_pow2 = results[,1];
errors_pow2 = results[,2]; 

results_pow3 = vector(4);
errors_pow3 = vector(4);
results = sum_primes_derlog_AP(5,0,3,prec);
results_pow3 = results[,1];
errors_pow3 = results[,2]; 
 

class1 = - ( 16*results_pow4[1] - 20*results_pow5[1] );
err_class1 = 16*errors_pow4[1]+20*errors_pow5[1];

corr_class1 = 16*log(11)/(11^4-1) -20*log(11)/(11^5-1);

class4 = 4*results_pow2[4];
err_class4 = 4* errors_pow2[4];

Aplusminusone = class4 + class1 + corr_class1;
err_Aplusminusone = err_class1+ err_class4;

class2 = 4*results_pow2[2] - 12*results_pow3[2] + 12*results_pow4[2];
err_class2 = 4*errors_pow2[2] +12*errors_pow3[2]+ 12*errors_pow4[2];

class3 = 4*results_pow2[3] - 12*results_pow3[3] + 12*results_pow4[3];
err_class3 = 4*errors_pow2[3] +12*errors_pow3[3]+ 12*errors_pow4[3];

Aplusminustwo = class2 + class3;
err_Aplusminustwo = err_class2 + err_class3;

print(" ***************************");
print("gammaK1 = ", gammaK1);
print("gammaK2 = ", gammaK2);
print("A+_1 = ", Aplusminusone);

print("A+_2 = ", Aplusminustwo);


several_gammas = 4 * Euler + gammaK1 - 2*gammaK2;
corr_summand =  - log(5) / 4;
gammaS = 1/4* (several_gammas + corr_summand - Aplusminusone - Aplusminustwo )  ;
print("gammaS verification using gammaK1 and gammaK2 =", gammaS);


toterr = err_Aplusminusone + err_Aplusminustwo;
default(format,e); 
print("error(A+_1) = ", err_Aplusminusone);
print("error(A+_2) = ", err_Aplusminustwo);
print("toterr = ", toterr);
}


{gammaN12() = local(ret_val, toterr, prec, class2, class3, class4, class5, 
results, results_pow3, errors_pow3, err_class2, err_class3, err_class4, 
err_class5, gamma71, gamma76);

prec = 100;
default(realprecision, prec+10);        \\ set working precision;  

q=12;
phiq=eulerphi(12);
qminusone = q-1;

qcoprimes = vector(phiq);
i=1;
qcoprimes[1]=1;
for(j=2, qminusone, 
	if (gcd(j,q) == 1, 
		i+=1;
		qcoprimes[i]=j
		)
	);
numqcoprimes = i;

results = matrix(numqcoprimes,2);
results_pow2 = vector(numqcoprimes);
errors_pow2 = vector(numqcoprimes); 
results = sum_primes_derlog_AP(12,0,2,prec); \\  \\ second parameter = 0 : is (-1)*(log der at 2)
results_pow2 = results[,1];
errors_pow2 = results[,2]; 


index5 = vecsearch(qcoprimes, 5);
index7 = vecsearch(qcoprimes, 7);
index11 = vecsearch(qcoprimes, 11);

class5 = results_pow2[index5];
class7 = results_pow2[index7];
class11 = results_pow2[index11];

err_class5 = errors_pow2[index5];
err_class7 = errors_pow2[index7];
err_class11 = errors_pow2[index11];

gamma121 = 0.780445251680074822178150748784085025947705319122330857926994720340219283582881987696103318559254512929001484176846749995;
err_gamma121 = 2.16790502288384663677784640770548835333128443989810823434226935388126414130531209337224428019372970302237583362578694653/10^(124);
ret_val = -2*log(2)/3 - log(3)/4 + gamma121 - 2*(class5+class7+class11);
toterr = 2*(err_class5 + err_class7 + err_class11)+err_gamma121;

print(" ***************************");
print("gammaN12 = ", ret_val);
beta_0_prime = 0.302316142357065637947769900480199715602412795189369645458867841288865448752410510899487467813979272708567765913272591067;
beta_1_prime = 0.75*(1-ret_val)*beta_0_prime;
print("beta_0_prime = ", beta_0_prime);
print("beta_1_prime = ", beta_1_prime);

gammaSerre = gamma121 - 2*(class5+class7+class11);
print("gammaSerre = ", gammaSerre);
beta_0 = 2/3*beta_0_prime;
print("beta_0 = ", beta_0);
print("beta_1 = ", 3/4 * beta_0 *(1- gammaSerre));

default(format,e); 
print("toterr = ", toterr);
default(format,f); 
}
						
/************


Last login: Sun May  5 13:43:51 on ttys007
languasc@zygalski ~ % cd /Users/languasc/Documents/LANGUASCO/matematica/lavori/\[80\]EK_for_AP/programs
languasc@zygalski programs % gp2c-run -pmy_ -g -W sumprimes_derlog_AP.gp 
Warning:sumprimes_derlog_AP.gp:269: variable undeclared
kjtimess
Warning:sumprimes_derlog_AP.gp:346: variable undeclared
e
Warning:sumprimes_derlog_AP.gp:349: variable undeclared
f
Reading GPRC: /Users/languasc/.gprc
GPRC Done.

                  GP/PARI CALCULATOR Version 2.15.5 (released)
          i386 running darwin (x86-64/GMP-6.3.0 kernel) 64-bit version
     compiled: Feb 24 2024, Apple clang version 15.0.0 (clang-1500.1.0.2.5)
                           threading engine: pthread
                 (readline v8.0 enabled, extended help enabled)

                     Copyright (C) 2000-2022 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 ?18 for how to get moral (and possibly technical) support.

parisizemax = 2048000000, primelimit = 500000, nbthreads = 12
? quit
Goodbye!
languasc@zygalski programs % gp2c-run -pmy_ -g -W sumprimes_derlog_AP.gp
Warning:sumprimes_derlog_AP.gp:346: variable undeclared
e
Warning:sumprimes_derlog_AP.gp:349: variable undeclared
f
Reading GPRC: /Users/languasc/.gprc
GPRC Done.

                  GP/PARI CALCULATOR Version 2.15.5 (released)
          i386 running darwin (x86-64/GMP-6.3.0 kernel) 64-bit version
     compiled: Feb 24 2024, Apple clang version 15.0.0 (clang-1500.1.0.2.5)
                           threading engine: pthread
                 (readline v8.0 enabled, extended help enabled)

                     Copyright (C) 2000-2022 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 ?18 for how to get moral (and possibly technical) support.

parisizemax = 2048000000, primelimit = 500000, nbthreads = 12
? sum_primes_derlog_AP(7,0,3,)
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 19 DECIMAL DIGITS *******
d = 7
u = 0
s = 3
-------------------------- 
Accuracy parameters: 
default precision is with prec = 19
prec = 19; A = 3200; P = 9600; K = 3; J = 3
Accuracy is 10^(-29)
err1 = 4.7889654860024839329812159700 E-44
err2 = 30720000/612709757329767363772415999999999999999999999999
toterr = 5.0185821412108255536937185385 E-41
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
derlog_zeta_AP(7;1;3) = -0.00020567431791426310111498171778
derlog_zeta_AP(7;2;3) = -0.099367711755813983198745313475
derlog_zeta_AP(7;3;3) = -0.042986047314514441959665780476
derlog_zeta_AP(7;4;3) = -0.0018534757509890860207543913306
derlog_zeta_AP(7;5;3) = -0.013479595613151653035071321294
derlog_zeta_AP(7;6;3) = -0.0012403816484221365404242008239
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 0 sec, 369 millisec
******  END PROGRAM   ********
?  sum_primes_derlog_AP(7,1,3,)
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 19 DECIMAL DIGITS *******
d = 7
u = 1
s = 3
-------------------------- 
Accuracy parameters: 
default precision is with prec = 19
prec = 19; A = 3200; P = 9600; K = 3; J = 3
Accuracy is 10^(-29)
err1 = 0.00000000000000000000000000000000000000045974068665623845756619673312
err2 = 294912000000/612709757329767363772415999999999999999999999999
toterr = 0.00000000000000000000000000000000000048178388555623925315459697970
-------------------------- 
          RESULTS          
-------------------------- 
d;a;u;s
sum_primes_AP (7;1;1;3) = 0.0085032416224231132294467959059
sum_primes_AP (7;2;1;3) = 0.20898683218785116439598278380
sum_primes_AP (7;3;1;3) = 0.14380640141595451869969236973
sum_primes_AP (7;4;1;3) = 0.023993380400057840149448058019
sum_primes_AP (7;5;1;3) = 0.078172800998601817362139917525
sum_primes_AP (7;6;1;3) = 0.019911283758551218432277428764
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 0 sec, 373 millisec
******  END PROGRAM   ********

*****************/
/***************
***********************************
VERIFICATION OF GAMMAS IN EK ARTIN USING GAMMA(5,1)
***********************************
?  gammaS_ver()
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 5
u = 0
s = 4
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 7; J = 7
Accuracy is 10^(-110)
err1 = 8.1260279111443532330346180829253600286325719272077762839910814821533213906367032748488894442699144823714365109 E-124
err2 = 8.5075207590815179272915716875662969080371319309509562294492523296297024080192246873541213908099341647230005090 E-121
toterr = 8.5156467869926622805246063056492222680657645028781640057332434111118557294098613906289702802542040792053719456 E-121
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (5;1;0;4) = 0.00016937744512014340027299710018830829049718273029339740204838688912281274822981602505484020111005951468414481948
sum_primes_AP (5;2;0;4) = 0.047057601122699647749854505722669374934563060794458749578353694071174342557861678260013654220399137791016238163
sum_primes_AP (5;3;0;4) = 0.013835611380580619930730165229688521332918906270073057800504061065825622956800618808102053765486298349562074180
sum_primes_AP (5;4;0;4) = 0.000027947583198118661173124089973726324226256971766183101908880378389209862689524494131759148923050982453424553252
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 4 sec, 101 millisec
******  END PROGRAM   ********
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 5
u = 0
s = 5
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 5; J = 5
Accuracy is 10^(-110)
err1 = 5.9911578583285080463408734716020367272840462603595271327411568767030239019600202745365969363006290262196070872 E-116
err2 = 6.2724249052556215374335299738088793843576166300515210088483447575893869913844139774924466190163482609669738153 E-113
toterr = 6.2784160631139500454798708472804814210849006763118805359810859144660900152863739977669832159526488899931934224 E-113
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (5;1;0;5) = 0.000015049081176738061596653764329510041753741550817654580693284874052200517861791370668558161705005447668046490488
sum_primes_AP (5;2;0;5) = 0.022477441751559857824951820476004112559067449178380133495900105689365031435018986024274954632136798347236769947
sum_primes_AP (5;3;0;5) = 0.0045471545870790090501973153875495982043052977761047854522880199281552068540878866139248340797375575194225944296
sum_primes_AP (5;4;0;5) = 0.0000013617805387771396831810339138828782990949162828203195308948812848407387680985084631032666957653616830087000289
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 2 sec, 113 millisec
******  END PROGRAM   ********
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 5
u = 0
s = 3
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 9; J = 9
Accuracy is 10^(-110)
err1 = 9.9852630972254662269889286902502330355295517539778642811169135029493910666633841618523299689193912398663687648 E-116
err2 = 1.0454041508759369229055883289681465640596027716752535014747241262648978318974023295820744365027247101611623026 E-112
toterr = 1.0464026771856594695282872218371715873631557268506512879028358176151927710040686679982596694996166492851489394 E-112
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (5;1;0;3) = 0.0020148854399831909866643013375462185093878175199592322633512965939733517461388501320530393450370409317274747763
sum_primes_AP (5;2;0;3) = 0.10542878535395337765344393977570609692827550316677732622227310419056760486930046740769648001881775731636496776
sum_primes_AP (5;3;0;3) = 0.043786170300197757462243565590019523317692403463141200591968999700026334897767196230328267369588310320635363277
sum_primes_AP (5;4;0;3) = 0.00061350306064210461155481927986300336825384535100520758517896992499241845150263035873320786481691499238561739347
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 5 sec, 825 millisec
******  END PROGRAM   ********
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 5
u = 0
s = 2
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 14; J = 14
Accuracy is 10^(-110)
err1 = 1.4977894808341830102016472711140385263282757148880811790807887067726631630287999288935538520656095259305321424 E-115
err2 = 1.5681062263139053843583824934522198460894041575128802522120861893973467478461034943731116547540870652417434538 E-112
toterr = 1.5696040157947395673685841407233338846157324332277683333911669781041194110091322943020052086061526747676739860 E-112
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (5;1;0;2) = 0.030187066190421240908509990749329047308514417895397393076146150336544128109883384981445802138883643371197373363
sum_primes_AP (5;2;0;2) = 0.28987367264898584871977847497133447370980254955516658164332526651036924267289945153101562331782432684212701867
sum_primes_AP (5;3;0;2) = 0.16595552485655846060048198979828232691219101715826611627052066109031205903575510126690548676338286001377855394
sum_primes_AP (5;4;0;2) = 0.016884816380479740562728932283026336159407573103579316821844953689267723060383595482245052952476607691447826493
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 14 sec, 604 millisec
******  END PROGRAM   ********
 ***************************
gammaS verification using gamma(5,4) =-0.0033929593299054971161341577386037819013398922685265956872251123824941717374782236692483746253425966981312270299
B1 = 0.20409356435860438176715327932352696935073282976569170231601654005163881011035553375679846306399517872560702719
B2 = 0.033769632760959481125457864566052672318815146207158633643689907378535446120767190964490105904953215382895652985
B = 0.17032393159764490064169541475747429703191768355853306867232663267310336398958834279230835715904196334271137420
toterr = 1.2556832228448172418558774402405213671743516782504558696727859887931361389028132348395396729733973053293287270 E-111? ##
  ***   last result: cpu time 26,324 ms, real time 7,183 ms.
******************/
/****************************

***********************************
VERIFICATION OF GAMMAS IN EK ARTIN USING GAMMAK5 and GAMMAK5plus
***********************************


? gammaS_ver_alt()
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 5
u = 0
s = 4
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 7; J = 7
Accuracy is 10^(-110)
err1 = 8.1260279111443532330346180829253600286325719272077762839910814821533213906367032748488894442699144823714365109 E-124
err2 = 8.5075207590815179272915716875662969080371319309509562294492523296297024080192246873541213908099341647230005090 E-121
toterr = 8.5156467869926622805246063056492222680657645028781640057332434111118557294098613906289702802542040792053719456 E-121
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (5;1;0;4) = 0.00016937744512014340027299710018830829049718273029339740204838688912281274822981602505484020111005951468414481948
sum_primes_AP (5;2;0;4) = 0.047057601122699647749854505722669374934563060794458749578353694071174342557861678260013654220399137791016238163
sum_primes_AP (5;3;0;4) = 0.013835611380580619930730165229688521332918906270073057800504061065825622956800618808102053765486298349562074180
sum_primes_AP (5;4;0;4) = 0.000027947583198118661173124089973726324226256971766183101908880378389209862689524494131759148923050982453424553252
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 4 sec, 117 millisec
******  END PROGRAM   ********
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 5
u = 0
s = 5
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 5; J = 5
Accuracy is 10^(-110)
err1 = 5.9911578583285080463408734716020367272840462603595271327411568767030239019600202745365969363006290262196070872 E-116
err2 = 6.2724249052556215374335299738088793843576166300515210088483447575893869913844139774924466190163482609669738153 E-113
toterr = 6.2784160631139500454798708472804814210849006763118805359810859144660900152863739977669832159526488899931934224 E-113
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (5;1;0;5) = 0.000015049081176738061596653764329510041753741550817654580693284874052200517861791370668558161705005447668046490488
sum_primes_AP (5;2;0;5) = 0.022477441751559857824951820476004112559067449178380133495900105689365031435018986024274954632136798347236769947
sum_primes_AP (5;3;0;5) = 0.0045471545870790090501973153875495982043052977761047854522880199281552068540878866139248340797375575194225944296
sum_primes_AP (5;4;0;5) = 0.0000013617805387771396831810339138828782990949162828203195308948812848407387680985084631032666957653616830087000289
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 2 sec, 57 millisec
******  END PROGRAM   ********
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 5
u = 0
s = 2
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 14; J = 14
Accuracy is 10^(-110)
err1 = 1.4977894808341830102016472711140385263282757148880811790807887067726631630287999288935538520656095259305321424 E-115
err2 = 1.5681062263139053843583824934522198460894041575128802522120861893973467478461034943731116547540870652417434538 E-112
toterr = 1.5696040157947395673685841407233338846157324332277683333911669781041194110091322943020052086061526747676739860 E-112
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (5;1;0;2) = 0.030187066190421240908509990749329047308514417895397393076146150336544128109883384981445802138883643371197373363
sum_primes_AP (5;2;0;2) = 0.28987367264898584871977847497133447370980254955516658164332526651036924267289945153101562331782432684212701867
sum_primes_AP (5;3;0;2) = 0.16595552485655846060048198979828232691219101715826611627052066109031205903575510126690548676338286001377855394
sum_primes_AP (5;4;0;2) = 0.016884816380479740562728932283026336159407573103579316821844953689267723060383595482245052952476607691447826493
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 14 sec, 728 millisec
******  END PROGRAM   ********
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 5
u = 0
s = 3
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 9; J = 9
Accuracy is 10^(-110)
err1 = 9.9852630972254662269889286902502330355295517539778642811169135029493910666633841618523299689193912398663687648 E-116
err2 = 1.0454041508759369229055883289681465640596027716752535014747241262648978318974023295820744365027247101611623026 E-112
toterr = 1.0464026771856594695282872218371715873631557268506512879028358176151927710040686679982596694996166492851489394 E-112
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (5;1;0;3) = 0.0020148854399831909866643013375462185093878175199592322633512965939733517461388501320530393450370409317274747763
sum_primes_AP (5;2;0;3) = 0.10542878535395337765344393977570609692827550316677732622227310419056760486930046740769648001881775731636496776
sum_primes_AP (5;3;0;3) = 0.043786170300197757462243565590019523317692403463141200591968999700026334897767196230328267369588310320635363277
sum_primes_AP (5;4;0;3) = 0.00061350306064210461155481927986300336825384535100520758517896992499241845150263035873320786481691499238561739347
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 6 sec, 272 millisec
******  END PROGRAM   ********
 ***************************
gammaK1 = 1.7206242125134047616957287886491072983521953749102677795716239642335799261299871857296633346415432163620997662
gammaK2 = 1.4048951416170377485975590797597760777960232886148925534510825180146208883437815265879518244256964898942433246
A+_1 = 0.067453075884830773681197940988978569906602379878063861732494819045643480770327457468232000440631549258114172189
A+_2 = 0.76345587221172682805980784611805451474614299206909015843077152535959751580575381235277596749458116946655806609
gammaS verification using gammaK1 and gammaK2 =-0.0033929593299054971161341577386037819013398922685265956872251123824941717374782236692483746253425966981312270096
error(A+_1) = 1.8835248325657206952316674745847999271020287374587066451577463833080552252398968843299769516393432963124348056 E-111
error(A+_2) = 3.7670496583189266695451461182469340509222931343816125645873331585195423658855243892210947111935307707085864335 E-111
toterr = 5.6505744908846473647768135928317339780243218718403192097450795418275975911254212735510716628328740670210212391 E-111

****************************/

/****************************
***********************************
VERIFICATION OF GAMMA7 IN EK for primes in AP
***********************************
? gammaK7()
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 7
u = 0
s = 3
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 9; J = 9
Accuracy is 10^(-110)
err1 = 9.9852630972254662269889286902502330355295517539778642811169135029493910666633841618523299689193912398663687648 E-116
err2 = 1.0454041508759369229055883289681465640596027716752535014747241262648978318974023295820744365027247101611623026 E-112
toterr = 1.0464026771856594695282872218371715873631557268506512879028358176151927710040686679982596694996166492851489394 E-112
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (7;1;0;3) = 0.00020567431791426310111498171778140625565076116797872805077284349471875474934616568533612911630838700954598157627
sum_primes_AP (7;2;0;3) = 0.099367711755813983198745313474625100312300840664234369320760729392201906536062673389647963026507582803050087504
sum_primes_AP (7;3;0;3) = 0.042986047314514441959665780476210118764530058373057538726128236439324324707236704429044907825456969626565901324
sum_primes_AP (7;4;0;3) = 0.0018534757509890860207543913306021068972543307467095876761482612589128517426384217508571831149929120106549398500
sum_primes_AP (7;5;0;3) = 0.013479595613151653035071321293754872113902662260246198946641582825965489657634694241213107626031205825661619137
sum_primes_AP (7;6;0;3) = 0.0012403816484221365404242008239190638090870537318673125125235372279192469090561506933096130684331454392485710844
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 10 sec, 616 millisec
******  END PROGRAM   ********
 ***************************
gammaK7 = 1.9571564544497147527138218614254566264775388945426607376494545598379949989204133082397134632509851932705328733
toterr = 1.2556832126227913634339446662046059048357868722207815454834029811382313252048824015979116033995399791421787273 E-111
verification K7 = 1.9571564544497147527138218614254566264775388945426607376494545598379949989204133082397134632509851932705328733 + 0.000000000000000000000000000000000000
? ##
  ***   last result: cpu time 10,758 ms, real time 3,004 ms.

? gammaN12()
****** COMPUTATION OF sum_{p = a mod d} (p^u log p)/ (p^s-1));   Re(s)> 1; in AP *******
****** WITH A PRECISION OF AT LEAST 100 DECIMAL DIGITS *******
d = 12
u = 0
s = 2
-------------------------- 
Accuracy parameters: 
prec = 100; A = 3200; P = 9600; K = 14; J = 14
Accuracy is 10^(-110)
err1 = 1.4977894808341830102016472711140385263282757148880811790807887067726631630287999288935538520656095259305321424 E-115
err2 = 1.5681062263139053843583824934522198460894041575128802522120861893973467478461034943731116547540870652417434538 E-112
toterr = 1.5696040157947395673685841407233338846157324332277683333911669781041194110091322943020052086061526747676739860 E-112
-------------------------- 
          RESULTS          
-------------------------- 
d;a;s
sum_primes_AP (12;1;0;2) = 0.022292145528363059037841182710355366098664997270141512426066123893661785420872667763959779988138634998709631577
sum_primes_AP (12;5;0;2) = 0.087883434185488853244264403750269804303221667005861856641827499744106487605715532219429265775027198470761222653
sum_primes_AP (12;7;0;2) = 0.058694498218847279921838460698174768912445075064937804377994587117002327655967724085975133723046616560952633263
sum_primes_AP (12;11;0;2) = 0.032715318891671466299103951092888827316380843618061039938579021480662016041960323324013618923048064725629133622
-------------------------- 
-------------------------- 
******  COMPUTATION TIME   ********
Computation time: 0 min, 6 sec, 135 millisec
******  END PROGRAM   ********
 ***************************
gammaN12 = -0.31489244345226467254589560583466424666192924795124807633726775274829170030885389035171220831736806047734465618
beta_0_prime = 0.30231614235706563794776990048019971560241279518936964545886784128886544875241051089948746781397927270856776591
beta_1_prime = 0.29813490833920854459071499161910339901119112670998131249524270466672035077289255966515508230249495397142786410
gammaSerre = 0.42185874908805962324773711770141822488361014774460945601019250365667762097559482843726728171701075341431550510
beta_0 = 0.20154409490471042529851326698679981040160853012624643030591189419257696583494034059965831187598618180571184394
beta_1 = 0.087390716356593085507282150858840987765244584774866372734975910755936786918849862809983750604929833617230852147
toterr = 9.4176240947706053092343886909767811541021000877199412847868999768590587354086750299533365637302882928862376455 E-112

************************/