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// This file mainly due to Kevin McGown, with modifications by Michael Rubinstein
#ifndef Lexplicit_formula_H
#define Lexplicit_formula_H
inline Complex xxx_phi(Double A, Double x_0, Complex x); // the phi in the explicit formula
inline Complex xxx_phi_hat(Double A, Double x_0, Complex x); //its fourier transform
/***************************************************************************************************/
inline Complex xxx_phi(Double A, Double x_0, Complex x)
{
return exp(-A*(x-x_0)*(x-x_0));
}
inline Complex xxx_phi_hat(Double A, Double x_0, Complex x)
{
return sqrt(Pi/A) * exp(-2*Pi*I*x_0*x - Pi*Pi*x*x/A);
}
/***************************************************************************************************/
template <class ttype>
int L_function <ttype>::
dirichlet_coeffs_log_diff(int num_coeffs, Complex *c)
{
std::vector<Complex> b(num_coeffs+1);
int j, n, d1, ind;
Complex total, total2, temp;
if (what_type_L != 1 && what_type_L != -1
&& num_coeffs > number_of_dirichlet_coefficients)
{
cout << "Don't have enough Dirichlet coefficients." << endl;
return 1;
}
b[1] = 1;
if(my_verbose!=0) cout << "Computing " << num_coeffs << " Dirichlet coefficients of the logarithmic derivative" << endl;
//XXXXXXXX this should be computed just once, not each time test is called
for (n=2;n<=num_coeffs;n++)
{
total = 0;
total2 = 0;
for (j=1;j<=n/2;j++)
if (n % j == 0)
{
d1 = n/j;
if (what_type_L == -1)
temp = b[j];
else if (what_type_L == 1)
{
ind = d1 % period;
if (ind == 0)
ind = period;
temp = dirichlet_coefficient[ind]*b[j];
}
else
temp = dirichlet_coefficient[d1]*b[j];
total -= temp;
total2 += temp*LOG(d1);
}
b[n] = total;
c[n] = total2;
if(my_verbose>5) cout << "c[" << n << "] = " << c[n] << endl;
}
return 0;
}
/************************************************************************************************/
template <class ttype>
int L_function <ttype>::
test_explicit_formula(Double A, Double x_0, Double *zero_table, int number_zeros, Complex *c, int num_coeffs)
{
Double t, t_begin, t_end, t_step, u;
Double D;
Double total;
Double term1, term2, term3;
int p, n, x, j;
Double temp;
Double LHS, RHS;
//Complex *c;
//int num_coeffs;
int flag;
//num_coeffs = 150; //XXXXXXXXXX should depend on test required
//c = new Complex[num_coeffs+1]; //XXXXXXX move to L.h
//dirichlet_coeffs_log_diff(num_coeffs, c);
//compute the possible contribution from poles
term1 = 0;
for (j=1;j<=number_of_poles;j++)
term1 += real(xxx_phi(A, x_0, (pole[j]-0.5)/I));
// compute the contribution from the Gamma factors (integral of the log diff)
t_step = 0.02;
D = ceil(sqrt(DIGITS*log(10.0)/A)/t_step) * t_step;
t_begin = x_0 - D;
t_end = x_0 + D;
total = 0;
for (t=t_begin;t<=t_end;t=t+t_step)
{
temp = 0;
for (j=1;j<=this->a;j++)
{
temp += 2*real(log_GAMMA(gamma[j]/2 + I*t*gamma[j]+lambda[j], 1)*gamma[j]); //1 here calls the logarithmic derivative
//temp += log_GAMMA(gamma[j]/2 - I*t*gamma[j]+conj(lambda[j]), 1)*gamma[j];
}
total = total + real(xxx_phi(A, x_0, t)) * temp;
//cout << t << " total " << total << endl;
}
term2 = t_step*total + 2*log(Q)*sqrt(Pi/A);
term2 = 1/(2*Pi) * term2;
//compute the contribution from the Dirichlet coefficients
x = num_coeffs;
//extend_prime_table(x);
j = 0;
p = get_prime(j);
term3 = 0;
while (p <= x)
{
n = p;
while (n <= x)
{
temp = 2*real(c[n]*xxx_phi_hat(A, x_0, LOG(n)/(2*Pi)));// + conj(c[n])*xxx_phi_hat(A, x_0, -LOG(n)/(2*Pi));
term3 += two_inverse_sqrt(n)*temp;
n = n * p;
//cout << n << "term3 " << term3 << endl;
}
j = j + 1;
p = get_prime(j);
}
term3 = 1/(4*Pi) * term3;
/*** COMPUTE RHS ***/
RHS = term1 + term2 - term3;
/*** COMPUTE LHS ***/
LHS = 0;
//we will want to truncate this automatically
for (j=0;j<=number_zeros-1;j++)
{
u=zero_table[j]-x_0;
u=u*u*A;
if(u<2.3*DIGITS*2) LHS += exp(-u);
//LHS += exp(-A*(zero_table[j]-x_0)*(zero_table[j]-x_0));
}
/*** Display Results ***/
//if (my_verbose > 3)
if (my_verbose==0)
{
cout << endl << endl;
cout << "*** Testing Explicit Formula for L ***" << endl;
cout << "A = " << A << endl;
cout << "x_0 = " << x_0 << endl;
cout << "D = " << D << endl;
cout << "TERM 1: " << term1 << endl;
cout << "TERM 2: " << term2 << endl;
cout << "TERM 3: " << term3 << endl;
cout << "RHS: " << RHS << endl;
cout << "LHS: " << LHS << endl;
cout << "LHS - RHS: " << LHS-RHS << endl;
cout << "CUTOFF: " << pow(10.0, -DIGITS/3) << endl;
}
//cout << "A = " << A << ", x_0 = " << x_0 << ", ";
cout << " x_0=" << x_0 << ",";
cout << "LHS = " << LHS << ", RHS = " << RHS << ", DIFF = " << abs((LHS - RHS)) << ", ";
// I arbitrarily chose DIGITS/3.
if (abs((LHS - RHS)) < pow(10.0,-DIGITS/3))
{
flag = 0;
//cout << "PASS." << endl;
}
else
{
flag = 1;
//cout << "FAIL!" << endl;
}
return flag;
}
#endif
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