//#include <Lfunction/L.h> #include "L.h" //a simple program illustrating a few features of the L_function class. int main (int argc, char *argv[]) { initialize_globals(); //initialize global variables. This *must* be called. Double x,y; Complex s; L_function<int> zeta; //default L-function is the Riemann zeta function L_function<int> L4; //will be assigned L(s,chi_{-4}), chi{-4} being the real quadratic character mod 4 L_function<Complex> L5; //will be assigned L(s,chi), with chi being a complex character mod 5 //==================== Initialize the L-functions ================================== //one drawback of arrays- the index starts at 0 rather than 1 //so each array below is declared to be one entry larger than it is. //I prefer this so that referring to the array elements is more //straightforward, for example coeff[1] refers //to the first Dirichlet cefficient rather than coeff[0]. But to make up //for this, one needs to have a bogus entry (such as 0) at the start of each array. int coeff_L4[5] = {0,1,0,-1,0}; //the Dirichlet coefficients, periodic of period 4. Double gamma_L4[2] = {0,.5}; //the gamma factor Gamma(gamma s + lambda) is Gamma(s/2+1/2) Complex lambda_L4[2] = {0,.5}; //the lambda Complex pole_L4[1] = {0}; //no pole Complex residue_L4[1] = {0}; //no residue L4=L_function<int>("L4",1,4,coeff_L4,4,sqrt(4/Pi),1,1,gamma_L4,lambda_L4,0,pole_L4,residue_L4); // "L4" is the name of the L-function // 1 - what_type, 1 stands for periodic Dirichlet coefficients // 4 - N_terms, number of Dirichlet coefficients given // coeff_L4 - array of Dirichlet coefficients // 4 - period (0 if coeffs are not periodic) // sqrt(4/Pi) - the Q^s that appears in the functional equation // 1 - sign of the functional equation // 1 - number of gamma factors of the form Gamma(gamma s + lambda), gamma = .5 or 1 // gamma_L4 - array of gamma's (each gamma is .5 or 1) // lambda_L4 - array of lambda's (given as complex numbers) // 0 - number of poles. Typically there won't be any poles. // pole_L4 - array of poles, in this case none // residue_L4 - array of residues, in this case none // Note: one can call the constructor without the last three arguements when number of poles = 0 // as in: L4 = L_function<int>("L4",1,4,coeff_L4,4,sqrt(4/Pi),1,1,gamma_L4,lambda_L4); Complex coeff_L5[6] = {0,1,I,-I,-1,0}; Complex gauss_sum=0; for(int n=1;n<=4; n++) gauss_sum=gauss_sum+coeff_L5[n]*exp(n*2*I*Pi/5); L5=L_function<Complex>("L5",1,5,coeff_L5,5,sqrt(5/Pi),gauss_sum/(I*sqrt(5.)),1,gamma_L4,lambda_L4); // "L5" is the name of the L-function // 1 - what_type, 1 stands for periodic Dirichlet coefficients // 5 - N_terms, number of Dirichlet coefficients given // coeff_L5 - array of Dirichlet coefficients // 5 - period (0 if coeffs are not periodic) // sqrt(5/Pi), the Q^s that appears in the functional equation // gauss_sum/sqrt(5) - omega of the functional equation // 1 - number of gamma factors of the form Gamma(gamma s + lambda), gamma = .5 or 1 // gamma_L4 - L5 has same gamma factor as L4 // lambda_L4 - ditto //=========== print basic data for the L-function ======================================== zeta.print_data_L(); L4.print_data_L(); L5.print_data_L(); //=========== print some L-values ======================================== x= .5; y =0; cout << "zeta" << x+I*y << " = " << zeta.value(x+I*y) << endl; cout << "L4" << x+I*y << " = " << L4.value(x+I*y) << endl; cout << "L5" << x+I*y << " = " << L5.value(x+I*y) << endl; x= 1; y =0; cout << "L4" << x+I*y << " = " << L4.value(x+I*y) << endl; cout << "L5" << x+I*y << " = " << L5.value(x+I*y) << endl; //=========== find and print some zeros ======================================== //find zeros of zeta up to height 100 taking steps of size .1, looking for sign //changes on the critical line. Some zeros can be missed in this fashion. //First column gives the imaginary part of the zeros. //Second column outputted is related to S(T) and should be small on average //(larger values means zeros were missed). zeta.find_zeros(0,100,.1); //find the first 100 zeros of zeta. This also verifies RH and does not omit zeros. zeta.find_zeros_via_N(100,false); //false, here, means do *not* look for zeros below the //real axis as they come in conjugate pairs //do same for L4 and L5 L4.find_zeros(0,100,.1); L4.find_zeros_via_N(100,false); L5.find_zeros(0,100,.1); L5.find_zeros_via_N(100,true); //true here means look above and below the real axis }

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