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prime_functions_in_terms_of_zeros_of_zeta.pl
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prime_functions_in_terms_of_zeros_of_zeta.pl
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#!/usr/bin/perl
# Approximate the Chebyshev function and the weighted prime counting function, using zeros of the Riemann zeta function.
# See also:
# https://oeis.org/A267712
# https://en.wikipedia.org/wiki/Chebyshev_function
# https://en.wikipedia.org/wiki/Logarithmic_integral_function
# https://en.wikipedia.org/wiki/Riemann_zeta_function
use utf8;
use 5.020;
use strict;
use warnings;
binmode(STDOUT, ':utf8');
use ntheory qw(forprimes prime_count);
use experimental qw(signatures);
use Math::AnyNum qw(:overload gamma complex tau ilog iroot log Li harmreal);
my @zeta_ρ = map { chomp; complex(1 / 2, $_) } <DATA>;
sub Li_approx ($x) {
my $sum = 0;
foreach my $k (0 .. 0) {
$sum += gamma($k + 1) / log($x)**$k;
}
return ($sum * ($x / log($x)));
}
sub chebyshev_ψ ($x) {
my $sum = 0;
forprimes {
$sum += ilog($x, $_) * log($_)
} $x;
return $sum;
}
sub weighted_prime_count ($x) {
my $sum = 0;
foreach my $k (1 .. ilog($x, 2)) {
$sum += Math::AnyNum->new(prime_count(iroot($x, $k))) / $k;
}
return $sum;
}
sub weighted_prime_count_from_zeta_zeros ($x) {
my $sum = Li($x);
foreach my $ρ (@zeta_ρ) {
$sum -= Li_approx($x**$ρ);
}
return abs($sum - log(2));
}
sub chebyshev_ψ_from_zeta_zeros($x) {
my $sum = $x - log(tau) - log(1 - $x**(-2)) / 2;
foreach my $ρ (@zeta_ρ) {
$sum -= $x**$ρ / $ρ;
}
return abs($sum);
}
my $x = 10**3;
say "ψ($x) = ", chebyshev_ψ($x); # 996.680912247175240263021765666421541665778436902
say "ψ($x) ≅ ", chebyshev_ψ_from_zeta_zeros($x); # 996.068434632130345546023799228964726756917555651
say "\n=> Weighted prime count approximation: ";
foreach my $k (10 .. 14) {
my $exact = weighted_prime_count(10**$k);
my $approx = weighted_prime_count_from_zeta_zeros(10**$k);
say "Π(10^$k) = ", $exact->as_dec, " ≅ ", $approx, ' -> ', abs($exact - $approx);
}
__DATA__
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