Using the Landau-Selberg-Delange method we determine the asymptotic preference factor $ ho(q)$ of the number of divisors function for non-zero squares modulo $q$ over non-squares in case $q$ is an odd prime. This factor is $1$ if $qequiv pm 3pmod*{8}$ and roughly of size $2^{n_q}$ otherwise, where $n_q$ is the smallest non-quadratic residue mod $q.$ We show that the twin primes conjecture is equivalent to the existence of a subsequence of primes $q$ for which $liminf n_q=infty$ and $ ho(q)<2^{2+n_q}/5.$
Quadratic residue bias of the divisor function and Fekete polynomials
Alessandro Languasco
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In corso di stampa
Abstract
Using the Landau-Selberg-Delange method we determine the asymptotic preference factor $ ho(q)$ of the number of divisors function for non-zero squares modulo $q$ over non-squares in case $q$ is an odd prime. This factor is $1$ if $qequiv pm 3pmod*{8}$ and roughly of size $2^{n_q}$ otherwise, where $n_q$ is the smallest non-quadratic residue mod $q.$ We show that the twin primes conjecture is equivalent to the existence of a subsequence of primes $q$ for which $liminf n_q=infty$ and $ ho(q)<2^{2+n_q}/5.$File in questo prodotto:
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