{Let $d(n)=\sum_{d\mid n}1$ be the divisor function. Using the Landau--Selberg--Delange method we determine the asymptotic preference factor $\rho(q)$ of $d(n)$ for non-zero squares modulo $q$ over non-squares in case $q$ is an odd prime. This factor is $1$ if $q\equiv \pm 3\pmod{8}$ and, using estimates for Fekete polynomials, it is seen to be roughly of size $2^{n_q}$ otherwise, where $n_q$ is the least quadratic non-residue mod $q.$ We also show that the twin primes conjecture is equivalent to the existence of a subsequence of primes $q$ for which $\liminf n_q=\infty$ and $\rho(q)<2^{2+n_q}/5.$
Quadratic residue bias of the divisor function, Fekete polynomials and prime gaps
Alessandro Languasco
;
In corso di stampa
Abstract
{Let $d(n)=\sum_{d\mid n}1$ be the divisor function. Using the Landau--Selberg--Delange method we determine the asymptotic preference factor $\rho(q)$ of $d(n)$ for non-zero squares modulo $q$ over non-squares in case $q$ is an odd prime. This factor is $1$ if $q\equiv \pm 3\pmod{8}$ and, using estimates for Fekete polynomials, it is seen to be roughly of size $2^{n_q}$ otherwise, where $n_q$ is the least quadratic non-residue mod $q.$ We also show that the twin primes conjecture is equivalent to the existence of a subsequence of primes $q$ for which $\liminf n_q=\infty$ and $\rho(q)<2^{2+n_q}/5.$
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