On the parity of the prime-counting function and related problems

Abstract

We study the parity of the prime-counting function $\pi (x)$, and more generally the distribution of its values in residue classes modulo $q$. We prove that for any integer $q\ge 2$ and $0\le a\le q-1$, the function $\pi (n)$ lies in the residue class $a \pmod q$ for a positive proportion of integers $n\ge 1$. Based on the numerical evidence, we conjecture that $\pi (n)$ should be equidistributed among the residue classes modulo $q$, and we prove an average version of this conjecture using the large sieve.