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

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. [ABSTRACT FROM AUTHOR]