On the parity of the prime-counting function and related problems
Let \(\pi(x)\) denote the number of primes not exceeding \(x\). The aim of the paper under review is to study the distribution of the values of \(\pi(x)\) in residue class modulo given \(q\geq 2\). The author defines \[ A(x;q,a)=\#\{n\leq x~:~\pi(n)\equiv a\pmod{q}\}, \] and shows that for \(0\leq a\leq q-1\) one has \(A(x;q,a)\geq((16q^2)^{-1}+o(1))x\). He performs some computations on \(\max_{a\pmod{q}}|x^{-1}A(x;q,a)-q^{-1}|\) for several values of \(x\) and \(q\), supporting the conjecture, asserting that for each integer \(q\geq 2\), the function \(\pi(n)\) is equidistributed among the residue classes modulo \(q\). The author applies the large sieve to prove some average estimates confirming the above mentioned conjecture in a certain range, indeed by showing the average estimate \[ \begin{aligned} \sum_{q\leq Q}\sum_{a=0}^{q-1}\left|\frac{q}{N}A(N;q,a)-1\right|^2 \ll \begin{cases} \frac{Q^3}{N}\log^3 N+Q\log^2 N & \text{assuming the \(RH\)},\\ \frac{Q^3}{N^{1-\varepsilon}}+QN^\varepsilon & \text{assuming the \(LH\)},\\ \frac{Q^3}{N^{\frac34-\varepsilon}}+QN^{\frac14+\varepsilon} & \text{unconditionally}, \end{cases} \end{aligned} \] where \(RH\) and \(LH\) are abbreviations for the Riemann hypothesis and the Lindelöf hypothesis, respectively. The author deduces from the above result that for almost all residue classes \(a\pmod{q}\) with \(q\leq Q\) we have \(A(x;q,a)\sim q^{-1}x\), uniformly for \(Q\) in the range \[ \begin{cases} \frac{Q}{\log^2 x}\to\infty \text{and} Q=o(x\log^{-3}x) & \text{assuming the \(RH\)},\\ x^\varepsilon\ll Q\ll x^{1-\varepsilon} & \text{assuming the \(LH\)},\\ x^{\frac14+\varepsilon}\ll Q\ll x^{\frac34-\varepsilon} & \text{unconditionally}.\end{cases} \]
scientific article; zbMATH DE number 6495139