On the parity of the prime-counting function and related problems
type: Item
instance of: scholarly article
title: On the parity of the prime-counting function and related problems
zbMATH Open document ID: 1328.11098
author: Mihai Alboiu
published in: The Ramanujan Journal
publication date: 2015-10-16T00:00:00Z
review text: 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} \]
reviewed by: Mehdi Hassani
Mathematics Subject Classification ID: 11N64
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 6495139
zbMATH Keywords: prime counting function
zbMATH Keywords: parity
zbMATH Keywords: sieve method
zbMATH Keywords: residue class
MaRDI profile type: Publication
full work available at URL: s11139-014-9596-1
OpenAlex ID: W2048113612
cites work: Q5402458
cites work: Deterministic methods to find primes
cites work: The large sieve
cites work: Q4052222
cites work: GAPS BETWEEN PRIMES
cites work: Q5845915
cites work: The Differences Between Consecutive Primes
DOI: 10.1007/S11139-014-9596-1
description: scientific article; zbMATH DE number 6495139