Residue classes containing an unexpected number of primes. (English) Zbl 1264.11081

Fiorilli, Daniel

Duke Math. J. 161, No. 15, 2923-2943 (2012).

Fix a non-zero integer \(a\), and positive real numbers \(B\) and \(\varepsilon\). For any \(M\in [1,(\log x)^B]\) let \(\mathcal{Q}\) be the set of moduli \(q\leq x/M\) which are coprime to \(a\). The main result of the paper is then that \[ \frac{1}{\# \mathcal{Q}}\sum_{q\in\mathcal{Q}} \left(\psi(x;q,a)-\frac{\psi(x)}{\phi(q)}\right)=\nu(a,M)+ O_{a,B,\varepsilon}(M^{\varepsilon-205/538}), \] for a certain \(\nu(a,M)\). Thus \(\psi(x;q,a)\) differs from its expected value \(\psi(x)/\phi(q)\) on average, by an amount \(\nu(a,M)\). If \(a\) were a positive prime power one might expect to have \(\nu(a,M)=\Lambda(a)\), so as to allow for the fact that the term \(\Lambda(a)\) is necessarily counted by \(\psi(x;q,a)\). However in fact it is shown that \(\nu(a,M)=\frac32\Lambda(a)\) whenever \(a\geq 2\). Similarly one has \(\nu(a,M)=\frac12\Lambda(|a|)\) if \(a\leq -2\); while if \(a=\pm 1\) one has \(\nu(\pm 1,M)=-\frac12\log M-C\) for a certain explicitly given constant \(C\).
The discrepancy between \(\psi(x;q,a)\) and its “expected” value, as revealed by these results, is a very small, and corresponds to less than one prime for each modulus \(q\); but it is non-zero none the less.
For the proof one uses a deep result of J. Friedlander, A. Granville, A. Hildebrand and H. Maier [J. Am. Math. Soc. 4, No. 1, 25–86 (1991; Zbl 0724.11040)], which shows that moduli \(q\leq xM^{-2}\) make a negligible contribution. The remaining moduli are then handled by replacing \(q\) by its “complementary divisor” in the usual way.

Reviewer: D. R. Heath-Brown (Oxford)

MSC:

11N13

Primes in congruence classes

Keywords:

primes; congruence class; average; discrepancy

Citations:

Zbl 0724.11040

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