On a generalization of the Selberg formula
type: Item
instance of: scholarly article
title: On a generalization of the Selberg formula
zbMATH Open document ID: 1130.11052
author: Emmanuel Knafo
published in: Journal of Number Theory
publication date: 2007-09-13T00:00:00Z
review text: Define the generalized von Mangoldt function \(\Lambda_k\) by \(\Lambda_k(n)= \sum_{d|n}\mu(d)(\log {n\over d})^k\), and for \((a, q)= 1\) let \(\psi_k(x;q, a)= \sum_{\substack{ n\leq x\\ n\equiv a\pmod q}} \Lambda_k(n)\). For the case \(k= 1\), the Siegel-Walfisz theorem yields the asymptotic formula \(\psi((x; q, a)\sim{x\over\varphi(q)}\), valid uniformly in the range \(q< (\log x)^A\) for any fixed \(A\). In [Recent progress in analytic number theory, Durham Symposium 1979. Vol. 1, 15--23 (1981; Zbl 0459.10029)], \textit{J. B. Friedlander} used a sieve method developed by E. Bombieri to prove that \(\psi_2(x;q, a)\sim{2x\over\psi(q)}\log x\) uniformly for \(\log q< \varepsilon(x)\log x\) for a fixed positive function \(\varepsilon(x)\to 0\) as \(x\to\infty\), and he showed that analytic methods yield this asymptotic formula in the smaller range of validity \(\log q< \varepsilon(x){\log x\over\log\log x}\). The present author generalizes Friedlander's lemmas 3.1 and 3.2, and then establishes an estimate for the sum \(\psi_k(x;\chi)= \sum_{n\leq x} \chi(n)\Lambda_k(n)\), where \(\chi\) denotes a character modulo \(q\). He deduces that for \(k\geq 2\) and \(\varepsilon(x)\) as above \[ \psi_k(x; q,a)\sim{kx\over\varphi(q)}(\log x)^{k- 1} \] uniformly for \(\log q<(\varepsilon(x))^{{1\over k-1}}{\log x\over\log\log x}\). He also derives some other related results.
reviewed by: Eira J. Scourfield
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M20
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 5190746
zbMATH Keywords: Selberg formula
zbMATH Keywords: generalized von Mangoldt function
zbMATH Keywords: almost primes
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.10.013
OpenAlex ID: W2040426974
cites work: Q3605192
cites work: Q4043139
cites work: Corrigendum to my paper "On twin almost primes" and an addendum on Selberg's sieve
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cites work: A zero-density theorem for the Riemann zeta-function
cites work: Q3417372
cites work: EFFECTIVE LOWER BOUND FOR THE VARIANCE OF DISTRIBUTION OF PRIMES IN ARITHMETIC PROGRESSIONS
cites work: Q3934736
DOI: 10.1016/J.JNT.2006.10.013
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description: scientific article; zbMATH DE number 5190746