Irregularities in the distribution of primes in arithmetic progressions. I
type: Item
instance of: scholarly article
title: Irregularities in the distribution of primes in arithmetic progressions. I
zbMATH Open document ID: 0558.10036
DOI: 10.1007/BF01190694
author: S. Salerno
author: János Pintz
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: Assuming that all Dirichlet L-functions (mod q) have no zeros in \(\sigma >\), \(| t| \leq cq^ 2 \log^ 6q\), where \(s=\sigma +it\), lower bounds are obtained for various definite integrals of \(| \Delta_ i(x)| /x\), \(i=2,4\), where \[ \Delta_ 4(x)=\sum_{n\leq x, n\equiv \ell_ 1 (q)}\Lambda (n)-\sum_{n\leq x, n\equiv \ell_ 2 (q)}\Lambda (n) \] for arbitrary \(\ell_ 1,\ell_ 2\) satisfying \((\ell_ 1,q)=(\ell_ 2,q)=1\), \(\ell_ 1\not\equiv \ell_ 2 (q)\) and where \(\Lambda_ 2(x)\) is the same difference with summands \(\Lambda\) (n)/log n. The results improve the previous bounds of Knapowski.
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 3889651
zbMATH Keywords: oscillatory properties
zbMATH Keywords: lower bounds
zbMATH Keywords: mean value
reviewed by: W. E. Briggs
MaRDI profile type: Publication
cites work: Contributions to the theory of the distribution of prime numbers in arithmetical progressions I
cites work: Contributions to the theory of the distribution of prime numbers in arithmetical progressions III
cites work: Accumulation theorems for primes in arithmetic progressions
full work available at URL: https://doi.org/10.1007/bf01190694
OpenAlex ID: W4246122579
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description: scientific article; zbMATH DE number 3889651