Lower bounds for sums of Barban-Davenport-Halberstam type. (Supplement)
type: Item
instance of: scholarly article
title: Lower bounds for sums of Barban-Davenport-Halberstam type. (Supplement)
zbMATH Open document ID: 0834.11036
DOI: 10.1007/BF02570468
author: Yanyan Li
published in: Manuscripta Mathematica
publication date: 1995-09-16T00:00:00Z
full work available at URL: https://eudml.org/doc/156081
review text: Let \[ S(Q, x)= \sum_{q\leq Q} \sum _{\substack{ 0< a\leq q\\ (a,q)=1 }} \biggl( \psi (x; q, a)- {x\over {\varphi (q)}} \biggr)^2, \quad S^* (Q, x)= \sum_{q\leq Q} \sum _{\substack{ 0< a\leq q\\ (b, q)=1 }} \biggl( \psi (x; q, a)- {{\psi(x, \chi^0)} \over {\varphi (q)}} \biggr)^2, \] where \(\varphi (q)\) is Euler's function, \(\psi (x; q, a)\) is the Chebyshev function which sums the von Mangoldt function \(\Lambda (n)\) over the arithmetic progression \(n\equiv a\pmod q\), \(n\leq x\), and \(\psi (x, \chi^0)= \sum_{n\leq x} \chi^0 (n) \Lambda (n)\) with \(\chi^0 \pmod q\) being the principal character. In this supplement to [J. Reine Angew. Math. 438, 163-174 (1993; Zbl 0776.11056)], the author proves that if \(c\) and \(x\) are positive numbers which are small and large, respectively, and \(x\exp (-c \log x/ \log \log x)\leq Q\leq x\), then \[ S^* (Q, x)\geq \bigl( {\textstyle {1\over 4}} +O(c) \bigr) Qx\log x. \] On applying the Vinogradov-Korobov version of the prime number theorem the author deduces that if \(Q\) further satisfies \[ x\exp \bigl(- B(\log x)^{3\over 5} (\log \log x)^{-{1\over 5}} \bigr)\leq Q\leq x \] for some positive constant \(B\), then \[ S(Q, x)\geq \bigl( {\textstyle {1\over 8}}+ O(c)\bigr) Qx\log x. \] Although there are asymptotic formulae for \(S(Q, x)\) in the range \(x/ (\log x)^A\leq Q\leq x\), the derivation of such formulae makes use of the Siegel-Walfisz theorem which renders them ineffective, whereas the estimates in this supplement are effective.
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 784068
zbMATH Keywords: lower bounds
zbMATH Keywords: sums of Barban-Davenport-Halberstam type
zbMATH Keywords: effective estimates
zbMATH Keywords: Chebyshev function
zbMATH Keywords: von Mangoldt function
zbMATH Keywords: arithmetic progression
reviewed by: Peter Shiu
MaRDI profile type: Publication
cites work: Q3900124
cites work: A large sieve density estimate near \(\sigma = 1\)
cites work: On the Barban-Davenport-Halberstam theorem. I.
cites work: On the Barban-Davenport-Halberstam Theorem-II
cites work: Lower bounds for sums of Barban-Davenport-Halberstam type.
cites work: Primes in arithmetic progressions
cites work: The exceptional set of Goldbach's problem
cites work: Q5640160
cites work: Q5523060
OpenAlex ID: W2119114239
Recommended article: Brun's fundamental lemma and exponential sums over primes
Recommended article: Congruences and rational exponential sums with the Euler function
Recommended article: A theorem of Bombieri–Vinogradov type with few exceptional moduli
Recommended article: Q4885308
Recommended article: EFFECTIVE LOWER BOUND FOR THE VARIANCE OF DISTRIBUTION OF PRIMES IN ARITHMETIC PROGRESSIONS
Recommended article: On the error term in the explicit formula of Riemann–von Mangoldt
Recommended article: On a Titchmarsh-Estermann sum.
Recommended article: ON THE MONTGOMERY–HOOLEY THEOREM IN SHORT INTERVALS
Recommended article: Theorems and problems on \(p^k/k\).
Recommended article: Q3472172
description: scientific article; zbMATH DE number 784068