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ON THE MONTGOMERY–HOOLEY THEOREM IN SHORT INTERVALS

Part of:Multiplicative number theoryAdditive number theory; partitions

Published online by Cambridge University Press:  12 May 2010

A. Languasco,
A. Perelli and
A. Zaccagnini
A. Languasco
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy (email: languasco@math.unipd.it)
A. Perelli
Affiliation:
Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy (email: perelli@dima.unige.it)
A. Zaccagnini
Affiliation:
Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze 53/a, Campus Universitario, 43124 Parma, Italy (email: alessandro.zaccagnini@unipr.it)

Abstract

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We prove two results about the asymptotic formula for \[ S(x,h,Q)=\sum _{q\le Q} \sum ^q_{\substack {a=1\\ (a,q)=1}} \biggl \vert \psi (x+h;q,a)-\psi (x;q,a)-\frac {h}{\varphi (q)}\biggr \vert ^2. \] The first result is for x7/12+ε≤h≤x and h/(log x)B≤Q≤h, where ε,B>0 are arbitrary constants. For the second result we assume that the Generalized Riemann Hypothesis holds and we obtain a stronger error term and a better uniformity on h.

MSC classification

Secondary:11N13: Primes in progressions11P55: Applications of the Hardy-Littlewood method

Information

Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010, pp. 231 - 243
Copyright
Copyright © University College London 2010

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