Functiones et Approximatio Commentarii Mathematici

Explicit relations between primes in short intervals and exponential sums over primes

Alessandro Languasco and Alessandro Zaccagnini

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Under the assumption of the Riemann Hypothesis, we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.

Article information

Source
Funct. Approx. Comment. Math., Volume 51, Number 2 (2014), 379 -391.

Dates
First available in Project Euclid: 26 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1417010860

Digital Object Identifier
doi:10.7169/facm/2014.51.2.9

Mathematical Reviews number (MathSciNet)
MR3282634

Zentralblatt MATH identifier
1357.11087

Subjects
Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Secondary: 11N05: Distribution of primes 11M45: Tauberian theorems [See also 40E05]

Keywords
exponential sum over primes primes in short intervals pair-correlation conjecture

Citation

Languasco, Alessandro; Zaccagnini, Alessandro. Explicit relations between primes in short intervals and exponential sums over primes. Funct. Approx. Comment. Math. 51 (2014), no. 2, 379 --391. doi:10.7169/facm/2014.51.2.9. https://projecteuclid.org/euclid.facm/1417010860


Export citation

References

  • T.H. Chan, More precise pair correlation of zeros and primes in short intervals, J. London Math. Soc. 68 (2003), 579–598.
  • D.A. Goldston and H.L. Montgomery, Pair correlation of zeros and primes in short intervals, Analytic Number Theory and Diophantine Problems (A.C. Adolphson et al., eds.), Proc. of a Conference at Oklahoma State University (1984), Birkhäuser Verlag, 1987, pp. 183–203.
  • I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, series, and products, Academic Press, 2007.
  • D.R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. Lon. Math. Soc. 38 (1979), 385–422.
  • A. Languasco, A note on primes and Goldbach numbers in short intervals, Acta Math. Hungar. 79 (1998), 191–206.
  • A. Languasco and A. Perelli, Pair correlation of zeros, primes in short intervals and exponential sums over primes, J. Number Theory 84 (2000), 292–304.
  • A. Languasco, A. Perelli, and A. Zaccagnini, Explicit relations between pair correlation of zeros and primes in short intervals, J. Math. Anal. Appl. 394 (2012), 761–771.
  • H.L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, Springer Verlag, 1971.
  • H.L. Montgomery, K. Soundararajan, Beyond pair correlation, in Paul Erdős and his Mathematics, Bolyai Soc. Math. Stud. 11 (2002), 507–514.
  • F.W.J. Olver, D.W. Lozer, R.F. Boisvert, and C.W. Clark, NIST Handbook of Mathematical Functions, Cambridge U. P., 2010.
  • A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), 87–105.