## Functiones et Approximatio Commentarii Mathematici

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

#### 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

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

#### 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

#### 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.