The mean value theorem for the Riemann zeta-function

D. R. Heath-Brown

doi:10.1112/S0025579300009414

Extract

In this paper we shall consider the well known mean value theorem,

where γ is Euler's constant. The error term E(T) has been estimated by various writers; in particular the bounds E(T) = o(T log T), E(T) ≪ T1/2 log T, E(T) ≪ T5/12(log T)2 and

where ε is any positive quantity, have been given by Hardy and Littlewood [7], Ingham [8], Titchmarsh [10], and Balasubramanian [2], respectively.

References

1.Atkinson, F. V.. “The mean value of the Riemann zeta function”. Acta Math., 81 (1949), 353–376.Google Scholar

2.Balasubramanian, R.. “An improvement on a theorem of Titchmarsh on the mean square of

”. Proc.London Math. Soc. (3), 36 (1938), 540–576.Google Scholar

3.Balasubramanian, R.. “An improvement on a theorem of Titchmarsh on the mean square of

”, unpublished.Google Scholar

4.Corrádi, K. and Kátai, I.. “A comment on K. S. Gangadharan's paper entitled ‘Two classical lattice point problems’”. Magyar Tud. Akad. Mat. Fiz. Oszt. Kozl, 17 (1967), 89–97.Google Scholar

5.Good, A.. “Ein Ω-Resultat für das quadratische Mittel der Riemannschen Zetafunktion auf der kritische Linie.” Invent. Math., 41 (1977), 233–251.CrossRef Google Scholar

6.Hardy, G. H.. “On Dirichlet's divisor problem.” Proc. London Math. Soc. (2) 15 (1916), 1–25.Google Scholar

7.Hardy, G. H. and Littlewood, J. E.. “The Riemann zeta-function and the theory of the distribution of primes.” Ada. Math., 41 (1918), 119–196.Google Scholar

8.Ingham, A. E.. “Mean-value theorems in the theory of the Riemann Zeta-function.” Proc. London Math. Soc. (2) 27 (1926), 273–300.Google Scholar

9.Kolesnik, G. A.. “On the estimation of some trigonometric sums.” Acta Arith., 25 (1973), 7–30.Google Scholar

10.Titchmarsh, E. C.. “On van der Corput's method and the zeta-function of Riemann (V).” Quart. J. Math. Oxford Ser., 5 (1934), 195–210.Google Scholar

11.Titchmarsh, E. C.. The theory of the Riemann Zeta-function (Oxford, 1951).Google Scholar

12.Tong, K.-C.. “On divisor problems, III.” Acta Math. Sinica, 6 (1956), 515–541.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Jutila, Matti 1983. Riemann’s zeta-function and the divisor problem. Arkiv för Matematik, Vol. 21, Issue. 1-2, p. 75.

Ivić, Aleksandar 1983. Large values of the error term in the divisor problem. Inventiones Mathematicae, Vol. 71, Issue. 3, p. 513.

Balasubramanian, R. and Ramachandra, K. 1984. Mean-value of the Riemann zeta-function on the critical line. Proceedings of the Indian Academy of Sciences - Section A, Vol. 93, Issue. 2-3, p. 101.

Motohashi, Yoichi 1986. A note on the mean value of the zeta and $L$-functions, IV. Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 62, Issue. 8,

Hafner, James Lee and Ivić, Aleksandar 1989. On the mean-square of the Riemann zeta-function on the critical line. Journal of Number Theory, Vol. 32, Issue. 2, p. 151.

MATSUMOTO, Kohji 1989. The mean square of the Riemann zeta-function in the critical strip. Japanese journal of mathematics. New series, Vol. 15, Issue. 1, p. 1.

Ivić, A. and te Riele, H. J. J. 1991. On the zeros of the error term for the mean square of |𝜁(\frac{1}2+𝑖𝑡)|. Mathematics of Computation, Vol. 56, Issue. 193, p. 303.

Laurinčikas, A. 1996. On the mean square of the riemann zeta-function. Lithuanian Mathematical Journal, Vol. 36, Issue. 3, p. 282.

Matsumoto, Kohji 2000. Number Theory. p. 241.

Sankaranarayanan, A. 2007. Higher moments of certain L-functions on the critical line. Lithuanian Mathematical Journal, Vol. 47, Issue. 3, p. 277.

LAU, YUK-KAM and TSANG, KAI-MAN 2009. On the mean square formula of the error term in the Dirichlet divisor problem. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 146, Issue. 2, p. 277.

Zhang, Q. 2010. On the Fourth Moment of the Riemann Zeta Function. International Mathematics Research Notices,

Tsang, Kai-Man 2010. Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function. Science China Mathematics, Vol. 53, Issue. 9, p. 2561.

Ishikawa, Hideaki and Matsumoto, Kohji 2011. An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial. Central European Journal of Mathematics, Vol. 9, Issue. 1, p. 102.

Ivić, Aleksandar 2014. Analytic Number Theory, Approximation Theory, and Special Functions. p. 3.

Jutila, Matti 2015. Riemann’s zeta-function and the divisor problem. III. Arkiv för Matematik, Vol. 53, Issue. 2, p. 303.

Kong, Kar-Lun and Tsang, Kai-Man 2017. On some mean values for the divisor function and the Riemann zeta-function. Proceedings of the Steklov Institute of Mathematics, Vol. 296, Issue. 1, p. 142.

Dona, Daniele Helfgott, Harald A. and Alterman, Sebastian Zuniga 2022. Explicit L 2 bounds for the Riemann ζ function. Journal de théorie des nombres de Bordeaux, Vol. 34, Issue. 1, p. 91.

Simonič, Aleksander and Starichkova, Valeriia V. 2023. Atkinson's formula for the mean square of ζ(s) with an explicit error term. Journal of Number Theory, Vol. 244, Issue. , p. 111.

Article contents

The mean value theorem for the Riemann zeta-function

Extract

Access options

References

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

The mean value theorem for the Riemann zeta-function

Extract

Access options

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests