Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T12:01:35.760Z Has data issue: false hasContentIssue false
  • English
  • Français

The mean value theorem for the Riemann zeta-function

Published online by Cambridge University Press:  26 February 2010

D. R. Heath-Brown
D. R. Heath-Brown
Affiliation:
Trinity College, Cambridge CB2 1TQ.
Get access

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.

Type
Research Article
Information
Mathematika , Volume 25 , Issue 2 , December 1978 , pp. 177 - 184
Copyright
Copyright © University College London 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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 “An improvement on a theorem of Titchmarsh on the mean square of ”, unpublished.Google Scholar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.CrossRefGoogle 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