We obtain formulas for computing mean values of Dirichlet polynomials that have more terms than the length of the integration range. These formulas allow one to compute the contribution of off-diagonal terms provided one knows the correlation functions for the coefficients of the Dirichlet polynomials. A smooth weight is used to control error terms, and this weight can in typical applications be removed from the final result. Similar results are obtained for the tails of Dirichlet series. Four examples of applications to the Riemann zeta-function are included.
@article{bwmeta1.element.bwnjournal-article-aav84i2p155bwm,
author = {D. A. Goldston and S. M. Gonek},
title = {Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series},
journal = {Acta Arithmetica},
volume = {84},
year = {1998},
pages = {155-192},
zbl = {0902.11033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav84i2p155bwm}
}
D. A. Goldston; S. M. Gonek. Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series. Acta Arithmetica, Tome 84 (1998) pp. 155-192. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav84i2p155bwm/
[000] [1] J. Bolanz, Über die Montgomery'sche Paarvermutung, Diplomarbeit Universität Freiburg, 1987, 1-131.
[001] [2] D. A. Goldston, S. M. Gonek, A. E. Özlük, and C. Snyder, On the pair correlation of the zeros of the Riemann zeta-function, to appear. | Zbl 1029.11046
[002] [3] H. L. Montgomery, The pair correlation of zeros of the zeta function, in: Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, R.I., 1973, 181-193. | Zbl 0268.10023
[003] [4] H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119-134. | Zbl 0296.10023
[004] [5] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., revised by D. R. Heath-Brown, Clarendon Press, Oxford, 1986. | Zbl 0601.10026