ArticleOriginal scientific text
Title
Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series
Authors 1, 2
Affiliations
- Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192, U.S.A.
- Department of Mathematics, University of Rochester, Rochester, New York 14627, U.S.A.
Abstract
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.
Bibliography
- J. Bolanz, Über die Montgomery'sche Paarvermutung, Diplomarbeit Universität Freiburg, 1987, 1-131.
- 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.
- 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.
- H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119-134.
- E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., revised by D. R. Heath-Brown, Clarendon Press, Oxford, 1986.
Pages:
155-192
Main language of publication
English
Received
1997-06-12
Published
1998
Exact and natural sciences