Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series

D. A. Goldston; S. M. Gonek

Acta Arithmetica (1998)

  • Volume: 84, Issue: 2, page 155-192
  • ISSN: 0065-1036

Abstract

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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.

How to cite

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D. A. Goldston, and S. M. Gonek. "Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series." Acta Arithmetica 84.2 (1998): 155-192. <http://eudml.org/doc/207141>.

@article{D1998,
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.},
author = {D. A. Goldston, S. M. Gonek},
journal = {Acta Arithmetica},
keywords = {Dirichlet series; Dirichlet polynomials; mean value theorem; arithmetical functions; coefficient correlation functions; Möbius function},
language = {eng},
number = {2},
pages = {155-192},
title = {Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series},
url = {http://eudml.org/doc/207141},
volume = {84},
year = {1998},
}

TY - JOUR
AU - D. A. Goldston
AU - S. M. Gonek
TI - Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series
JO - Acta Arithmetica
PY - 1998
VL - 84
IS - 2
SP - 155
EP - 192
AB - 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.
LA - eng
KW - Dirichlet series; Dirichlet polynomials; mean value theorem; arithmetical functions; coefficient correlation functions; Möbius function
UR - http://eudml.org/doc/207141
ER -

References

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  1. [1] J. Bolanz, Über die Montgomery'sche Paarvermutung, Diplomarbeit Universität Freiburg, 1987, 1-131. 
  2. [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. Zbl1029.11046
  3. [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. Zbl0268.10023
  4. [4] H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119-134. Zbl0296.10023
  5. [5] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., revised by D. R. Heath-Brown, Clarendon Press, Oxford, 1986. Zbl0601.10026

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