Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T05:13:15.564Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Mertens conjecture for cusp forms

Published online by Cambridge University Press:  26 February 2010

Robert J. Anderson
Robert J. Anderson
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA (U.S.A.), Tufts University, Medford, MA (U.S.A.).
Get access

Extract

Before stating the main theorem, we would like to recall the basic properties of the “zeta-functions” attached to cusp forms on SL (2, ℤ). Let k be an even integer ≥ 12 and f a cusp form of weight k on SL (2, ℤ) with q-expansion We shall assume that c1 = 1, and that f is an eigenfunction of the Hecke operators. Define φ(s) as the Dirichlet series The series and the product

over the primes are equal and absolutely convergent for Re (S) > ½(k + 1).

Type
Research Article
Information
Mathematika , Volume 26 , Issue 2 , December 1979 , pp. 236 - 249
Copyright
Copyright © University College London 1979

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.Goldstein, L. J.. “A necessary and sufficient condition for the Riemann hypothesis for zeta functions attached to eigenfunctions of the Hecke operators”, Ada. Aritk, 15 (1968/1969), 205–215.Google Scholar
2.Ingham, A. E.. “On two conjectures in the theory of numbers”, Amer. Jour, of Math., 64 (1942), 313–319.CrossRefGoogle Scholar
3.Jurkat, W.. “On the Mertens conjecture and related general Ω-theorems”, Proc. Symp. Pure Math., Analytic Number Theory, A.M.S., 24 (1973), 147–158.CrossRefGoogle Scholar
4.Titchmarsh, E. C., The theory of the Riemann zeta-function (Oxford, 1951).Google Scholar