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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.).
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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

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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
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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