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Zeros of Dirichlet L-series on the critical line

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Introduction. In 1974, N. Levinson showed that at least 1/3 of the zeros of the Riemann ζ-function are on the critical line ([19]). Today it is known (Conrey, [6]) that at least 40.77% of the zeros of ζ(s) are on the critical line and at least 40.1% are on the critical line and are simple. In [16] and [17], Hilano showed that Levinson's original result is also valid for Dirichlet L-series.
This paper is a shortened version of parts of the dissertation [3], the full details of which may be found at http://www.math.uni-frankfurt.de/~pbauer/diss.ps. We shall prove a mean value theorem for Dirichlet L-series and use this for proving some corollaries concerning the distribution of the zeros of L-series - amongst other results we improve the above mentioned bounds for Dirichlet L-series.
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  • Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, Robert-Mayer-Straße 6-10, 60054 Frankfurt/Main, Germany
Bibliografia
  • [1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1986.
  • [2] P. J. Bauer, Über den Anteil der Nullstellen der Riemannschen Zeta-Funktion auf der kritischen Geraden, Diploma thesis, Frankfurt a.M., 1992. (Available at http://www.math.uni-frankfurt.de/ pbauer/diplom.ps.)
  • [3] P. J. Bauer, Zur Verteilung der Nullstellen der Dirichletschen L-Reihen, Dissertation, Frankfurt a.M., 1997. (Available at http://www.math.uni-frankfurt.de/ pbauer/diss.ps.)
  • [4] D. A. Burgess, On character sums and L-series, Proc. London Math. Soc. 12 (1962), 193-206.
  • [5] J. B. Conrey, Zeros of derivatives of Riemann's Xi-function on the critical line, J. Number Theory 16 (1983), 49-74.
  • [6] J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. Reine Angew. Math. 399 (1989), 1-26.
  • [7] J. B. Conrey and A. Ghosh, A simpler proof of Levinson's theorem, Math. Proc. Cambridge Philos. Soc. 97 (1985), 385-395.
  • [8] J. B. Conrey, A. Ghosh and S. M. Gonek, Mean values of the Riemann zeta-function with application to the distribution of zeros, in: Number Theory, Trace Formulas and Discrete Groups, K. E. Aubert et al. (eds.), Academic Press, 1989, 185-199.
  • [9] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, 1980.
  • [10] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982), 219-288.
  • [11] J.-M. Deshouillers and H. Iwaniec, Power mean-values for Dirichlet's polynomials and the Riemann zeta-function II, Acta Arith. 43 (1984), 305-312.
  • [12] T. Estermann, On the representation of a number as the sum of two products, Proc. London Math. Soc. (2) 31 (1930), 123-133.
  • [13] D. W. Farmer, Mean value of Dirichlet series associated with holomorphic cusp forms, J. Number Theory 49 (1994), 209-245.
  • [14] P. X. Gallagher, A large sieve density estimate near σ=1, Invent. Math. 11 (1970), 329-339.
  • [15] S. M. Gonek, Mean values of the Riemann zeta-function and its derivatives, ibid. 75 (1984), 123-141.
  • [16] T. Hilano, On the distribution of zeros of Dirichlet's L-function on the line σ=1/2, Proc. Japan Acad. 52 (1976), 537-540.
  • [17] T. Hilano, On the distribution of zeros of Dirichlet's L-function on the line σ=1/2 (II), Tokyo J. Math. 1 (1978), 285-304.
  • [18] G. Kolesnik, On the order of Dirichlet L-functions, Pacific J. Math. 82 (1979), 479-484.
  • [19] N. Levinson, More than one third of the zeros of Riemann's zeta-function are on σ=1/2, Adv. Math. 13 (1974), 383-436.
  • [20] H. L. Montgomery, Topics in Multiplicative Number Theory, Springer, 1971.
  • [21] K. Prachar, Primzahlverteilung, Springer, 1957.
  • [22] A. Selberg, On the zeros of Riemann's zeta-function, Skr. Norske Videnskaps-Akad. Oslo, I, 10 (1942), 1-59.
  • [23] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Clarendon Press, 1988.
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