More than two fifths of the zeros of the Riemann zeta function are on the critical line.

  title={More than two fifths of the zeros of the Riemann zeta function are on the critical line.},
  author={J. Brian Conrey},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  pages={1 - 26}
  • J. Conrey
  • Published 1989
  • Mathematics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
In this paper we show that at least 2/5 of the zeros of the Riemann zeta-function are simple and on the critical line. Our method is a refinement of the method Levinson [11] used when he showed that at least 1/3 of the zeros are on the critical line (and are simple, äs observed by Heath-Brown [10] and, independently, by Seiberg). The main new element here is the use of a mollifier of length y=T with = 4/7 — whereas in Levinson's theorem the mollifier has 0 = 1/2 — . The work [6] of Deshouillers… 
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More than one third of the zeros of Riemann's zeta-function are on = 1/2
  • Adv. Math. 13
  • 1974
Titchmarsh, The Theory of the Riemann Zeta-Function
  • Second Ed., Oxford 1986. Oklahoma State University,
  • 1988
On the Representations of a Number as the Sum of Two Products
Kloosterman sums and Fourier coefficients of cusp forms
On the zeros of Riemann's zeta-function
Deduction of Semi-Optimal Mollifier for Obtaining Lower Bound for N(0)(T) for Riemann's Zeta-Function.
  • N. Levinson
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1975
A mollifier played a key role in showing N(0)(T) > 1/3N(T) for large T in ref. 1, and a deductive procedure is given for finding a mollifiers that actually minimizes the larger term.
Über die Anzahl der Primzahlen unter eine gegebener Grosse
  • Monatsber. Akad. Berlin
  • 1859