2000/09/01 by K. Soundararajan · 17 citations
Mathematics · #Analytic Number Theory Research #Finite Group Theory Research #Advanced Algebra and Geometry
paper · doi:10.2307/2661390
The Generalized Riemann Hypothesis (GRH) states that all nontrivial zeros of Dirichlet L-functions lie on the line Re(s) = 1 2 .Further, it is believed that there are no Q-linear relations among the nonnegative ordinates of these zeros.In particular, it is expected that L( 1 2 , χ) = 0 for all primitive characters χ, but this remains unproved.It appears to have been conjectured first by S. D. Chowla [5] in the case when χ is a quadratic character.In addition to numerical evidence (see [16] and [17]) the philosophy of N.Katz and P. Sarnak [13] lends theoretical support to this belief.Assuming the GRH, they proved that (oral communication) for at least (19Independently, A. E. Özluk and C. Snyder [15] showed, also assuming GRH, that L( 1 2 , χ d ) = 0 for at least 15 16 of the fundamental discriminants |d| ≤ X.Katz and Sarnak also developed conjectures on the low-lying zeros in this family of L-functions (analogous to the Pair Correlation conjecture regarding the vertical distribution of zeros of ζ(s)) which imply that L( 1 2 , d • ) = 0 for almost all fundamental discriminants d.In a different vein, R. Balasubramanian and V. K. Murty [1] showed that for a (small) positive proportion of the characters (mod q), L( 1 2 , χ) = 0. Recently, H. Iwaniec and P. Sarnak [10] have demonstrated that this proportion is at least one third.For integers d ≡ 0, or 1 (mod 4) we put χ d (n) = d n .Notice that χ d is a real character with conductor ≤ |d|.If d is an odd, positive, square-free integer then χ 8d is a real, primitive character with conductor 8d, and with χ 8d (-1) = 1.In [19], we considered the family of quadratic twists of a fixed Dirichlet L-function L(s, ψ).Precisely, we considered the family L(s, ψ ⊗ χ 8d ) for odd, positive, square-free integers d.When ψ is not quadratic we showed that at least 1 5 of these L-functions are not zero at s = 1 2 , and indicated how this proportion may be improved to 1 3 .The most interesting case when