Spacing of zeros of Hecke L-functions and the class number problem

2001/11/01 by J. Brian Conrey, Henryk Iwaniec, Conrey, J. Brian +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.math/0111012

61 pages

arxiv created 2001/11/01 · arxiv updated 2009/11/30

Abstract

We derive strong and effective lower bounds for the class number h(q) of the imaginary quadratic field Q(√(-q)), conditionally subject to the existence of many small (subnormal) gaps between zeros of the L-function associated with a character of the class group associated with this field. In particular, we prove that if the gap between consecutive zeros of the L-function is somewhat smaller than the average for sufficiently many pairs of zeros on the critical line, then h >> √ q (log q)^-A for some constant A > 0. For the trivial character, the L-function is the Dedekind zeta-function of the number field and so contains the Riemann zeta-function as a factor. Thus, as a corollary to our main result, we prove that this lower bound for h follows from the hypothesis that there are sufficiently many pairs of adjacent zeros of the Riemann zeta-function on the critical line whose spacing is slightly smaller than 1/2 of the average spacing.

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