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

By J. Brian Conrey, Henryk Iwaniec. 2001

77 citations recorded by the source corpus.

We derive strong and effective lower bounds for the class number h(q) of the imaginary quadratic field Q(\sqrt{-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 >> \sqrt 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/708e49c8e1653f314596c6840363ccea7f3eb2a8

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Citations in the corpus

Extreme values of ζ′(ρ) (2007, unknown): “This idea was further explored by Conrey and Iwaniec in [3].”
Summation formulas, from Poisson and Voronoi to the present (2003, methodology): “[2, 11, 15] If the Schwartz function f ∈ S(R) vanishes to infinite order at the origin, ∑ n6=0 an e −2π i na/c f(n) = |c| ∑ n6=0 an |n| e 2π in ā/c F (n/c(2)) ,”
Automorphic distributions, L-functions, and Voronoi summation for GL(3) (2004, methodology): “The changes necessary to treat the case of a congruence subgroup can easily be adapted from [ 8 ,21], for example.”
Shifted convolution sums and subconvexity bounds for automorphic L-functions (2004, unknown): “Harcos’ paper [10] yields such a result for h ≤ M while variants of the δ-method (see [1, 5]) give uniformity up to h ≤M3/2−ε.”