Newman's conjecture in function fields

Abstract

TextDe Bruijn and Newman introduced a deformation of the completed Riemann zeta function ζ, and proved there is a real constant Λ which encodes the movement of the nontrivial zeros of ζ under the deformation. The Riemann hypothesis is equivalent to the assertion that Λ≤0. Newman, however, conjectured that Λ≥0, remarking, “the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so”. Andrade, Chang and Miller extended the machinery developed by Newman and Pólya to L-functions for function fields. In this setting we must consider a modified Newman's conjecture: supf∈F⁡Λf≥0, for F a family of L-functions.We extend their results by proving this modified Newman's conjecture for several families of L-functions. In contrast with previous work, we are able to exhibit specific L-functions for which ΛD=0, and thereby prove a stronger statement: maxL∈F⁡ΛL=0. Using geometric techniques, we show a certain deformed L-function must have a double root, which implies Λ=0. For a different family, we construct particular elliptic curves with p+1 points over Fp. By the Weil conjectures, this has either the maximum or minimum possible number of points over Fp2n. The fact that #E(Fp2n) attains the bound tells us that the associated L-function satisfies Λ=0. VideoFor a video summary of this paper, please visit http://youtu.be/hM6-pjq7Gi0.