Newman's conjecture in function fields.

Text De 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: sup f ∈ F ⁡ Λ f ≥ 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 Λ D = 0 , and thereby prove a stronger statement: max L ∈ F ⁡ Λ L = 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 F p . By the Weil conjectures, this has either the maximum or minimum possible number of points over F p 2 n . The fact that # E ( F p 2 n ) attains the bound tells us that the associated L -function satisfies Λ = 0 . Video For a video summary of this paper, please visit http://youtu.be/hM6-pjq7Gi0 . [ABSTRACT FROM AUTHOR]