Newman’s Conjecture in Function Fields

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 Polya to L-functions for function fields. In this setting we must consider a modified Newman’s conjecture: supfPF 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: maxLPF 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 #EpFp2nq attains the bound tells us that the associated L-function satisfies Λ 0.