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: supf∈FΛf ≥ 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∈FΛL = 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. Video. For a video summary of this paper, please visit http://youtu.be/hM6-pjq7Gi0.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Function fields
- Newman's conjecture
- Zeros of the L-functions