TY - JOUR
T1 - Newman's conjecture in function fields
AU - Chang, Alan
AU - Mehrle, David
AU - Miller, Steven J.
AU - Reiter, Tomer
AU - Stahl, Joseph
AU - Yott, Dylan
N1 - Publisher Copyright:
© 2015 Elsevier Inc.
PY - 2015/7/4
Y1 - 2015/7/4
N2 - 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.
AB - 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.
KW - Function fields
KW - Newman's conjecture
KW - Zeros of the L-functions
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U2 - 10.1016/j.jnt.2015.04.028
DO - 10.1016/j.jnt.2015.04.028
M3 - Article
AN - SCOPUS:84947243564
SN - 0022-314X
VL - 157
SP - 154
EP - 169
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -