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 - Funding Information: This work was supported by NSF grants DMS-1347804 , DMS-1265673 , Williams College , and the PROMYS program. The authors thank Noam Elkies, David Geraghty, Rob Pollack, Glenn Stevens, and Keith Conrad for their insightful comments and support. We would also like to thank the referees for their careful reading of drafts and their helpful suggestions. 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 - https://doi.org/10.1016/j.jnt.2015.04.028
DO - https://doi.org/10.1016/j.jnt.2015.04.028
M3 - Article
SN - 0022-314X
VL - 157
SP - 154
EP - 169
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -