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Newman's conjecture, zeros of the L-functions, function fields | | Alan Chang
; David Mehrle
; Steven J. Miller
; Tomer Reiter
; Joseph Stahl
; Dylan Yott
; | | Date: |
8 Nov 2014 | | Abstract: | De Bruijn and Newman introduced a deformation of the completed Riemann zeta
function $zeta$, and proved there is a real constant $Lambda$ which encodes
the movement of the nontrivial zeros of $zeta$ under the deformation. The
Riemann hypothesis is equivalent to the assertion that $Lambdaleq 0$. Newman,
however, conjectured that $Lambdageq 0$, 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: $sup_{finmathcal{F}} Lambda_f geq
0$, for $mathcal{F}$ 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 $Lambda_D = 0$, and thereby prove
a stronger statement: $max_{Linmathcal{F}} Lambda_L = 0$. Using geometric
techniques, we show a certain deformed $L$-function must have a double root,
which implies $Lambda = 0$. For a different family, we construct particular
elliptic curves with $p + 1$ points over $mathbb{F}_p$. By the Weil
conjectures, this has either the maximum or minimum possible number of points
over $mathbb{F}_{p^{2n}}$. The fact that $#E(mathbb{F}_{p^{2n}})$ attains
the bound tells us that the associated $L$-function satisfies $Lambda = 0$. | | Source: | arXiv, 1411.2071 | | Services: | Forum | Review | PDF | Favorites | |
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