Newman's conjecture, zeros of the L-functions, function fields
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2014
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| _version_ | 1866929630766694400 |
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| author | Chang, Alan Mehrle, David Miller, Steven J. Reiter, Tomer Stahl, Joseph Yott, Dylan |
| author_facet | Chang, Alan Mehrle, David Miller, Steven J. Reiter, Tomer Stahl, Joseph Yott, Dylan |
| contents | 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 $Λ\leq 0$. Newman, however, conjectured that $Λ\geq 0$, 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: $\sup_{f\in\mathcal{F}} Λ_f \geq 0$, 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 $Λ_D = 0$, and thereby prove a stronger statement: $\max_{L\in\mathcal{F}} Λ_L = 0$. 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 $\mathbb{F}_p$. 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 $Λ= 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1411_2071 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Newman's conjecture, zeros of the L-functions, function fields Chang, Alan Mehrle, David Miller, Steven J. Reiter, Tomer Stahl, Joseph Yott, Dylan Number Theory 11M20, 11M26, 11Y35, 11Y60, 14G10 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 $Λ\leq 0$. Newman, however, conjectured that $Λ\geq 0$, 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: $\sup_{f\in\mathcal{F}} Λ_f \geq 0$, 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 $Λ_D = 0$, and thereby prove a stronger statement: $\max_{L\in\mathcal{F}} Λ_L = 0$. 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 $\mathbb{F}_p$. 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 $Λ= 0$. |
| title | Newman's conjecture, zeros of the L-functions, function fields |
| topic | Number Theory 11M20, 11M26, 11Y35, 11Y60, 14G10 |
| url | https://arxiv.org/abs/1411.2071 |