Non-vanishing of high derivatives of Dirichlet \(L\)-functions at the central point
type: Item
instance of: scholarly article
title: Non-vanishing of high derivatives of Dirichlet \(L\)-functions at the central point
zbMATH Open document ID: 1001.11032
published in: Journal of Number Theory
publication date: 2002-12-17T00:00:00Z
review text: Let \(\chi\) run over all even Dirichlet characters to modulus \(q\), and let \(\Lambda(s,\chi)\) be the `completed' \(L\)-function. It is shown that if \(q\) is large and \(k\) is a fixed positive integer, then a proportion at least \(c_k\) of the derivatives \(\Lambda^{(k)} (\frac 12,\chi)\) are nonzero, with constants \(c_k= \frac 23- O(k^{-2})\). The authors say that a similar result can be proved for odd characters. One can think of this as being analogous to \textit{B. Conrey}'s result [J. Number Theory 16, 49-74 (1983; Zbl 0502.10022)] for the Riemann \(\xi\)-function, which states that a proportion \(1- O(k^{-2})\) of the zeros of \(\xi^{(k)}(s)\) lie on the critical line. It is however disappointing that the constants \(c_k\) only tend to \(\frac 23\) in the present case. The proof depends on a comparison of the first and second moments of \(\Lambda^{(k)} (\frac 12,\chi) M_k (\frac 12,\chi)\), where \(M(s,\chi)\) is a mollifier, taken to have the form \[ M^*(s,\chi)+ (-1)^k \frac{\overline{\tau(\chi)}} {\sqrt{q}} M^*(1-s, \overline{\chi}) \] for a suitable Dirichlet polynomial \(M^*(s,\chi)\).
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11M20
zbMATH DE Number: 1420268
zbMATH Keywords: Dirichlet \(L\)-functions
zbMATH Keywords: higher derivatives
zbMATH Keywords: nonvanishing
zbMATH Keywords: central point
zbMATH Keywords: mollifier
author: Philippe Michel
author: Jeffrey M. Vanderkam
MaRDI profile type: Publication
full work available at URL: jnth.1999.2460
OpenAlex ID: W2021919514
Wikidata QID: Q57268263
cites work: Zeros of Dirichlet $L$-functions
cites work: Zeros of derivatives of Riemann's xi-function on the critical line
cites work: Q4252239
cites work: The non-vanishing of central values of automorphic \(L\)-functions and Landau-Siegel zeros
cites work: Zeroes of zeta functions and symmetry
cites work: Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip
cites work: Mean‐values of the Riemann zeta‐function
cites work: The rank of quotients of \(J_0(N)\)
DOI: 10.1006/JNTH.1999.2460
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description: scientific article; zbMATH DE number 1420268