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Non-vanishing of high derivatives of Dirichlet \(L\)-functions at the central point. (English) Zbl 1001.11032

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 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 \(M^*(s,\chi)\).

MSC:

11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)

Citations:

Zbl 0502.10022
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References:

[1] Balasubramanian, R.; Murty, V. K., Zeros of Dirichlet \(L\) functions, Ann. Sci. École. Norm. Sup., 25, 567-615 (1992) · Zbl 0771.11033
[2] Conrey, B., Zeros of derivatives of Riemann’s \(ξ\)-function on the critical line, J. Number Theory, 16, 49-74 (1983) · Zbl 0502.10022
[3] H. Iwaniec, and, P. Sarnak, Dirichlet \(L\); H. Iwaniec, and, P. Sarnak, Dirichlet \(L\) · Zbl 0929.11025
[4] H. Iwaniec, and, P. Sarnak, Non-vanishing of central values of automorphic \(L\); H. Iwaniec, and, P. Sarnak, Non-vanishing of central values of automorphic \(L\) · Zbl 0992.11037
[5] Katz, N.; Sarnak, P., Zeros of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.), 36, 1-26 (1999) · Zbl 0921.11047
[6] E. Kowalski, and, P. Michel, The analytic rank of \(J_0qL\); E. Kowalski, and, P. Michel, The analytic rank of \(J_0qL\)
[7] E. Kowalski, and, P. Michel, A lower bound for the rank of \(J_0q\); E. Kowalski, and, P. Michel, A lower bound for the rank of \(J_0q\)
[8] E. Kowalski, P. Michel, and, J. VanderKam, Non-vanishing of high derivatives of automorphic \(L\); E. Kowalski, P. Michel, and, J. VanderKam, Non-vanishing of high derivatives of automorphic \(L\) · Zbl 1020.11033
[9] Soundararajan, K., Mean values of the Riemann zeta-function, Mathematika, 42, 158-174 (1995) · Zbl 0830.11032
[10] J. VanderKam, The rank of quotients of \(J_0N\); J. VanderKam, The rank of quotients of \(J_0N\)
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