Articles

PROPORTION OF DISTINCT ZEROS OF A DIRICHLET L-SERIES

  • Xinhua QIN ,
  • Xiaosheng WU
Expand
  • School of Mathematics, Hefei University of Technology, Hefei 230009, China

Received date: 2022-07-14

  Online published: 2022-12-16

Supported by

This work was supported in part by NSFC (11871187) and the Fundamental Research Funds for the Central Universities of China.

Abstract

By studying the distribution of zeros of combinations of a Dirichlet L-function and its first-order derivative, we prove that every Dirichlet L-function has more than 66.7934% distinct zeros.

Cite this article

Xinhua QIN , Xiaosheng WU . PROPORTION OF DISTINCT ZEROS OF A DIRICHLET L-SERIES[J]. Acta mathematica scientia, Series B, 2022 , 42(6) : 2367 -2376 . DOI: 10.1007/s10473-022-0610-1

References

[1] Anderson R J. Simple zeros of the Riemann zeta-function. J Number Theory, 1983, 17: 176-182
[2] Bauer P. Zeros of Dirichlet L-series on the critical line. Acta Arith, 2000, 93: 37-52
[3] Balasubramanian R, Conrey J B, Heath-Brown D R. Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. J Reine Angew Math, 1985, 357: 161-181
[4] Bui H, Conrey J B, Young M. More than 41% of the zeros of the zeta function are on the critical line. Acta Arith, 2011, 150(1): 35-64
[5] Conrey J B. More than two fifths of the zeros of the Riemann zeta function are on the critical line. J Reine Angew Math, 1989, 339: 1-26
[6] Conrey J B, Iwaniec H, Soundararajan K. Asymptotic large sieve. arXiv: 1105.1176
[7] Conrey J B, Iwaniec H, Soundararajan K. Critical zeros of Dirichlet L-function. J Reine Angew Math, 2013, 681: 175-198
[8] Farmer D. Counting distinct zeros of the Riemann zeta-function. Electron J Combin, 1995, 2: Art 1
[9] Feng S. Zeros of the Riemann zeta function on the critical line. J Number Theory, 2012, 132(4): 511-542
[10] Heath-Brown D R. Simple zeros of the Riemann zeta-function on the critical line. Bull Lond Math Soc, 1979, 11: 17-18
[11] Levinson N. More than one third of zeros of Riemann’s zeta-function are on σ = 1/2. Adv Math, 1974, 13: 383-436
[12] Littlewood J E. On the zeros of the Riemann Zeta-function. Cambridge Phil Soc Proc, 1924, 22: 295-318
[13] Pratt K, Robles N, Zaharescu A, Zeindler D. More than five-twelfths of the zeros of are on the critical line. Res Math Sci, 2020, 7(1): Art 2
[14] Titchmarsh E C. The theory of the Riemann zeta-function. revised by D. R. Heath-Brown. 2nd Ed. Oxford: Clarendon Press, 1986
[15] Wu X. Distinct zeros of the Riemann Zeta-function. Quart J Math, 2015, 66: 759-771
[16] Wu X. Distinct zeros and simple zeros for the family of Dirichlet L-functions. Quart J Math, 2016, 67: 757-779
[17] Wu X. The twisted mean square and critical zeros of Dirichlet L-functions. Math Z, 2019, 293: 825-865
Options
Outlines

/