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Mat. Zametki, 2010, Volume 88, Issue 3, Pages 456–475 (Mi mz8817)  

This article is cited in 6 scientific papers (total in 6 papers)

On the Zeros on the Critical Line of $L$-Functions Corresponding to Automorphic Cusp Forms

I. S. Rezvyakova

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider an automorphic cusp form of integer weight $k\ge1$, which is the eigenfunction of all Hecke operators. It is proved that, for the $L$-series whose coefficients correspond to the Fourier coefficients of such an automorphic form, the positive fraction of nontrivial zeros lies on the critical line.

Keywords: automorphic cusp form, Riemann zeta function, Riemann hypothesis, Hecke operator, $L$-function, Jutila's circle method

DOI: https://doi.org/10.4213/mzm8817

Full text: PDF file (583 kB)
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English version:
Mathematical Notes, 2010, 88:3, 423–439

Bibliographic databases:

Document Type: Article
UDC: 511
Received: 21.09.2009

Citation: I. S. Rezvyakova, “On the Zeros on the Critical Line of $L$-Functions Corresponding to Automorphic Cusp Forms”, Mat. Zametki, 88:3 (2010), 456–475; Math. Notes, 88:3 (2010), 423–439

Citation in format AMSBIB
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\pages 423--439
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. S. Rezvyakova, “Zeros of linear combinations of Hecke $L$-functions on the critical line”, Izv. Math., 74:6 (2010), 1277–1314  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Milinovich M.B., Ng N., “Simple Zeros of Modular l-Functions”, Proc. London Math. Soc., 109:6 (2014), 1465–1506  crossref  mathscinet  zmath  isi  scopus
    3. I. S. Rezvyakova, “On the zeros of the Epstein zeta-function on the critical line”, Russian Math. Surveys, 70:4 (2015), 785–787  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Rezvyakova I.S., “Selberg'S Method in the Problem About the Zeros of Linear Combinations of l-Functions on the Critical Line”, Dokl. Math., 92:1 (2015), 448–451  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    5. I. S. Rezvyakova, “On the zeros of linear combinations of L-functions of degree two on the critical line. Selberg's approach”, Izv. Math., 80:3 (2016), 602–622  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. M. A. Korolev, “On Anatolii Alekseevich Karatsuba's works written in the 1990s and 2000s”, Proc. Steklov Inst. Math., 299 (2017), 1–43  mathnet  crossref  crossref  isi  elib  elib
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