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On the zeros of the Epstein zeta-function on the critical line. (English. Russian original) Zbl 1329.11099
Russ. Math. Surv. 70, No. 4, 785-787 (2015); translation from Usp. Mat. Nauk 70, No. 4, 213-214 (2015).
From the text: Let \(Q(n,m)=an^2+bnm+c m^2\) be a positive-definite quadratic form with integer coefficients, where the fundamental discriminant \(-D=b^2-4ac\) is \(<0\). We consider the Epstein zeta-function \(\zeta_{Q}(s)\), which is given for \(\operatorname{Re}(s)> 1\) by the series
\[ \zeta_{Q}(s)= \sum_{(m,n) \neq (0,0)} (Q(n,m))^{-s}. \]
It is well known that \(\zeta_{Q}(s)\) can be represented by a linear combination of \(L\)-functions with ideal class group Hecke characters of the imaginary quadratic extension \(\mathbb Q (\sqrt{-D})\) of the rational field. Namely,
\[ \zeta_{Q}(s)= \frac{\varepsilon_{-D}}{h(-D)} \sum_{j=1}^h \overline{\psi_j(I_Q)}L(s, \psi_j), \]
where \(h=h(-D)\) is the class number, \(\varepsilon_{-D}\) is the number of units, \(I_Q = (a, b-i\sqrt{D}/2)\), and the Hecke \(L\)-functions are given by the following Euler product over all prime ideals:
\[ L(s, \psi_j) = \prod_{\mathfrak p} \left(1 - \frac{\psi_j(\mathfrak p)}{N\mathfrak p^s} \right)^{-1}. \]
We prove the following unconditional result.
Theorem 1. A positive proportion of non-trivial zeros of \(\zeta_{Q}(s)\) lie on the critical line \(\operatorname{Re}(s)= 1/2\).

MSC:
11M41 Other Dirichlet series and zeta functions
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[1] H. Davenport and H. Heilbronn 1936 J. Lond. Math. Soc.S1-11 3 181–185 · Zbl 0014.21601
[2] H. Davenport and H. Heilbronn 1936 J. Lond. Math. Soc.S1-11 4 307–312 · Zbl 0015.19802
[3] С. М. Воронин 1976 Теория чисел, математический анализ и их приложения Тр. МИАН СССР 142 135–147 · Zbl 1241.68050
[4] English transl. S. M. Voronin 1979 Proc. Steklov Inst. Math.142 143–155
[5] A. Selberg 1942 Skr. Norske Vid. Akad. Oslo I1942 10 59 pp.
[6] J. L. Hafner 1983 Math. Ann.264 21–37 · Zbl 0497.10018
[7] И. С. Резвякова 2010 Матем. заметки88 3 456–475
[8] English transl. I. S. Rezvyakova 2010 Math. Notes88 3 423–439 · Zbl 1257.11083
[9] A. Selberg 1999 Linear combinations of L-functions and zeros on the critical line 18 pp. http://www.msri.org/realvideo/ln/msri/1999/random/selberg/1/main.html
[10] Э. Бомбьери, А. Гош 2011 УМН66 2(398) 15–66
[11] English transl. E. Bombieri and A. Ghosh 2011 Russian Math. Surveys66 2 221–270
[12] I. S. Rezvyakova 2015 On the zeros of linear combinations of degree two L-functions on the critical line. Selberg’s approach, Manuscript 22 pp.
[13] A. Selberg 1946 Arch. Math. Naturvid.48 5 89–155
[14] A. Ghosh 1983 J. Number Theory17 1 93–102 · Zbl 0511.10030
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