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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 \(\operatorname{Re}(s)= 1/2\).

11M41 Other Dirichlet series and zeta functions
Full Text: DOI
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