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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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##### References:
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