On the zeros of the Epstein zeta functions
The Epstein zeta function is a special case of an Eisenstein series for the general linear group. Given a positive definite quadratic form \(Q(x)\), the Epstein zeta function \(E(Q,s)\) is the sum of \(Q(x)^{-s}\) over non-zero \(x\) in the integer lattice in Euclidean \(n\)-space, assuming \(\text{Re}(s) > n/2\). It is interesting to compare Epstein and Riemann zeta functions. For example, \textit{H. Davenport} and \textit{H. Heilbronn} [J. Lond. Math. Soc. 11, 181-185, 307-312 (1936; Zbl 0014.21601, Zbl 0015.19802)] show that in certain cases (when \(n=2\)), the Epstein zeta function has infinitely many zeros in \(\text{Re}(s)>1\); i.e., outside the critical strip. Nevertheless \textit{H. M. Stark} [Mathematika 14, 47-55 (1967; Zbl 0242.12012)] has shown that when \(n=2\) and \(Q(x,y)=ax^2+bxy+cy^2\), for large \(k=\sqrt{|b^2-4ac|}/2a\), one has a \(k\)-analogue of the Riemann-von Mangoldt formula for the number of zeros of \(E(Q,s)\) in a rectangular box. This paper mostly concerns the case \(G_d(s)=E(Q_d,s)\), where \(n=4\) and \(Q_d(x,y,u,v) = x^2+y^2 +d(u^2 + v^2)\). Theorem 1 is a Chowla-Selberg formula of the sort that comes out of a Fourier expansion of an Eisenstein series (see \textit{A. Terras} [Harmonic analysis on symmetric spaces and applications, Vol. I (1985; Zbl 0574.10029), p. 209, Vol. II (1988; Zbl 0668.10033), p. 223 ff.]). Theorem 4 says that if \(d\) is sufficiently large and \(k=d^{1/2}\) then all zeros of \(G_d(s)\) with \(-1/2 < \text{Re}(s) < 5/2\) and \(-k\leq \text{Im}(s) \leq k\), are simple, and with the exception of two real zeros between \(0\) and \(2\), all are on the line \(\text{Re}(s)=1\). It had earlier been noted that \(G_d(s)\) has no zeros in \(\text{Re}(s) \geq 5/2\), when \(k > 2.94\). Theorem 5 gives a Riemann-von Mangoldt type formula for \(G_d(s)\) when \(d\) is sufficiently large. Theorem 6 shows the failure of the GUE law or Berry conjecture in the sense of \textit{M. V. Berry} [Nonlinearity 1, 399-407 (1988; Zbl 0664.10022)], for \(G_d(s)\) when \(d\) is sufficiently large. There are some other references for behavior of Epstein zeta functions on the real line. The case \(n=3\) is considered in [\textit{A. Terras}, Ill. J. Math. 23, 1-14 (1979; Zbl 0392.10024)] and the general case in [\textit{A. Terras}, J. Number Theory 12, 258-272 (1980; Zbl 0432.10010)].
Eisenstein series for the general linear group
scientific article; zbMATH DE number 1062463
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