Number variance of the zeros of the Epstein zeta functions
Let \(\zeta (s,Q)\) be the Epstein zeta-function of an integral form \(Q= ax^ 2+ bxy+ cy^ 2\), of discriminant \(d\), and let \(k= \sqrt{d} /2a\). It was shown by \textit{H. M. Stark} [Mathematika 14, 47-55 (1967; Zbl 0242.12010)] that the zeros of \(\zeta (s,Q)\) are very well behaved for \(| \text{Im}(s)| \leq 2k\). In particular if \(N(T)\) is the number of zeros with \(-1<\sigma <2\) and \(0\leq t\leq T\), then \(N(T)= {T\over \pi}\log ({{kT} \over {\pi e}})+ S(T)\), with \(S(T)= {1\over\pi} \arg \zeta({1\over 2}+ 2iT)+ O(1)\) for \(T\leq 2k\). The present paper shows that \(\int_ T^{2T-2} \{S(t+ \alpha)- S(t)\}^ 2 dt \ll T\) uniformly for \(T\leq k\) and \(\alpha\leq 1\). If the GUE law for the distribution of the zeros held for \(T\leq k\) the above integral would have been asymptotically \(CT\log (kT)\) for \(1\ll \alpha\leq 1\). In fact the paper proves an asymptotic formula for \[ \int_ T^{2T} \biggl( \arg {{\zeta (\sigma+ i(t+ h))} \over {\zeta (\sigma+ it)}} \biggr)^ 2 dt, \] uniformly for \(h\leq T\), for any fixed \(\sigma\in ({1\over 2}, 1]\).
scientific article; zbMATH DE number 701469
scientific article