## On the value distribution of the Epstein zeta function in the critical strip.(English)Zbl 1285.11065

The Epstein zeta function of a lattice $$L \subset \mathbb{R}^n$$ of covolume $$1$$ is defined as the meromorphic continuation to all $$s \in \mathbb{C}$$ of $E_n(L, s) = \sum_{v \in L, v \neq 0} \|v\|^{-2s}$ which converges absolutely for $$\mathrm{Re}(s) > n/2$$. The only pole of $$E_n(L, s)$$ is at $$s=n/2$$ (simple) with residue $$\pi^{n/2} \Gamma(n/2)^{-1}$$. Moreover it satisfies a function equation under $$s \mapsto n/2 - s$$. Call $$\{ s : 0 < \text{Re}(s) < n/2 \}$$ the critical strip for $$E_n(L, s)$$. The first goal of this paper is to study the value distribution of $$E_n(L, cn)$$ as $$n \to \infty$$, where $$0 < c < 1/2$$; here $$L$$ is viewed as a random lattice of covolume $$1$$.
The first result (Theorem 1.1) of this paper describes the limit distribution of $$V_n^{-2c} E_n(\cdot, cn)$$ as $$n \to \infty$$ (regarded as a random variable), where $$c$$ lies in a closed interval in $$(1/4, 1/2)$$ and $$V_n := \mathrm{vol}(\mathbb{S}^n)$$: it equals $H(c) := \int_{V > 0} V^{-2c} \,dR(V)$ where $$R(V)$$ is explicitly given in terms of a Poisson process on $$\mathbb{R}_{>0}$$ with constant intensity $$1/2$$. As a byproduct (Theorem 1.3), the author obtains $$R_n(V) \ll V^{1/2} (\log V)^{\varepsilon + 3/2}$$ for almost every $$L$$, where $$R_n$$ is essentially the remainder term in the generalized circle problem.
By subtracting the singularities from both $$E_n$$ and $$H$$ at $$c=1/2$$, the author extends the aforementioned result to intervals of the form $$[1/4 + \varepsilon, 1/2]$$ (Theorem 6.2). These results are closely related to P. Sarnak and A. Strömbergsson [Invent. Math. 165, No. 1, 115–151 (2006; Zbl 1145.11033)] on the heights $$h_n(L)$$ of flat tori $$\mathbb{R}^n/L$$ and the minima of $$E_n(\cdot, s)$$. For example, the Theorem 1.4 in this paper sharpens the description of $$h_n(L)$$ as $$n \to \infty$$ in Theorem 3 of loc. cit. Such results are also of interest in arithmetic.

### MSC:

 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) 11P21 Lattice points in specified regions 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

### Keywords:

Epstein zeta function; random lattice

Zbl 1145.11033
Full Text:

### References:

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