# On the value distribution of the Epstein zeta function in the critical strip

@inproceedings{Sodergren2011OnTV,
title={On the value distribution of the Epstein zeta function in the critical strip},
author={Anders Sodergren},
year={2011}
}
We study the value distribution of the Epstein zeta function En(L, s) for 0 < s < n 2 and a random lattice L of large dimension n. For any fixed c ∈ ( 1 4 , 1 2 ) and n → ∞, we prove that the random variable V −2c n En(·, cn) has a limit distribution, which we give explicitly (here Vn is the volume of the ndimensional unit ball). More generally, for any fixed ε > 0 we determine the limit distribution of the random function c 7→ V −2c n En(·, cn), c ∈ [ 1 4 + ε, 1 2 − ε]. After compensating for…
3 Citations

## Figures from this paper

. We formulate and prove the extension of the Rogers integral formula ([23]) to the adeles of number ﬁelds. We also prove the second moment formulas for a few important cases, enabling a number of

### Random lattice vectors in a set of size O(n)

We adopt the sieve ideas of Schmidt and S\"odergren in order to study the statistics of vectors of a random lattice of dimension n contained in a set of volume O(n). We also give some sporadic

### On the universality of the Epstein zeta function

• Mathematics
Commentarii Mathematici Helvetici
• 2020
We study universality properties of the Epstein zeta function $E_n(L,s)$ for lattices $L$ of large dimension $n$ and suitable regions of complex numbers $s$. Our main result is that, as $n\to\infty$,

## References

SHOWING 1-10 OF 28 REFERENCES

### On the Poisson distribution of lengths of lattice vectors in a random lattice

We prove that the volumes determined by the lengths of the non-zero vectors ±x in a random lattice L of covolume 1 define a stochastic process that, as the dimension n tends to infinity, converges

### Lattice point problems and distribution of values of quadratic forms

• Mathematics
• 1999
For d-dimensional irrational ellipsoids E with d > 9 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order o(rd-2). The

### Convergence of probability measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the

### Stable non-Gaussian random processes

The asymptotic behaviour of (Yn, n e N) is of fundamental importance in probability theory. Indeed, if the Xj have common mean fi and variance a, then by taking each an = n/u and b„ = n a, the

### On the zeros of Epstein's zeta function

Let Q ( x, y ) = ax 2 + bxy + cy 2 be a positive definite quadratic form with discriminant d = b 2 – 4 ac . The Epstein zeta function associated with Q is given by where Σ′ means the sum is over all

### Lattice point problems and values of quadratic forms

For d-dimensional ellipsoids E with d≥5 we show that the number of lattice points in rE is approximated by the volume of rE, as r tends to infinity, up to an error of order $\mathcal{O}(r^{d-2})$ for

### Mean values over the space of lattices

t . Various methods have been used for calculating the mean value of a function, defined for all lattices of determinant 1, over some or all the lattices of determinant 1. I t is accepted tha t the