Anders Södergren

On the distribution of angles between the N shortest vectors in a random lattice

We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random n-dimensional lattice as n→∞. Moreover, we interpret the result in terms of eig ...

On the uniform equidistribution of closed horospheres in hyperbolic manifolds

We prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. This extends earlier results by Hejhal [10] and Strömbergsson [32] in dimension 2. Our proofs use spectral methods, and lead to precise estimates on the rate of convergence to equidistribution.

On the value distribution and moments of the Epstein zeta function to the right of the critical strip

We study the Epstein zeta function E-n(L, S) for s > n/2 and a random lattice L of large dimension n. For any fixed c > 1/2 we determine the value distribution and moments of E-n(., cn) (suit ...

Low-lying zeros of quadratic Dirichlet L-functions: lower order terms for extended support

We study the 1-level density of low-lying zeros of Dirichlet L-functions attached to real primitive characters of conductor at most X. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of log X, which is valid when the support of the Fourier transform of the corresponding even test function phi is contained in (-2, 2). We uncover a phase transition when the supremum sigma of the support of (phi) over cap reaches 1, both in the main term and in the lower order terms. A new lower order term appearing at sigma = 1 involves the quantity (phi) over cap (1), and is analogous to a lower order term which was isolated by Rudnick in the function field case.

Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture

We study the low-lying zeros of L -functions attached to quadratic twists of a given elliptic curve E defined over

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

\mathbb{Q}

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

. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L -functions Ratios Conjecture.

Angles in hyperbolic lattices: The pair correlation density

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 the pole at c = 1 2 we even obtain a limit result on the whole interval [ 1 4 + ε, 1 2 ], and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson [15] concerning the height function hn(L) of the flat torus R/L: The random variable n { hn(L)− (log(4π)− γ+1) } + log n has a limit distribution as n → ∞, which we give explicitly. Finally we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice L ⊂ R for which En(L, s) has no zeros in (0,∞).We study the value distribution of the Epstein zeta function E-n (L, s) for 0 infinity, we prove that the random variable V-n(-2c) E-n(.,cn) has a limit distribution, which we give explicitly (here V-n is the volume of the n-dimensional unit ball). More generally, for any fixed epsilon > 0, we determine the limit distribution of the random function c bar right arrow V-n(-2c) E-n(., cn), c epsilon [1/4 + epsilon, 1/2 - epsilon]. After compensating for the pole at c = 1/2, we even obtain a limit result on the whole interval [1/4 + epsilon, 1/2], and as a special case we deduce the following strengthening of a result by Sarnak and Strombergsson concerning the height function h(n) (L) of the flat torus R-n/L: the random variable n{h(n) (L) - (log(4 pi) - gamma + 1)} + log n has a limit distribution as n -> infinity, which we give explicitly. Finally, we discuss a question posed by Sarnak and Strombergsson as to whether there exists a lattice L subset of R-n for which E-n(L, s) has no zeros in (0, infinity).