We study the one-level density of low-lying zeros in the family of Maass form L-functions of prime level N tending to infinity. Generalizing the influential work of Iwaniec, Luo and Sarnak to this context, Alpoge et al. have proven the Katz-Sarnak prediction for test functions whose Fourier transform is supported in (-3/2,3/2). In this paper, we extend the unconditional admissible support to (-15/8,15/8). The key tools in our approach are analytic estimates for integrals appearing in the Kutznetsov trace formula, as well as a reduction to bounds on Dirichlet polynomials, which eventually are obtained from the large sieve and the fourth moment bound for Dirichlet L-functions. Assuming the Grand Density Conjecture, we extend the admissible support to (-2,2). In addition, we show that the same techniques also allow for an unconditional improvement of the admissible support in the corresponding family of L-functions attached to holomorphic forms.

We study the harmonically weighted one-level density of low-lying zeros of $L$-functions in the family of holomorpic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz--Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(-\tfrac32,\tfrac32)$. In this paper, we extend this admissible support to $(-\Theta_k,\Theta_k)$, where $\Theta_2 = 1.866\dots$ and $\Theta_k$ tends monotonically to $2$ as $k$ tends to infinity. This is asymptotically as good as the best known GRH result. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$-functions.

The Dedekind zeta functions of infinitely many non-Galois cubic fields have negative central values.

We prove a generalized version of Rogers' mean value formula in the space X_n of unimodular lattices in R^n, which gives the mean value of a multiple sum over a lattice L and its dual L^*. As an application, we prove that for L random with respect to the SL(n,R)-invariant probability measure, in the limit of large dimension n, the volumes determined by the lengths of the non-zero vectors ± x in L on the one hand, and the non-zero vectors ± x' in L^* on the other hand, converge weakly to two independent Poisson processes on the positive real line, both with intensity 1/2.

In this paper, we obtain a precise formula for the one-level density of L -functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of this shape has appeared in previously studied families. The presence of this new term allows us to deduce an omega result for cubic field counting functions, under the assumption of the Generalised Riemann Hypothesis. We also investigate the associated L -functions Ratios Conjecture and find that it does not predict this new lower-order term. Taking into account the secondary term in Roberts’s conjecture, we refine the Ratios Conjecture to one which captures this new term. Finally, we show that any improvement in the exponent of the error term of the recent Bhargava–Taniguchi–Thorne cubic field counting estimate would imply that the best possible error term in the refined Ratios Conjecture is $O_\varepsilon (X^{-\frac 13+\varepsilon })$ . This is in opposition with all previously studied families in which the expected error in the Ratios Conjecture prediction for the one-level density is $O_\varepsilon (X^{-\frac 12+\varepsilon })$ .

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over Fq(T ). In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.

We study low-lying zeros of $L$-functions attached to holomorphic cusp forms of level $1$ and large weight. In this family, the Katz--Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions $\phi$ satisfying the condition supp$(\widehat \phi) \subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\widehat \phi$ reaches the point $1$. In particular the first of these terms involves the quantity $\widehat \phi(1)$ which appeared in previous work of Fouvry--Iwaniec and Rudnick in symplectic families. Our approach involves a careful analysis of the Petersson formula and circumvents the assumption of GRH for $\text{GL}(2)$ automorphic $L$-functions. Finally, when supp$(\widehat \phi)\subset (-1,1)$ we obtain an unconditional estimate which is significantly more precise than the prediction of the $L$-functions Ratios Conjecture.

In this paper, we consider the family $\{L_j(s)\}_{j=1}^{\infty}$ of $L$-functions associated to an orthonormal basis $\{u_j\}_{j=1}^{\infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues $\{\lambda_j=\kappa_{j}^{2}+1/4\}_{j=1}^{\infty}$. We prove the following effective non-vanishing result: At least $50 \%$ of the central values $L_j(1/2)$ with $\kappa_j \leq T$ do not vanish as $T\rightarrow \infty$. Furthermore, we establish effective non-vanishing results in short intervals.

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 -> infinity, E-n(L,s) is universal in the right half of the critical strip as L varies over all n-dimensional lattices L. The proof uses a novel combination of an approximation result for Dirichlet polynomials, a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same approach we also prove that, as n -> infinity, E-n(L-1,s) - E-n(L-2,s) is universal in the full half-plane to the right of the critical line as E-n(L,s) varies over all pairs of n-dimensional lattices. Finally, we prove a more classical universality result for E-n(L,s) in the s-variable valid for almost all lattices L of dimension n. As part of the proof we obtain a strong bound of E-n(L,s) on the critical line that is subconvex for n >= 5 and almost all n-dimensional lattices L.

We study various families of Artin L-functions attached to geometric parametrizations of number fields. In each case we find the Sato-Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.

In this note we study the error term R_{n,L}(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f(n) from the positive integers to the positive real line, tending to infinity with n but with subexponential growth. Then, the random function t -> (2f(n))^{-1/2} R_{n,L}(t f(n)) on the interval [0,1] converges in distribution to one-dimensional Brownian motion as n tends to infinity. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers' mean value formula. For the individual k:th moment of the variable (2f(n))^{-1/2} R_{n,L}(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)<

We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the lower order terms when the support of the Fourier transform of the corresponding test function reaches the point $1$. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case.

In this note we study, for a random lattice L of large dimension n, the supremum of the real parts of the zeros of the Epstein zeta function \(E_n(L,s)\) and prove that this random variable scaled by \(n^{-1}\) has a limit distribution, which we give explicitly. This limit distribution is studied in some detail; in particular we give an explicit formula for its distribution function. Furthermore, we obtain a limit distribution for the frequency of zeros of \(E_n(L,s)\) in vertical strips contained in the half-plane \(\mathfrak {R}s>\frac{n}{2}\).

It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in such hyperbolic lattices. Using this formula we determine, among other things, the asymptotic behavior of the density function in both the small and large variable limits. This extends earlier results by Boca, Pasol, Popa and Zaharescu and Kelmer and Kontorovich in dimension 2 to general dimension $n$. Our proofs use the decay of matrix coefficients together with a number of careful estimates, and lead to effective results with explicit rates.

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 $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.

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

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 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.

We study the Epstein zeta function $E_n(L,s)$ for $s>\frac{n}{2}$ and determine for fixed $c>\frac{1}{2}$ the value distribution and moments of $E_n(\cdot,cn)$ (suitably normalized) as $n\to\infty$. We further discuss the random function $c\mapsto E_n(\cdot,cn)$ for $c\in[A,B]$ with $\frac{1}{2}