Selberg Eigenvalue Conjecture Research Articles

Euler products of Selberg zeta functions in the critical strip

For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary Re s = 1, if they are attached with a nontrivial irreducible unitary representation. The region is determined by the size of the lowest eigenvalue of the Laplacian, and it extends to Re s $\geqslant$ 3/4 under Selberg's eigenvalue conjecture. More generally, for any unitary representation we establish the relation between the behavior of partial Euler products in the critical strip and the estimate of the error term in the prime geodesic theorem. For the trivial representation, the proof essentially exploits the idea of the celebrated work of Ramanujan.

A quadratic divisor problem and moments of the Riemann zeta-function

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length T^{\frac14 - \varepsilon} . Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length T^{\frac{1}{11}-\varepsilon} were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta-function, zero spacing and lower bounds for moments. Along the way we obtain an asymptotic formula for a quadratic divisor problem, where the condition am_1m_2 - bn_1n_2 = h is summed with smooth averaging on the variables m_1, m_2, n_1, n_2, h and arbitrary weights in the average on a,b . Using Watt's work allows us to exploit all averages simultaneously. It turns out that averaging over m_1, m_2, n_1, n_2, h right away in the quadratic divisor problem considerably simplifies the combinatorics of the main terms in the twisted fourth moment.

Twist‐minimal trace formulas and the Selberg eigenvalue conjecture

We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore, of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley from 1985.

Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables

We obtain new estimates on the level of distribution of the set \{Q(n)\} where Q\in\mathbb Z[X] is irreducible quadratic, for well-factorable moduli, improving a result due to Iwaniec. As a by-product of our arguments, we study the Chebyshev problem of estimating max \{P^+(n^2-D), n\leq x\} and make explicit, in Deshouillers–Iwaniec's state-of-the-art result, the dependence on the Selberg eigenvalue conjecture. Combined with the construction of an upper-bound sieve for numbers free of large factors, we obtain new upper bounds for the quantity \Psi_Q(x, y) = |\{n\leq x: p\mid Q(n)\Rightarrow p\leq y\}| for Q\in\mathbb Z[X] linear or quadratic.

The Selberg trace formula as a Dirichlet series

Abstract We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N > 1 ${N>1}$ .

Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$

We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a~Hecke congruence subgroup $\Gamma\leq SL(2,{\mathbb Z}[i])$, and correspond to exceptional eigenvalues of the Laplace operator on the space $L^2(\Gamma\backslash SL(2,{\mathbb C})/SU(2))$. These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a~proof of one such application, which is an upper bound for a~sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-functions with groessencharakters). Our proofs make extensive use of Lokvenec-Guleska's generalisation of the Bruggeman-Motohashi summation formulae for $PSL(2,{\mathbb Z}[i])\backslash PSL(2,{\mathbb C})$. We also employ a~bound of Kim and Shahidi for the first eigenvalues of the relevant Laplace operators, and an `unweighted' spectral large sieve inequality (our proof of which is to appear separately).

Non-split sums of coefficients of GL(2)-automorphic forms

Given a cuspidal automorphic form π on GL2, we study smoothed sums of the form $$\sum\nolimits_n {{a_\pi }({n^2} + d)V({n \over x})} $$ . The error term we get is sharp in that it is uniform in both d and Y and depends directly on bounds towards Ramanujan for forms of half-integral weight and Selberg eigenvalue conjecture. Moreover, we identify (at least in the case where the level is square-free) the main term as a simple factor times the residue as s = 1 of the symmetric square L-function L(s, sym2 π). In particular there is no main term unless d > 0 and π is a dihedral form.

On Selberg's small eigenvalue conjecture and residual eigenvalues

We show that Selberg's eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbation theory and number theory.

Numerical computations with the trace formula and the Selberg eigenvalue conjecture

We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [M. N. Huxley, Introduction to Kloostermania, in: Elementary and analytic theory of numbers, Banach Center Publ. 17, Warsaw (1985), 217–306.]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1/4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois representations of small squarefree conductor.