Yuk-Kam Lau

A density theorem on automorphic L-functions and some applications

We establish a density theorem of automorphic L-functions and give some applications on the extreme values of these L-functions at s =1 and the distribution of the Hecke eigenvalue of holomorphic cusp forms. Contents

On modular signs

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first signchange of Hecke eigenvalues. Here we improve the recent estimate of Iwaniec, Kohnen and Sengupta.

A new bound K2/3+ɛ for Rankin-Selberg ℒ-functions for Hecke congruence subgroups

Let f be a holomorphic Hecke eigenform for Γ0(N ) of weight k, or a Maass eigenform for Γ0(N ) with Laplace eigenvalue 1/4 + k. Let g be a fixed holomorphic or Maass cusp form for Γ0(N ). A subconvexity bound for central values of the Rankin-Selberg L-function L(s, f ⊗ g) is proved in the k-aspect: L(1/2+ it, f ⊗ g) N ,g,t,e k, while a convexity bound is only k. This new bound improves earlier subconvexity bounds for these Rankin-Selberg L-functions by Sarnak, the authors, and Blomer. Techniques used include a result of Good, spectral large sieve, meromorphic continuation of a shifted convolution sum to −1/2 passing through all Laplace eigenvalues, and a weighted stationary phase argument.

On the mean square formula of the error term in the Dirichlet divisor problem

by Preissmann[17]. Further progress towards its size may be hard because, as shown in [11], F (x) = Ω−(x log x). In other words, the room for further improvement is not more than log x. It is anticipated that the omega result is closer to the true magnitude, in virtue of the almost all result in [21]. Very recently, though unable to improve the bound for F (x), Nowak[16] pushed forth the record in a closely analogous situation the circle problem. Let P (t) denote the error term in the circle problem. Owing to the similarity between their Voronoi series, the methodology (based on these series) to handle ∆(t) usually applies to P (t) and vice versa. Let

On the least quadratic non-residue

We prove that for almost all real primitive characters χd of modulus |d|, the least positive integer nχd at which χd takes a value not equal to 0 and 1 satisfies nχd ≪ log|d|, and give a quite precise estimate on the size of the exceptional set.

ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS

We show that, for every x exceeding some explicit bound depending only on k and N , there are at least C ( k , N ) x /log 17 x positive and negative coefficients a ( n ) with n ≤ x in the Fourier expansion of any non-zero cuspidal Hecke eigenform of even integral weight k ≥2 and squarefree level N that is a newform, where C ( k , N ) depends only on k and N . From this we deduce the existence of a sign change in a short interval.

On mean values of random multiplicative functions

Let P denote the set of primes and {f (p)} p∈P be a sequence of independent Bernoulli random variables taking values ±1 with probability 1/2. Extending f by multiplicativity to a random multiplicative function f supported on the set of squarefree integers, we prove that, for any e > 0, the estimate nx f (n) √ x (log log x) 3/2+e holds almost surely—thus qualitatively matching the law of iterated logarithm, valid for independent variables. This improves on corresponding results by Wintner, Erd˝ os and Halasz .

Non-vanishing of symmetric square L-functions

Given a complex number s with 0 < Res < 1, we study the existence of a cusp form of large even weight for the full modular group such that its associated symmetric square L-function L(sym 2 f, s) does not vanish. This problem is also considered in other articles.

Sign of Fourier coefficients of modular forms of half integral weight

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

Sign changes of the coefficients of automorphic L-functions

This survey gives an account of background and the recent development concerning sign changes of Fourier coefficients of modular forms, which includes the great contributions of other authors. We are attempting to elucidate interesting viewpoints, ideas and methods which, even unspecified, may not originate from us -- the present authors. Moreover, we formulate some questions for future study.