Kannan Soundararajan

Nonvanishing of quadratic Dirichlet L-functions at s=1/2

We show that for a positive proportion of fundamental discriminants d, L(1/2,chi_d) != 0. Here chi_d is the primitive quadratic Dirichlet character of conductor d.

Mean-values of the Riemann zeta-function

Let Asymptotic formulae for I k ( T ) have been established for the cases k =1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of I k ( T ) remains unknown for any other value of k (except the trivial k = 0, of course). Heath-Brown, [6], and Ramachandra, [10], [11], independently established that, assuming the Riemann Hypothesis, when 0≤ K ≤2, I k ( T ) is of the order T (log T ) k 2 One believes that this is the right order of magnitude for I k ( T ) even when k = 2 and indeed expects an asymptotic formula of the form where C k is a suitable positive constant. It is not clear what the value of C k should be.

Large character sums: Pretentious characters and the Pólya-Vinogradov theorem

In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters

Large character sums

Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n infinity and q -> infinity (q is the size of the finite field).

Lower bounds for moments of L-functions

The moments of central values of families of L-functions have recently attracted much attention and, with the work of Keating and Snaith [(2000) Commun. Math. Phys. 214, 57-89 and 91-110], there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of L-functions. As an example we work out the case of the family of all Dirichlet L-functions to a prime modulus.

Omega results for the divisor and circle problems

We obtain new omega results for the error terms in two classical lattice point problems. These results are likely to be the best possible.

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.

Decay of Mean Values of Multiplicative Functions

For given multiplicative function f, with | f(n)| ≤ 1 for all n, we are interested in how fast its mean value (1/x) P nx f(n) converges. Halshowed that this depends on the minimum M (over y ∈ R) of P px 1 − Re( f(p)p i y ) � /p, and subsequent authors gave the upper bound ≪ (1 + M)e M. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Hallemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of y that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the k-th powers mod p.

Bounding |ζ(½+it)| on the Riemann hypothesis

In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |\zeta(1/2+it)| \ll \exp(C\log t/\log \log t). In this note we show how the problem of bounding |\zeta(1/2+it)| may be framed in terms of minorizing the function \log ((4+x^2)/x^2) by functions whose Fourier transforms are supported in a given interval, and drawing upon recent work of Carneiro and Vaaler we find the optimal such minorant. Thus we establish that any C> (\log 2)/2 is permissible in Littlewoods result.

BENFORD'S LAW FOR THE 3x + 1 FUNCTION

We show that for most choices of an initial seed