We give new bounds for a,m,n α m β n ν a e am n where α m , β n and ν a are arbitrary coefficients, improving upon a result of Duke, Friedlander and Iwaniec [DFI97]. We also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations.

Abstract. Improving earlier work of Balasubramanian, Conrey and Heath-Brown [BCHB85], we obtain an asymptotic formula for the mean-square of the Riemann zetafunction times an arbitrary Dirichlet polynomial of length T 1/2+δ , with δ = 0.01515 . . .. As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec [DI84], obtaining asymptotic estimates in place of bounds. Using the work of Watt [Wat95], we compute the mean-square of the Riemann zetafunction times a Dirichlet polynomial of length going up to T 3/4 provided that the Dirichlet polynomial assumes a special shape. Finally, we exhibit a conjectural estimate for trilinear sums of Kloosterman fractions which implies the Lindelöf Hypothesis.

We investigate the period function of ∞ n=1 σ a (n)e (nz), showing it can be analytically continued to | arg z| < π and studying its Taylor series. We use these results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moment of the Riemann zeta function. Moreover, we introduce a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula.

Following the work of Conrey, Rubinstein and Snaith [11] and Forrester and Witte [16] we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U (N ) (also known as the CUE) and relate the moment to the solution of a Painlevé differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.DIMA -DIPARTIMENTO DI MATEMATICA VIA DODECANESO,

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length T 1 4 −ε. 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 1 11 −ε were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zetafunction, zero spacing and lower bounds for moments. Along the way we obtain an asymptotic formula for a quadratic divisor problem, where the condition am 1 m 2 − bn 1 n 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 simplifies considerably the combinatorics of the main terms in the twisted fourth moment.