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.

Let S(t) = 1 π arg ζ 1 2 + it be the argument of the Riemann zeta-function at the point 1 2 + it. For n ≥ 1 and t > 0 define its iterateswhere δn is a specific constant depending on n and S 0 (t) := S(t). In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that Sn(t) = O(log t/(log log t) n+1 ). The order of magnitude of this estimate was never improved up to this date. The best bounds for S(t) and S 1 (t) are currently due to Carneiro, Chandee and Milinovich. In this paper we establish, under RH, an explicit form of this estimate 2010 Mathematics Subject Classification. 11M06, 11M26, 41A30.