The mean square of the product of the Riemann zeta-function with Dirichlet polynomials

Abstract

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.

Perturbed moments and a longer mollifier for critical zeros of $$\zeta $$ ζ

“…[1,3,7,8,21]. The applications of I are very deep, as one may use asymptotic estimates for I to make sense of the distribution of values of L-functions, the location of their critical zeros, as well as upper and lower bounds for the size of L-functions (see, among many examples, [9][10][11]16,19,23,24]).…”

“…For example, it is known that if one could take θ = 1 − ε, then the Lindelöf hypothesis follows (see e.g. [3]). Moreover, as shown in [4], if one could take θ = ∞ in the Conrey-Levinson mollifier (see below), then the Riemann hypothesis would follow.…”

“…This is because when θ < 1 2 only the "diagonal" terms contribute to the main term, and the rest is absorbed in the error; for θ > 1 2 there is a nontrivial contribution from the "off-diagonal" terms. Bettin, Chandee, and Radziwiłł [3] succeeded in breaking the 1 2 barrier for an arbitrary Dirichlet polynomial. They showed that if θ < 1 2 + δ with δ = 1 66 , then Observe that 1 2 + δ = 17 33 .…”

“…Suppose that μ(n) and (n) denote the usual Möbius and von Mangoldt functions, respectively. Using the insights and methods of [3], we study the more general twisted second moment 5) where α, β (log T ) −1 . We consider three cases for the coefficients a n , namely a n = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ O(n ε ), for θ < 17 33 , μ 2 (n)(μ * * k )(n)f (n), with f (n) ∈ F and for θ < 6 11 , μ(n)f (n), with f (n) ∈ F and for θ < 4 7 .…”

“…The strength of the result appearing in [3] is the generality of a n . However, often times applications of I or I(α, β) allow for specialization of the shape of a n .…”