Abstract. We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of µ(n) in almost all intervals of the form [x, x + ψ(x)] with ψ(x) → ∞ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of x ǫ -smooth numbers in intervals of the form [x, x + c(ε) √ x], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n + 1), with λ(n) Liouville's function, is non-trivially bounded in absolute value by 1−δ for some δ > 0. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function f has a positive proportion of sign changes if and only if f is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

Abstract. Let λ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers h 1 , . . . , h k , one has 1≤n≤X λ(n + h 1 ) · · · λ(n + h k ) = o(X) as X → ∞. This conjecture remains unproven for any h 1 , . . . , h k with k ≥ 2. In this paper, using the recent results of the first two authors on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain-Sarnak-Ziegler, we establish an averaged version of this conjecture, namelyas X → ∞ whenever H = H(X) ≤ X goes to infinity as X → ∞, and k is fixed. Related to this, we give the exponential sum estimateas X → ∞ uniformly for all α ∈ R, with H as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of log log H log H ), and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

We prove the leading order of a conjecture by Fyodorov, Hiary, and Keating about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as T → ∞ for a set of t ∊ [T, 2T] of measure (1–o(1)) T, we have max|t−u|≤1log|ζ(12+iu)|=(1+o(1))loglogT. © 2018 Wiley Periodicals, Inc.

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.