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.

We investigate the distribution of the Riemann zeta-function on the line Re(s) = σ. For 1 2 < σ ≤ 1 we obtain an upper bound on the discrepancy between the distribution of ζ(s) and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of ζ(s) inside of the critical strip, strengthening a previous result of the first author.As an application of these results we obtain the first effective error term for the number of solutions to ζ(s) = a in a strip 1 2 < σ 1 < σ 2 < 1. Previously in the strip 1 2 < σ < 1 only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg) and to the right of the abscissa of absolute convergence (due to Matsumoto). In general our results are an improvement of the classical Bohr-Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function.

Let λ and µ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for (λ(n), λ(n + 1), λ(n + 2)) occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand's result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for (µ(n), µ(n + 1)). A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

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We study asymptotics of sums of the form 0true∑X0, where σ:=833=0.2424⋯, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. This improves upon results of Mikawa and Baier–Browning–Marasingha–Zhao, who obtained statements of this form with σ replaced by 13. We obtain an analogous result for the fourth sum for most N in an interval of the form [X,X+H] with Xσ+ε⩽H⩽X1−ε. Our method starts with a variant of an argument from a paper of Zhan, using the circle method and some oscillatory integral estimates to reduce matters to establishing some mean‐value estimates for certain Dirichlet polynomials associated to ‘Type d3’ and ‘Type d4’ sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type d3 sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type d4 sum is treated similarly using the classical L2 mean value theorem and the classical van der Corput exponential sum estimates. In a sequel to this paper we will obtain related results for the correlations involving dkfalse(nfalse) for much smaller values of H but with weaker bounds.