We establish several results concerning the expected general phenomenon
We introduce a simple approach to study partial sums of multiplicative functions which are close to their mean value. As a first application, we show that a completely multiplicative function f :with c = 0 if and only if f (p) = 1 for all but finitely many primes and |f (p)| < 1 for the remaining primes. This answers a question of Imre Ruzsa.For the case c = 0, we show, under the additional hypothesis p:|f (p)|<1 1/p < ∞, that f has bounded partial sums if and only if f (p) = χ(p)p it for some non-principal Dirichlet character χ modulo q and t ∈ R except on a finite set of primes that contains the primes dividing q, wherein |f (p)| < 1. This essentially resolves another problem of Ruzsa and generalizes previous work of the first and the second author on Chudakov's conjecture.We also consider some other applications, which include a proof of a recent conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions.
We study for bounded multiplicative functions f sums of the form n≤x n≡a (mod q) f (n), establishing a theorem stating that their variance over residue classes a (mod q) is small as soon as q = o(x), for almost all moduli q, with a nearly power-saving exceptional set of q. This substantially improves on previous results of Hooley on Barban-Davenport-Halberstam-type theorems for such f , and moreover our exceptional set is essentially optimal unless one is able to make progress on certain well-known conjectures. We are nevertheless able to prove stronger bounds for the number of the exceptional moduli q in the cases where q is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every q.These results are special cases of a "hybrid result" that we establish that works for sums of f (n) over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki-Radziwi l l theorem on multiplicative functions in short intervals.We also consider the maximal deviation of f (n) over all residue classes a (mod q) in the square root range q ≤ x 1/2−ε , and show that it is small for "smooth-supported" f , again apart from a nearly power-saving set of exceptional q, thus providing a smaller exceptional set than what follows from Bombieri-Vinogradov-type theorems.As an application of our methods, we consider the analogue of Linnik's theorem on the least prime in an arithmetic progression for products of exactly three primes, and prove the exponent 2 + o(1) for this problem for all smooth values of q.
We extend the Matomäki-Radziwi l l theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length h(log X) c , with h = h(X) → ∞ and where c = c f ≥ 0 is determined by the distribution of {|f (p)|}p in an explicit way. We give two applications. Our first application shows that the classical Rankin-Selberg-type asymptotic formula for partial sums of |λ f (n)| 2 , where {λ f (n)}n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length h log X, if h = h(X) → ∞.Our second application shows that the (non-multiplicative) Hooley ∆-function has average value ≫ log log X in typical short intervals of length (log X) 1/2+η , where η > 0 is fixed.
Let 2 ≤ y ≤ x such that β := log x log y → ∞. Let ω y (n) denote the number of distinct prime factors p of n such that p ≤ y, and let µ y (n) := µ 2 (n)(−1) ωy(n) , where µ is the Möbius function. We prove that if β is not too large (in terms of x) then for each fixed a ∈ N,This can be seen as a partial result towards the binary Chowla conjecture. Our main input is a quantitative bivariate analogue of the Erdős-Kac theorem regarding the distribution of the pairs (ω(n), ω(n + a)), where n and n + a both belong to any subset of the positive integers with suitable sieving properties; moreover, we show that the set of squarefree integers is an example of such a set. We end with a further application of this probabilistic result related to a problem of Erdős and Mirsky on the number of integers n ≤ x such that τ (n) = τ (n + 1).1 8 .
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