Let g 0 , . . . , g k : N → D be 1-bounded multiplicative functions, and let h 0 , . . . , h k ∈ Z be shifts. We consider correlation sequences f : N → Z of the formwhere 1 ≤ ω m ≤ x m are numbers going to infinity as m → ∞, and lim is a generalised limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely that these sequences f are the uniform limit of periodic sequences f i . Furthermore, if the multiplicative function g 0 . . . g k "weakly pretends" to be a Dirichlet character χ, the periodic functions f i can be chosen to be χ-isotypic in the sense that f i (ab) = f i (a)χ(b) whenever b is coprime to the periods of f i and χ, while if g 0 . . . g k does not weakly pretend to be any Dirichlet character, then f must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three, and of the Möbius function of length up to four.
Let E k be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log 1+ε x] contain E 3 numbers, and almost all intervals [x, x + log 3.51 x] contain E 2 numbers. By this we mean that there are only o(X) integers 1 ≤ x ≤ X for which the mentioned intervals do not contain such numbers. The result for E 3 numbers is optimal up to the ε in the exponent. The theorem on E 2 numbers improves a result of Harman, which had the exponent 7 + ε in place of 3.51. We also consider general E k numbers, and find them on intervals whose lengths approach log x as k → ∞.
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.
A famous conjecture of Chowla states that the Liouville function λ(n) has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically averaged Elliott conjecture on correlations of multiplicative functions, which in turn implied all the odd order cases of the logarithmically averaged Chowla conjecture. In this note, we give a new and shorter proof of the odd order cases of the logarithmically averaged Chowla conjecture. In particular, this proof avoids all mention of ergodic theory, which had an important role in the previous proof.
We study the asymptotic behaviour of higher order correlations E n≤X/d g 1 (n + ah 1 ) · · · g k (n + ah k ) as a function of the parameters a and d, where g 1 , . . . , g k are bounded multiplicative functions, h 1 , . . . , h k are integer shifts, and X is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all X if g 1 · · · g k does not (weakly) pretend to be a twisted Dirichlet character n → χ(n)n it , and behave asymptotically like a multiple of d −it χ(a) otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the d parameter is averaged out and one can set t = 0. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the k-point Chowla conjecture E n≤X λ(n + h 1 ) · · · λ(n + h k ) = o(1) for k odd or equal to 2 for all scales X outside of a set of zero logarithmic density.
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