Multiplicative functions in short intervals

Abstract

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.

Sign changes of Hecke eigenvalues

“…Remark 1.4. While this paper was being refereed, we have been able to obtain (by completely different means) a rather general result on multiplicative function in short intervals [17] which implies among other things that a multiplicative function f : N → R has a positive proportion of sign changes as soon as f (m) < 0 for some integer m and f (n) = 0 for a positive proportion of integers n. This recovers Theorem 1.2 in the holomorphic case, but not in the case of Maass forms, since for a Maass form it is currently not ruled out that λ f (n) = 0 for almost all integers n. In addition as pointed out by the anonymous referee, the method developed in this paper is general and will work for any multiplicative function satisfying reasonable estimates for the associated shifted convolution problem, which is especially interesting when the function vanishes on many primes.…”

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

“…Similarly for the variant x/ω<n x n it (n + 1) −2it (n + 2) it n of (50). This suggests that in order to establish cancellation in (50) and (51), one must somehow go beyond the techniques in [21,23], as these techniques do not exclude the problematic multiplicative functions n → n it for t between x and x 2 .…”

“…In [16] it was shown that the (+1, +1) and (−1, −1) patterns occur at least ( 1 60 + o(1))x times, and the (+1, −1) and (−1, +1) patterns occur x log −7−ε x times for ε > 0. In the recent paper [21] it was shown that in fact all four sign patterns occur x times, so in particular n x λ(n)λ(n + 1)…”

“…This latter expression can then be controlled in turn by an application of the Hardy-Littlewood circle method and an estimate for short sums of a modulated Liouville function established recently by Matomäki et al [23], which is based in turn on the results of Matomäki and Radziwiłł in [21]. The arguments in this paper extend to other bounded multiplicative functions than the Liouville function, though as they rely in an essential fashion on multiplicativity at small primes, they unfortunately do not appear to have any bearing as yet on twin prime-type sums such as (2).…”

“…Using Vinogradov-Korobov error term zero-free region for L-functions (see [25,Section 9.5]), it is not difficult to establish (8) when g is the Liouville function; see [22,Lemma 2] for a closely related calculation. Thus Corollary 5 implies Theorem 2.…”