Multiplicative functions in arithmetic progressions

Balog, Antal; Granville, Andrew; Soundararajan, K.

doi:10.1007/s40316-013-0001-z

Cited by 32 publications

(124 citation statements)

References 19 publications

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“…For quantitative statements, it is known for the Liouville λ-function and any positive constant A that n x λ(qn + a) q,A x/(log x) A , (see Brüdern [4]; compare also Iwaniec and Kowalski's book [18,Corollary 5.29]). For more general multiplicative functions, there are some quantitative results by Elliott [10] and Balog, Granville and Soundararajan [2]. However, such bounds would appear to give a rather weak upper bound on d only.…”

Section: Resultsmentioning

confidence: 99%

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Patterns and complexity of multiplicative functions

Buttkewitz

Elsholtz²

2011

Journal of the London Mathematical Society

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I. Schur and G. Schur proved that, for all completely multiplicative functions f : N → {−1, 1}, with the exception of two character-like functions, there is always a solution of f (n) = f (n + 1) = f (n + 2) = 1. Hildebrand proved that for the Liouville λ-function each of the eight possible sign combinations (λ(n), λ(n + 1), λ(n + 2)) occurs infinitely often. We prove for completely multiplicative functions f : N → {−1, 1}, satisfying certain necessary conditions, that any sign pattern ( 1, 2, 3, 4), i ∈ {−1, 1}, occurs for infinitely many 4-term arithmetic progressions (f (n), f(n + d), f(n + 2d), f(n + 3d)).The proof introduces graph theory and new combinatorial methods to the subject.

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Section: Resultsmentioning

confidence: 99%

“…There are two separate cases, depending on the sign of f (2). The case f (2) = −1 is discussed below; the other case f (2) = 1 can be proved along the same lines.…”

Section: At Least Four Patterns Ofmentioning

confidence: 90%

Patterns and complexity of multiplicative functions

Buttkewitz

Elsholtz²

2011

Journal of the London Mathematical Society

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“…, X}. Is #A ≫ X 1−o (1) ? The connection between this problem and the distribution of multiplicative functions on progressions is discussed in [9].…”

Section: On Allowing the Residue Class To Varymentioning

confidence: 99%

“…There is a considerable and deep literature concerning the distribution of primes in progressions a(mod q) with q > X 1/2 . However, these works typically require q to be "smooth" or "well-factorable" [3,8,15] or else "only" beat the X 1/2 barrier by a smaller term X o (1) [4,5].…”

Section: Introductionmentioning

confidence: 99%

A note on multiplicative functions on progressions to large moduli

Green¹

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…1 For the definitions of the standard multiplicative functions used in this paper, see Subsection 1.6. 2 If one generalises the Chowla conjecture by using affine forms a i n + h i instead of shifts n + h i , then a simple sieving argument can be used to show the equivalence of such generalised Chowla conjectures for the Liouville function and their counterparts for the Möbius function; we leave the details to the interested reader. Theorem 1.5 (Special case of logarithmically averaged Elliott).…”

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confidence: 99%

The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

Tao

Teräväinen

2019

Alg. Number Th.

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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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Multiplicative functions in arithmetic progressions

Abstract: We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.

Cited by 32 publications

References 19 publications

Patterns and complexity of multiplicative functions

Patterns and complexity of multiplicative functions

A note on multiplicative functions on progressions to large moduli

The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures

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