A note on multiplicative functions on progressions to large moduli

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class a we extend such averages out to moduli ≤ x 20 39 −δ .

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“…Since F (pm) is supported on pm ≥ Z 9 , we may add the restriction m ≥ Z 8 to the sum. By Lemma 2.1 of [21] we have…”

Section: 1mentioning

confidence: 93%

“…Breaking the x 1/2 -barrier. The main method used in our proofs is a modification of that developed by Green in [21]; see also [29] for using a similar argument to deal with higher Gowers norms. Green proved (a more general result which implies) that…”

Section: 4mentioning

confidence: 99%

Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2-barrier

Granville

Shao

2019

Advances in Mathematics

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“…We note in passing that in the literature there are numerous results, valid in the middle range q ≤ x θ , that go beyond θ = 1/2 (especially for f (n) = Λ(n) or f (n) = µ(n)), provided that one restricts 5 to a fixed 6 residue class a (mod q), and in some cases also adds a "wellfactorable" weight to the sum (as in [3]), or specializes to smooth q (as in [47], [37]). A result of this shape for general multiplicative functions was proved by Granville and Shao [13,Theorem 1.8] (generalizing work of Green [15]), who showed that if a ∈ [1, 2Q] is fixed, then for all but Q/(log x)…”

Section: Introduction and Resultsmentioning

confidence: 91%

Multiplicative functions in short arithmetic progressions

Klurman¹,

Mangerel²,

Teräväinen³

2019

Preprint

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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.

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“…Unfortunately we cannot directly apply [4, Proposition 2.2], since for example our function F is not necessarily bounded. In the remainder of this section we reproduce the argument from [4] with suitable modifications to prove (3.1). Recall the definition of Ramaré's weight function:…”

Section: The Minor Arc Case: Proof Of Proposition 21mentioning

confidence: 99%

Gowers norms of multiplicative functions in progressions on average

Shao

2017

Alg. Number Th.

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Let $\mu$ be the M\"{o}bius function and let $k \geq 1$. We prove that the Gowers $U^k$-norm of $\mu$ restricted to progressions $\{n \leq X: n\equiv a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-\sigma}$ for any $\sigma > 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$. This generalizes the Bombieri-Vinogradov inequality for $\mu$, which corresponds to the special case $k=1$.Comment: 20 page

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A note on multiplicative functions on progressions to large moduli

Abstract: Abstract. Let f : N → C be a bounded multiplicative function. Let a be a fixed nonzero integer (say a = 1). Then f is welldistributed on the progression n ≡ a(mod q) ⊂ {1, . . . , X}, for almost all primes q ∈ [Q, 2Q], for Q as large as X 1 2 + 1 78 −o(1) .

Cited by 11 publications

References 13 publications

Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2-barrier

Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2-barrier

Multiplicative functions in short arithmetic progressions

Gowers norms of multiplicative functions in progressions on average

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