Klurman, Oleksiy and Mangerel, Alexander P. and Pohoata, Cosmin and Teräväinen, Joni
(2019)
*Multiplicative functions that are close to their mean.*
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(Unpublished)
https://resolver.caltech.edu/CaltechAUTHORS:20200110-150317815

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## Abstract

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 : N→C satisfies ∑ f(n)_ (n≤x) = cx+O(1) 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.

Item Type: | Report or Paper (Discussion Paper) | ||||||
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Additional Information: | The authors would like to warmly thank Marco Aymone for suggesting that our initial argument could lead to a proof of Theorem 1.5. | ||||||

Record Number: | CaltechAUTHORS:20200110-150317815 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20200110-150317815 | ||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 100641 | ||||||

Collection: | CaltechAUTHORS | ||||||

Deposited By: | Tony Diaz | ||||||

Deposited On: | 11 Jan 2020 00:51 | ||||||

Last Modified: | 11 Jan 2020 00:51 |

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