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Multiplicative functions that are close to their mean

Klurman, Oleksiy and Mangerel, Alexander P. and Pohoata, Cosmin and Teräväinen, Joni (2019) Multiplicative functions that are close to their mean. . (Unpublished)

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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
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100641
Deposited By: Tony Diaz
Deposited On:11 Jan 2020 00:51
Last Modified:11 Jan 2020 00:51

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