On the proximity of multiplicative functions to the number of distinct prime factors function

2018 ◽  
Vol 68 (3) ◽  
pp. 513-526
Author(s):  
Jean-Marie De Koninck ◽  
Nicolas Doyon ◽  
François Laniel
Keyword(s):  
Additive Function ◽  
Multiplicative Function ◽  
Multiplicative Functions ◽  
Counting Multiplicity ◽  
Prime Factors

Abstract Given an additive function f and a multiplicative function g, let E(f, g;x) = #{n ≤ x: f(n) = g(n)}. We study the size of E(ω,g;x) and E(Ω,g;x), where ω(n) stands for the number of distinct prime factors of n and Ω(n) stands for the number of prime factors of n counting multiplicity. In particular, we show that E(ω,g;x) and E(Ω,g;x) are $\begin{array}{} \displaystyle O\left(\frac{x}{\sqrt{\log\log x}}\right) \end{array}$ for any integer valued multiplicative function g. This improves an earlier result of De Koninck, Doyon and Letendre.

scholarly journals Some characterizations of totients

1996 ◽  
Vol 19 (2) ◽  
pp. 209-217 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Multiplicative Function ◽  
Arithmetical Function ◽  
Multiplicative Functions ◽  
Dirichlet Convolution

An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler's phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we present several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.

scholarly journals Correlations of multiplicative functions and applications

2017 ◽  
Vol 153 (8) ◽  
pp. 1622-1657 ◽  
Author(s):  
Oleksiy Klurman
Keyword(s):  
Asymptotic Formula ◽  
Multiplicative Function ◽  
Divided Difference ◽  
Circle Method ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Image Position ◽  
Bounded Partial Sums

We give an asymptotic formula for correlations $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$ where $f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\rightarrow \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either $f(n)=n^{s}$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of $n=a+b$, where $a,b$ belong to some multiplicative subsets of $\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.

Multiplicative functions at consecutive integers

1986 ◽  
Vol 100 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Random Sequence ◽  
Multiplicative Functions ◽  
Liouville Function ◽  
Prime Factors ◽  
Consecutive Integers ◽  
Image Position

Let λ(n) denote the Liouville function, i.e. λ(n) = 1 if n has an even number of prime factors, and λ(n) = − 1 otherwise. It is natural to expect that the sequence λ(n) (n ≥ 1) behaves like a random sequence of ± signs. In particular, it seems highly plausible that for any choice of εi = ± 1 (i = 0,…, k) we have

scholarly journals WHEN DOES THE BOMBIERI–VINOGRADOV THEOREM HOLD FOR A GIVEN MULTIPLICATIVE FUNCTION?

2018 ◽  
Vol 6 ◽  
Author(s):  
ANDREW GRANVILLE ◽  
XUANCHENG SHAO
Keyword(s):  
Multiplicative Function ◽  
Multiplicative Functions

Let $f$ and $g$ be 1-bounded multiplicative functions for which $f\ast g=1_{.=1}$. The Bombieri–Vinogradov theorem holds for both $f$ and $g$ if and only if the Siegel–Walfisz criterion holds for both $f$ and $g$, and the Bombieri–Vinogradov theorem holds for $f$ restricted to the primes.

Additive uniqueness of PRIMES − 1 for multiplicative functions

2020 ◽  
Vol 16 (06) ◽  
pp. 1369-1376
Author(s):  
Poo-Sung Park
Keyword(s):  
Multiplicative Function ◽  
Constant Function ◽  
Multiplicative Functions ◽  
Identity Function

Let [Formula: see text] be the set of all primes. A function [Formula: see text] is called multiplicative if [Formula: see text] and [Formula: see text] when [Formula: see text]. We show that a multiplicative function [Formula: see text] which satisfies [Formula: see text] satisfies one of the following: (1) [Formula: see text] is the identity function, (2) [Formula: see text] is the constant function with [Formula: see text], (3) [Formula: see text] for [Formula: see text] unless [Formula: see text] is odd and squareful. As a consequence, a multiplicative function which satisfies [Formula: see text] is the identity function.

scholarly journals MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS

2020 ◽  
Vol 8 ◽  
Author(s):  
ADAM J. HARPER
Keyword(s):  
Large Deviations ◽  
Multiplicative Function ◽  
Multiplicative Functions ◽  
Order Of Magnitude ◽  
Introductory Discussion ◽  
First Moment ◽  
Better Than

We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ , where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$ . In the Steinhaus case, this is equivalent to determining the order of $\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$ . In particular, we find that $\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$ . This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of $\sum _{n\leqslant x}f(n)$ . The proofs develop a connection between $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ and the $q$ th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.

scholarly journals A note on multiplicative functions on progressions to large moduli

2017 ◽  
Vol 148 (1) ◽  
pp. 63-77 ◽  
Author(s):  
Ben Green
Keyword(s):  
Multiplicative Function ◽  
Multiplicative Functions ◽  
Almost All

Let f : ℕ → ℂ be a bounded multiplicative function. Let a be a fixed non-zero integer (say a = 1). Then f is well distributed on the progression n ≡ a (mod q) ⊂ {1,…, X}, for almost all primes q ∈ [Q, 2Q], for Q as large as X1/2+1/78−o(1).

scholarly journals Some characterizations of specially multiplicative functions

2003 ◽  
Vol 2003 (37) ◽  
pp. 2335-2344 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Multiplicative Function ◽  
Multiplicative Functions ◽  
Divisor Functions

A multiplicative functionfis said to be specially multiplicative if there is a completely multiplicative functionfAsuch thatf(m)f(n)=∑d|(m,n)f(mn/d2)fA(d)for allmandn. For example, the divisor functions and Ramanujan'sτ-function are specially multiplicative functions. Some characterizations of specially multiplicative functions are given in the literature. In this paper, we provide some further characterizations of specially multiplicative functions.

scholarly journals A Structure Theorem for Level Sets of Multiplicative Functions and Applications

2018 ◽  
Vol 2020 (5) ◽  
pp. 1300-1345 ◽  
Author(s):  
Vitaly Bergelson ◽  
Joanna Kułaga-Przymus ◽  
Mariusz Lemańczyk ◽  
Florian K Richter
Keyword(s):  
Level Set ◽  
Multiplicative Function ◽  
Level Sets ◽  
Structure Theorem ◽  
Almost Periodic ◽  
Multiplicative Functions ◽  
Multiple Recurrence ◽  
Random Part

Abstract Given a level set E of an arbitrary multiplicative function f, we establish, by building on the fundamental work of Frantzikinakis and Host [14, 15], a structure theorem that gives a decomposition of $1_{E}$ into an almost periodic and a pseudo-random part. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence.

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