Sums of nonnegative multiplicative functions over integers without large prime factors I

2001 ◽  
Vol 97 (4) ◽  
pp. 329-351 ◽  
Author(s):  
Joung Min Song
Keyword(s):  
Multiplicative Functions ◽  
Prime Factors

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 A Proof of an Identity for Multiplicative Functions

1979 ◽  
Vol 22 (3) ◽  
pp. 299-304
Author(s):  
K. Krishna
Keyword(s):  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Multiplicative Functions ◽  
Prime Factors ◽  
Dirichlet Convolution

An arithmetic function f is said to be multiplicative if f(mn) = f(m)f(n), whenever (m, n) = 1 and f(1) = 1. The Dirichlet convolution of two arithmetic functions f and g, denoted by f • g, is defined by f • g(n) = Σd|nf(d)g(n/d). Let w(n) denote the product of the distinct prime factors of n, with w(l) = 1. R.

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.

Cancellation in algebraic twisted sums on GL_ m

2021 ◽  
Vol 33 (4) ◽  
pp. 1061-1082
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Dirichlet Series ◽  
Central Character ◽  
Multiplicative Functions ◽  
Cuspidal Representation ◽  
Series Coefficient

Abstract Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε > 0 {\varepsilon>0} .

On the prime factors of the iterates of the Ramanujan τ–function

2020 ◽  
Vol 63 (4) ◽  
pp. 1031-1047
Author(s):  
Florian Luca ◽  
Sibusiso Mabaso ◽  
Pantelimon Stănică
Keyword(s):  
Positive Integer ◽  
Prime Factors

AbstractIn this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

scholarly journals Sarnak’s conjecture for sequences of almost quadratic word growth

2020 ◽  
pp. 1-56
Author(s):  
REDMOND MCNAMARA
Keyword(s):  
Box Dimension ◽  
Multiplicative Functions ◽  
Logarithmic Density ◽  
Liouville Function ◽  
Sign Patterns

Abstract We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

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