Topic
Completely multiplicative function
About: Completely multiplicative function is a research topic. Over the lifetime, 38 publications have been published within this topic receiving 325 citations. The topic is also known as: totally multiplicative functions.
Papers
TL;DR: In this article, it was shown that for any multiple closed set S and for any divisor chain S (i.e. x 1j :::jxn), if f is a completely multiplicative function such that (f )(d) is a nonzero integer whenever lcm(S) is the least common multiple of all elements in S, then the matrix (f(xi;xj)) havingf evaluated at the greatest common divisors ofxi and xj as its i;j-entry divides the matrix having f(xi,x
Abstract: Let f be an arithmetical function. A set S =fx1;:::;xng of n distinct positive integers is called multiple closed if y2 S whenever xjyj lcm(S) for any x2 S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x1j :::jxn), if f is a completely multiplicative function such that (f )(d) is a nonzero integer whenever dj lcm(S), then the matrix (f(xi;xj)) havingf evaluated at the greatest common divisor (xi;xj) ofxi and xj as its i;j-entry divides the matrix (f(xi;xj)) having f evaluated at the least common multiple (xi;xj) of xi and xj as its i;j-entry in the ring Mn(Z) of n n matrices over the integers. But such a factorization is no longer true if f is multiplicative.
43 citations
TL;DR: Hong et al. as mentioned in this paper showed that the nonsingularity of LCM matrices and reciprocal GCD matrices is nonsingular if and only if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f satisfies 0 f (p ) ⩽ 1 p (resp., f(p)⩾ p) for any prime p) and S is lcm-closed.
31 citations
28 Oct 2008
TL;DR: In this article, it was shown that for any non-trivial completely multiplicative function from N to {-1,1], the series Σ ∞ n=1 f(n)z n is transcendental over Z(z), where f is Liouville's function.
Abstract: We give a new proof of Fatou's theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function. This result is applied to show that for any non-trivial completely multiplicative function from N to {-1,1}, the series Σ ∞ n=1 f(n)z n is transcendental over Z(z); in particular, Σ ∞ n=1 λ(n)z n is transcendental, where λ is Liouville's function. The transcendence of Σ ∞ n=1 μ(n)z n is also proved.
25 citations
TL;DR: In this article, it was shown that if f is a completely multiplicative function that assumes values inside the unit disc, then either f(p) is small on average or pretends to be\(n)n^{it}} for some t.
Abstract: Let f be a completely multiplicative function that assumes values inside the unit disc. We show that if \({\sum_{n \leq x}f(n)\ll x/(\rm log x)^A}\) , \({x \geq 2}\) , for some A > 2, then either f(p) is small on average or fpretends to be\({\mu(n)n^{it}}\) for some t.
22 citations
Posted Content•
TL;DR: In this article, it was shown that for any non-trivial completely multiplicative function from n = 1 to n = 2, the series is transcendent over n, where n is Liouville's function.
Abstract: We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely multiplicative function from $\mathbb{N}$ to $\{-1,1\}$, the series $\sum_{n=1}^\infty f(n)z^n$ is transcendental over $\mathbb{Z}[z]$; in particular, $\sum_{n=1}^\infty \lambda(n)z^n$ is transcendental, where $\lambda$ is Liouville's function. The transcendence of $\sum_{n=1}^\infty \mu(n)z^n$ is also proved.
20 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 3 |
| 2019 | 4 |
| 2018 | 5 |
| 2017 | 4 |
| 2016 | 2 |