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.

MORE ON A CERTAIN ARITHMETICAL DETERMINANT

2017 ◽  
Vol 97 (1) ◽  
pp. 15-25 ◽  
Author(s):  
ZONGBING LIN ◽  
SIAO HONG
Keyword(s):  
Linear Algebra ◽  
Acta Math ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Mobius Function ◽  
Dirichlet Convolution ◽  
Positive Integers

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

ASYMPTOTIC BEHAVIOR OF LARGEST EIGENVALUE OF MATRICES ASSOCIATED WITH COMPLETELY EVEN FUNCTIONS (MOD r)

2008 ◽  
Vol 01 (02) ◽  
pp. 225-235 ◽  
Author(s):  
Shaofang Hong
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Multiplicative Function ◽  
Infinite Sequence ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Dirichlet Convolution ◽  
Positive Integers

Given an arbitrary strictly increasing infinite sequence [Formula: see text] of positive integers, let Sn = {x1,…, xn} for any integer n ≥ 1. Let q ≥ 1 be a given integer and f an arithmetical function. Let [Formula: see text] be the eigenvalues of the matrix (f(xi, xj)) having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its i, j-entry. We obtain a lower bound depending only on x1 and n for [Formula: see text] if (f * μ)(d) < 0 whenever d|x for any x ∈ Sn, where g * μ is the Dirichlet convolution of f and μ. Consequently we show that for any sequence [Formula: see text], if f is a multiplicative function satisfying that f(pk) → ∞ as pk → ∞, f(2) > 1 and f(pm) ≥ f(2)f(pm−1) for any prime p and any integer m ≥ 1 and (f *μ)(d) > 0 whenever d|x for any [Formula: see text], then [Formula: see text] approaches infinity when n goes to infinity.

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.

scholarly journals An Asymptotic Formula for Reciprocals of Logarithms of Certain Multiplicative Functions

1978 ◽  
Vol 21 (4) ◽  
pp. 409-413 ◽  
Author(s):  
Jean-Marie de Koninck ◽  
Aleksandar Ivić
Keyword(s):  
Asymptotic Formula ◽  
Abelian Groups ◽  
Arithmetical Function ◽  
Multiplicative Functions ◽  
Class Of Functions

Sums of the form where f(n) is a multiplicative arithmetical function and denotes summation over those values of n for which f(n)>0 and f(n) ≠1, were studied by De Koninck [2], De Koninck and Galambos [3], Brinitzer [1] and Ivič [5]. The aim of this note is to give an asymptotic formula for a certain class of multiplicative, positive, primeindependent functions (an arithmetical function is prime-independent if f(pv) = g(v) for all primes p and v = 1, 2, …). This class of functions includes, among others, the functions a(n) and τ(e)(n), which represent the number of nonisomorphic abelian groups of order n and the number of exponential divisors of n respectively, and none of the estimates of the above-mentioned papers may be applied to this class of functions. We prove the following.

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.

scholarly journals On a subgroup of the group of multiplicative arithmetic functions

1975 ◽  
Vol 20 (3) ◽  
pp. 348-358 ◽  
Author(s):  
T. B. Carroll ◽  
A. A. Gioia
Keyword(s):  
Abelian Group ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Dirichlet Convolution ◽  
Image Position ◽  
Positive Integers ◽  
Multiplicative Arithmetic

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

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.

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