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About 6,680 papers for “Multiplicative function”

Applicable Multiplicative Calculus Using Multiplicative Modulus Function

C. Ganesa Moorthy · Fundamental Journal of Mathematics and Applications · 2019

The classical calculus is viewed as additive calculus based on addition in the real line. Another known multiplicative calculus corresponding to multiplication in the positive real axis has been precisely introduced. Abstract multiplicative integration through positive measures has been newly introduced. Results of multiplicative differentiation and integration have been obtained for completion, when some of them have been obtained through multiplicative modulus function. Results have been obtained also for abstract multiplicative measure integration.

Cited by 0Open access

THE SMARANDACHE MULTIPLICATIVE FUNCTION

Ma Jinping · 2014

Keywords: For any positive integer n, we define f(n) as a Smarandache multiplicative function, if f(ab) = max(f(a), f(b)), (a, b) = 1. Now for any prime p and any positive integer α, we take f(p α) = αp. It is clear that f(n) is a Smarandache multiplicative function. In this paper, we study the mean value properties of f(n), and give an interesting mean value formula for it. Smarandache multiplicative function; Mean Value; Asymptotic formula. §1 Introduction and results For any positive integer n, we define f(n) as a Smarandache multiplicative function, if f(ab) = max(f(a), f(b)), (a, b) = 1. Now for any prime p and any positive integer α, we take f(pα) = αp. It is clear that f(n) is a new Smarandache multiplicative function, and if n = p α1

Cited by 0

On the multiplicative partition function

A. Korchevskiy, Ya. Vorobyov · Researches in Mathematics and Mechanics · 2018

We study the number of representations $n=s_1\cdots s _m$, where $s_j$ are sonor numbers, i.e. for every $s_j$ there do not exist the natural numbers $n$ and $k$ such that $s_j=n^k$, $k\geqslant2$. The counting function $f(n)$ of such representation is the multiplicative analogue of the additive partitions of $n$. We construct the asymptotic formula for summatory function of $f(n)$ and investigate the distribution of values of the generalized divisor function $L(n)$ (as the number of representations $n$ factoring two sonor numbers).

Cited by 0Open access

Function theory and multiplicative linear functionals

Kenneth A. Hoffman, Hugo Rossi · Transactions of the American Mathematical Society · 1965

The principal result of this paper is that HO is a logmodular algebra on the maximal ideal space of Lw(m), i.e., that each real-valued in L(m) is the logarithm of the modulus of an invertible in the algebra H'. This enables us to deduce from (a) and (b) the bulk of the generalized analytic which is valid for logmodular algebras [ 8]. Srinivasan and Wang [ 10] have shown that this function theory follows if one assumes (a), (b), and (c) A + A is dense in L2(m). Lumer [9] has demonstrated the results under the assumption that A is an algebra of continuous functions on a compact Hausdorff space, m is a Borel measure on X which is multiplicative on A, and no other positive measure on X induces the same linear functional on A as does m. In ?5, we shall comment on this algebra setting.

Cited by 12

Multiplicative operators on analytic function spaces

Kanha Behera, Jun‐Ming Liu, Perumal Muthukumar · ArXiv.org · 2025

H. J. Schwartz proved in his thesis (1969) that a nonzero bounded operator on Hardy spaces $(H^p, 1\leq p\leq\infty)$ is almost multiplicative if and only if it is a composition operator. But, his proof has a gap. In this article, we show that his result is not correct for $H^\infty$ and we fill the gap for $H^p, 1\leq p<\infty.$ Further, we prove that on several classical spaces such as the Bloch space, the little Bloch space, Besov spaces $B_p$ for $p>1$, and weighted Bergman spaces an operator is almost multiplicative if and only if it is a composition operator. Finally, we give a complete characterization of those composition operators that are multiplicative with respect to the Duhamel product of analytic functions.

Cited by 0Open access

Multiplicative Ramanujan coefficients of null-function

Giovanni Coppola, Luca Ghidelli · arXiv (Cornell University) · 2020

The null-function $0(a):=0$, $\forall a\in $N, has Ramanujan expansions: $0(a)=\sum_{q=1}^{\infty}(1/q)c_q(a)$ (where $c_q(a):=$ Ramanujan sum), given by Ramanujan, and $0(a)=\sum_{q=1}^{\infty}(1/φ(q))c_q(a)$, given by Hardy ($φ:=$ Euler's totient function). Both converge pointwise (not absolutely) in N. A $G:$N $\rightarrow $C is called a Ramanujan coefficient, abbrev. R.c., iff (if and only if) $\sum_{q=1}^{\infty}G(q)c_q(a)$ converges in all $a\in $N; given $F:$N $\rightarrow $C, we call $$, the set of its R.c.s, the Ramanujan cloud of $F$. Our Main Theorem in arxiv:1910.14640, for Ramanujan expansions and finite Euler products, implies a complete Classification for multiplicative Ramanujan coefficients of $0$. Ramanujan's $G_R(q):=1/q$ is a normal arithmetic function $G$, i.e., multiplicative with $G(p)\neq 1$ on all primes $p$; while Hardy's $G_H(q):=1/φ(q)$ is a sporadic $G$, namely multiplicative, $G(p)=1$ for a finite set of $p$, but there's no $p$ with $G(p^K)=1$ on all integers $K\ge 0$ (Hardy's has $G_H(p)=1$ iff $p=2$). The $G:$N $\rightarrow $C multiplicative, such that there's at least a prime $p$ with $G(p^K)=1$, on all $K\ge 0$, are defined to be exotic. This definition completes the cases for multiplicative $0-$Ramanujan coefficients. The exotic ones are a kind of new phenomenon in the $0-$cloud (i.e., $<0>$): exotic Ramanujan coefficients represent $0$ only with a convergence hypothesis. The not exotic, apart from the convergence hypothesis, require in addition $\sum_{q=1}^{\infty}G(q)μ(q)=0$ for normal $G\in <0>$, while sporadic $G\in <0>$ need $\sum_{(q,P(G))=1}G(q)μ(q)=0$, $P(G):=$product of all $p$ making $G(p)=1$. We give many examples of R.c.s $G\in <0>$; we also prove that the only $G\in <0>$ with absolute convergence are the exotic ones; actually, these generalize to the weakly exotic, not necessarily multiplicative.

Cited by 1Open access

Peripherally-Multiplicative Spectral Preservers Between Function Algebras

Jeffrey Verlyn Johnson · The Mathematics Enthusiast · 2013

General sufficient conditions are established for maps between function algebras to be com-position or weighted composition operators, which extend previous results regarding spectral conditions for maps between uniform algebras. Let X and Y be a locally compact Hausdorff spaces, where A ⊂ C(X) and B ⊂ C(Y) are function algebras, not necessarily with unit. Also let ∂A be the Shilov boundary of A, δA the Choquet boundary of A, and p(A) the set of p-points of A. A map T ∶A → B is called weakly peripherally-multiplicative if the peripheral spec-tra of fg and TfTg have non-empty intersection for all f,g in A. (i.e. σpi(fg)∩σpi(TfTg) ≠ ∅ for all f, g in A) The map is said to be almost peripherally-multiplicative if the peripheral spectrum of fg is contained in the peripheral spectrum of TfTg (or if the peripheral spectrum of TfTg is contained in the peripheral spectrum of fg) for all f, g in A. Let X be a locally compact Hausdorff space and A ⊂ C(X) be a dense subalgebra of a function algebra, not necessarily with unit, such that δA = p(A). We show that if T ∶A → B is a surjective map onto a function algebra B ⊂ C(Y) that is almost peripherally-multiplicative, then there is a homeomorphism ψ ∶ δB → δA and a function α on δB so that (Tf)(y) = α(y) f(ψ(y)) for all f ∈ A and y ∈ δB, i.e. T is a weighted composition operator where the

Cited by 2Open access

Correlations of multiplicative functions in function fields

Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen · Mathematika · 2022

Abstract We develop an approach to study character sums, weighted by a multiplicative function , of the form where χ is a Dirichlet character and ξ is a short interval character over . We then deduce versions of the Matomäki–Radziwiłł theorem and Tao's two‐point logarithmic Elliott conjecture over function fields , where q is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin–Shusterman on correlations of the Möbius function for various values of q . Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that q is a power of 2. As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the existing one in the integer case. In a companion paper, we use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a “corrected” form of the Erdős discrepancy problem over .

Cited by 4Open access

Multiplicative potential energy function for swarm control

Saing Paul Hou, Chien Chern Cheah · 2009

This paper presents a novel method for shape control of a swarm of robots based on region control concept. Multiplicative potential energy function is used to form the overall desired shape of the entire swarm. The shape formed using this method is a union of all the regions defined by corresponding inequality functions. This proposed method is a complement to our previous method where the additive potential energy is used to form the desired shape. By combining the multiplicative and additive potential energies, a variety of complicated shapes can be formed. Lyapunov-like function is presented for convergence analysis of the multi-robot systems. Simulation results are presented to illustrate the performance of the proposed method.

Cited by 19

PROFINITE GROUPS WITH MULTIPLICATIVE PROBABILISTIC ZETA FUNCTION

Eloisa Detomi, Andrea Lucchini · Journal of the London Mathematical Society · 2004

To a finitely generated profinite group G, a formal Dirichlet series PG(s)=∑nan/ns is associated, where an = ∑|G:H|=n μG(H). It is proved that G is prosoluble if and only if the sequence {an}n∈N\sis multiplicative, that is, ars = aras for any pair of coprime positive integers r and s. This extends the analogous result on the probabilistic zeta function of finite groups.

Cited by 10