Multiplicative functions at consecutive integers. II

1988 ◽  
Vol 103 (3) ◽  
pp. 389-398 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Sufficient Conditions ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Multiplicative Functions ◽  
Global Behaviour ◽  
Consecutive Integers ◽  
The Mean ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Multiplicative Arithmetic

The global behaviour of multiplicative arithmetic functions has been extensively studied and is now well understood for a large class of multiplicative functions. In particular, Halász [5] completely determined the asymptotic behaviour of the meansfor multiplicative functions g satisfying |g| ≤ 1, and gave necessary and sufficient conditions for the existence of the ‘mean value’

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.

scholarly journals MEAN VALUE TYPE INEQUALITIES FOR QUASINEARLY SUBHARMONIC FUNCTIONS

2013 ◽  
Vol 55 (2) ◽  
pp. 349-368 ◽  
Author(s):  
OLEKSIY DOVGOSHEY ◽  
JUHANI RIIHENTAUS
Keyword(s):  
Sufficient Conditions ◽  
Continuous Functions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Subharmonic Functions ◽  
Mean Value Inequality ◽  
The Mean ◽  
Mean Value Inequalities ◽  
Necessary And Sufficient ◽  
Bounded Sets

AbstractThe mean value inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalise subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the characterisation of non-negative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalised mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the radius of this ball.

scholarly journals Two probability theorems and their application to some first passage problems

1964 ◽  
Vol 4 (2) ◽  
pp. 214-222 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
First Passage ◽  
Sums Of Random Variables ◽  
Large Numbers ◽  
The Mean ◽  
Image Position ◽  
Necessary And Sufficient

Let Xi, i = 1, 2, 3,··· be a sequence of independent and identically distributed random variables and write Sn = X1+X2+…+Xn. If the mean of Xi is finite and positive, we have Pr(Sn ≦ x) → 0 as n → ∞ for all x1 – ∞ < x < ∞ using the weak law of large numbers. It is our purpose in this paper to study the rate of convergence of Pr(Sn ≦ x) to zero. Necessary and sufficient conditions are established for the convergence of the two series where k is a non-negative integer, and where r > 0. These conditions are applied to some first passage problems for sums of random variables. The former is also used in correcting a queueing Theorem of Finch [4].

scholarly journals Mean Value theorems for multiplicative functions bounded in mean α-power, α >1

1980 ◽  
Vol 29 (2) ◽  
pp. 177-205 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Real Line ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Multiplicative Functions ◽  
The Real ◽  
Mean Value Theorems ◽  
Necessary And Sufficient

AbstractOn analogy with functions if Lebesuge class Lα on the real line the author considers those multiplicative arthmetic functions which are bounded in mean α>1. Necessary and sufficient conditions are obtained in order that they should have a mean-value, zero or non-zero. An application is made to Ramanujan's τ-function.

Multiplicative Functions in Short Intervals

1987 ◽  
Vol 39 (3) ◽  
pp. 646-672 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Number Theory ◽  
Mean Value ◽  
Considerable Progress ◽  
Central Problem ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Short Intervals ◽  
Probabilistic Number ◽  
The Mean ◽  
Image Position

A central problem in probabilistic number theory is to evaluate asymptotically the partial sumsof multiplicative functions f and, in particular, to find conditions for the existence of the “mean value”1.1In the last two decades considerable progress has been made on this problem, and the results obtained are very satisfactory.

Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables

Integers ◽  
2012 ◽  
Vol 12 (5) ◽  
Author(s):  
Noboru Ushiroya
Keyword(s):  
Infinite Product ◽  
Prime Numbers ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorems ◽  
Several Variables ◽  
The Mean ◽  
Multiplicative Arithmetic

Abstract.Letif this limit exists. We first generalize the Wintner theorem and then consider the multiplicative case by expressing the mean-value as an infinite product over all prime numbers. In addition, we study the mean-value of a function of the form

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

scholarly journals Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

2021 ◽  
Vol 56 (1) ◽  
pp. 79-94
Author(s):  
Nikola Lelas ◽  
Keyword(s):  
Rational Function ◽  
Finite Fields ◽  
Function Fields ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Irreducible Polynomials ◽  
Liouville Function ◽  
The Mean

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

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