With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomaki-Radziwill theorem that allow us to approximate the average of a general additive function over a typical short i...
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With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomaki-Radziwill theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a corresponding long average. As part of this treatment, we use a variant of the Matomaki-Radziwill theorem for divisor-bounded multiplicative functions recently proven in Mangerel (Divisor-bounded multiplicative functions in short intervals. arXiv: 2108.11401). We consider two sets of applications of these methods. Our first application shows that for an additive function g:N -> C any non-trivial savings in the size of the average gap vertical bar g(n) - g(n - 1)vertical bar implies that g must have a small first centred moment i.e. the discrepancy of g(n) from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erdos relating to characterizing constant multiples of log n as the only almost everywhere increasing additive functions. We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere well approximated by a constant times a logarithm. We also show that if the set {n is an element of N : g(n) < g(n -1)} is sufficiently sparse, and if g is not extremely large too often on the primes (in a precise sense), then g is identically equal to a constant times a logarithm.
A real-valued arithmetic function f is said to cluster around a pointr if the upper density of inputs n for which f(n) is within delta of r does not tend to zero as delta goes to zero. If f does not cluster around any...
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A real-valued arithmetic function f is said to cluster around a pointr if the upper density of inputs n for which f(n) is within delta of r does not tend to zero as delta goes to zero. If f does not cluster around any real number, then we say that f is nonclustering. We show that the product of nonclustering additive functions is nonclustering and provide a generalization for polynomials of nonclustering additive functions. We then use these results to prove that products of additive functions possessing continuous distribution functions also possess continuous distribution functions.
Let C be an affine plane curve. We consider additive functions for which , whenever . We show that if and C is the hyperbola with defining equation , then there exist nonzero additive functions with this property. Mor...
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Let C be an affine plane curve. We consider additive functions for which , whenever . We show that if and C is the hyperbola with defining equation , then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero f exists for a field K if and only if K is transcendental over or over , the finite field with p elements. We also consider the general question when K is a finite field. We show that if the degree of the curve C is large enough compared to the characteristic of K, then f must be identically zero.
We prove that the class of additive perfectly everywhere surjective functions contains (with the exception of the zero function) a vector space of maximal possible dimension (2(c)). Additionally, we show under the ass...
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We prove that the class of additive perfectly everywhere surjective functions contains (with the exception of the zero function) a vector space of maximal possible dimension (2(c)). Additionally, we show under the assumption of regularity of c that the family of additive everywhere surjective functions that are not strongly everywhere surjective contains (with the exception of the zero function) a vector space of dimension c(+).
Given an additive function f and a multiplicative function g, let E (f, g;x) = #{n 1. In particular, we show that for those additive functions f whose values f (n) are concentrated around their mean value lambda(n), ...
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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(f, g;x) for functions f such that f(n) not equal 0 for at least one integer n > 1. In particular, we show that for those additive functions f whose values f (n) are concentrated around their mean value lambda(n), one can find a multiplicative function g such that, given any epsilon > 0, then E (f,g;x) >> x/lambda(x)(1 + epsilon). We also show that given any additive function satisfying certain regularity conditions, no multiplicative function can coincide with it on a set of positive density. It follows that if omega(n) stands for the number of distinct prime factors of n, then, given any epsilon > 0, there exists a multiplicative function g such that E (omega, g;x) >> x/(log log x)(1+epsilon) , while for all multiplicative functions g, we have E (omega,g;x) = o(x) as x -> infinity.
We determine the solutions on various intervals in [0,infinity[ to the functional equation f(x(m)) = rf(x) for real r and positive m. Explicit formulas, involving periodic functions, are given for the set S of all sol...
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We determine the solutions on various intervals in [0,infinity[ to the functional equation f(x(m)) = rf(x) for real r and positive m. Explicit formulas, involving periodic functions, are given for the set S of all solutions. The formulas for r < 0 are more complicated. An approach to Swith the help of the axiom of choice is also given. A special attention is laidon solutions that are continuous on [0,infinity[ or on various open subintervals. We also describe solutions satisfying some asymptotic properties at the boundary of these intervals.
Given an integer k >= 2, let w( k) ( n ) denote the number of primes that divide n with multiplicity exactly k . We compute the density e k , m of those integers n for which w k ( n ) = m for every integer m >= ...
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Given an integer k >= 2, let w( k) ( n ) denote the number of primes that divide n with multiplicity exactly k . We compute the density e k , m of those integers n for which w k ( n ) = m for every integer m >= 0. We also show that the generating function Sigma infinity (m =u) e (k , m) z (m) is an entire function that can be written in the form Pi(p)(1 + ( p- 1)(z-1)Ip(k+i));from this representation we show how to both numerically calculate the e k , m to high precision and provide an asymptotic upper bound for the e k , m . We further show how to generalize these results to all additive functions of the form Sigma(infinity) (j =2 )a (j )w( j) ( n );when a (j )= j- 1 this recovers a classical result of R & eacute;nyi on the distribution of Omega(n)- w ( n ) .
Srinivasa Ramanujan provided series expansions of certain arithmetical functions in terms of the exponential sum defined by c(r) (n) = (r)Sigma(m=1(m,r)=1) l(2 pi imn/r) in (Trans Cambridge Philos Soc 22(13):259-276, ...
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Srinivasa Ramanujan provided series expansions of certain arithmetical functions in terms of the exponential sum defined by c(r) (n) = (r)Sigma(m=1(m,r)=1) l(2 pi imn/r) in (Trans Cambridge Philos Soc 22(13):259-276, 1918). Here we give similar type of expansions in terms of the Cohen-Ramanujan sum defined by Cohen (Duke Math J 16(85-90):2, 1949) by c(r)(s) (n) = (rs)Sigma(s)(h=1(h,r)()s=1) e(2 pi inh)/(rs). We also provide some necessary and sufficient conditions for such expansions to exist.
We will address the questions and conjectures left by the recent papers by Ferreira et al. (Bollettino dell'Unione Matematica Italiana, 2023, https://***/10.1007/s40574-023-00402-7), (J Algebra 638:488-505, 2024)....
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We will address the questions and conjectures left by the recent papers by Ferreira et al. (Bollettino dell'Unione Matematica Italiana, 2023, https://***/10.1007/s40574-023-00402-7), (J Algebra 638:488-505, 2024). We also present an example showing the necessity of the conditions of the results that answer the conjectures. It leads us to the generalized Vukman equation: f(x)+xng(x-1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(x)+x<^>ng(x<^>{-1})=0, \end{aligned}$$\end{document}for every invertible x, where n is a nonnegative integer and f, g are additive maps on an alternative ring D. We will study this equation for the split octonion algebras, the alternative division rings, and the field Z2(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_2(t)$$\end{document} of rational functions over the field Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_2$$\end{document} with two elements.
Let f be a transformation on the Euclidean space such that f(0) = 0, and f preserves distances. Then f is linear, and this makes it easier to analyze f. The Mazur-Ulam theorem generalizes this to maps between real nor...
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Let f be a transformation on the Euclidean space such that f(0) = 0, and f preserves distances. Then f is linear, and this makes it easier to analyze f. The Mazur-Ulam theorem generalizes this to maps between real normed linear spaces. We discuss this theorem and its proofs.
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