Mean Values Of Multiplicative Functions Research Articles

An Averaged Chowla and Elliott Conjecture Along Independent Polynomials

We generalize a result of Matomaki, Radziwill, and Tao, by proving an averaged version of a conjecture of Chowla and a conjecture of Elliott regarding correlations of the Liouville function, or more general bounded multiplicative functions, with shifts given by independent polynomials in several variables. A new feature is that we recast the problem in ergodic terms and use a multiple ergodic theorem to prove it; its hypothesis is verified using recent results by Matomaki and Radziwill on mean values of multiplicative functions on typical short intervals. We deduce several consequences about patterns that can be found on the range of various arithmetic sequences along shifts of independent polynomials.

SIGN PATTERNS OF THE LIOUVILLE AND MÖBIUS FUNCTIONS

Let ${\it\lambda}$ and ${\it\mu}$ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for $({\it\lambda}(n),{\it\lambda}(n+1),{\it\lambda}(n+2))$ occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand’s result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $({\it\mu}(n),{\it\mu}(n+1))$. A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

Mean values of multiplicative functions over function fields

We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors’ new proof of Halasz’s theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.

Voronoi summation formulae and multiplicative functions on permutations

We prove a Tauberian theorem for the Voronoi summation method of divergent series with an estimate of the remainder term. The results on the Voronoi summability are then applied to analyze the mean values of multiplicative functions on random permutations.

A Tauberian theorem for the Ingham summation method

The aim of this work is to prove a Tauberian theorem for the Ingham summability method. The Tauberian theorem we prove is then applied to analyze asymptotics of mean values of multiplicative functions on natural numbers.

Distribution modulo 1 of a linear sequence associated to a multiplicative function evaluated at polynomial arguments

In two previous papers, the first named author jointly with Florian Luca and Henryk Iwaniec, have studied the distribution modulo 1 of sequences which have linear growth and are mean values of multiplicative functions on the set of all the integers. In this note, we give a first result concerning sequences with linear growth associated to the mean values of multiplicative functions on a set of polynomial values, proving the density modulo 1 of the sequence $$ \left( {\sum\limits_{m \leqslant n} {\frac{{\phi (m^2 + 1)}} {{m^2 + 1}}} } \right)_n . $$ This result is but an illustration of the theme which is currently being developed in the PhD thesis of the second named author.

Decay of Mean Values of Multiplicative Functions

AbstractFor given multiplicative function f , with |f(n)| ≤ 1 for all n, we are interested in how fast its mean value (1/x) Σn≤xf(n) converges. Halász showed that this depends on the minimum M (over y ∈ ℝ) of Σp≤x (1 – Re(f(p)p–iy )/p, and subsequent authors gave the upper bound ⪡ (1 + M)e–M. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Halász-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of y that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the k-th powers mod p.

Mean Values of Multiplicative Functions with Weight

We study the characteristics of the mean values of multiplicative functions, given the values of the functions on primes.

Tauberian theorems and mean values of multiplicative functions

One gives an application of Tauberian theorems to the problem of connection between the mean values of multiplicative functions for the cases when the argument runs through all natural numbers and all prime numbers, respectively.

Mean-Values of Multiplicative Functions and Natural Boundaries of Power Series with Multiplicative Coefficients

Mean-Values of Multiplicative Functions and Natural Boundaries of Power Series with Multiplicative Coefficients

A Probabilistic Approach to mean Values of Multiplicative Functions

Journal of the London Mathematical SocietyVolume 2, Issue Part_3 p. 405-419 Notes and papers A Probabilistic Approach to mean Values of Multiplicative Functions J. Galambos, J. Galambos Department of Mathematics, University of Ibadan, Ibadan, NigeriaSearch for more papers by this author J. Galambos, J. Galambos Department of Mathematics, University of Ibadan, Ibadan, NigeriaSearch for more papers by this author First published: July 1970 https://doi.org/10.1112/jlms/2.Part_3.405Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume2, IssuePart_3July 1970Pages 405-419 RelatedInformation