On pseudorandom properties of multiplicative functions

In an earlier paper [3] Cassaigne et al studied the pseudorandom properties of the Liouville function. In this paper some of their results are generalized and sharpened considerably. This revised version was published online in June 2006 with corrections to the Cover Date.     

Summation of multiplicative functions

One sharpens the asymptotic formula of Wirsing's theorem in the case τ=0.     

Some properties of g-convex functions

In this paper,we obtain that eachg-convex function is continuous and convex,and we also extend Jia and Peng’s result on the characterization of g-convex function without the bounded assumption of the value of g at the origin.     

Some properties of Wilson functions

We prove three theorems on Wilson functions (these special functions were introduced by Groenevelt in 2003). These theorems were stated without proof and applied for the proof of a summation formula related to automorphic forms in our paper?[2].     

Some properties of porouscontinuous functions

The notion of porouscontinuous function will be introduced on the base of porous set and relations between porouscontinuous, continuous and quasicontinuous functions will be investigated.     

Some properties of almostS-continuous functions

Noiri et al. in [16] introduced and investigated the notion of almosts-continuous functions. The object of this paper is to investigate some more properties of this type of functions.     

Some properties of Shannon functions

Necessary and sufficient conditions are derived for an arbitrary numerical function to be equivalent to some Shannon function.Translated from Matematicheskie Zametki, Vol. 8, No. 5, pp. 663–674, November, 1970.The author wishes to express her gratitude to O. B. Lupanov who directed this work.     

Mean values of multiplicative functions

Let f(n) be a totally multiplicative function such that | ƒ (n)|⪯ 1 for all n, and let F(s) = ∑ ∞ n=1 ƒ(n)n —∞ be the associated Dirichlet series. A variant of Halász"s method is developed, by means of which estimates for ∑ N n=1 ƒ(n)/n are obtained in terms of the size of | F(s) | for s near 1 with ℜs >1. The result obtained has a number of consequences, particularly concerning the zeros of the partial sum U N (s) =∑ N n=1 n-s s of the series for the Riemann zeta function.     

Diophantine approximation with multiplicative functions

Let σ(n) denote the sum of divisors function. Our main result shows that, given any real α > 1 there are infinitely many integers n such that