On multiplicative functions which are additive on sums of primes

Let ${f : \mathbb{N} \to \mathbb{C}}$ be a multiplicative function satisfying f(p ₀) ≠ 0 for at least one prime number p ₀, and let k ≥ 2 be an integer. We show that if the function f satisfies f(p ₁ + p ₂ + . . . + p _k ) = f(p ₁) + f(p ₂) + . . . + f(p _k ) for any prime numbers p ₁, p ₂, . . . ,p _k then f must be the identity f(n) = n for each ${n \in \mathbb{N}}$ . This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k ≥ 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.     

On additive and multiplicative Hilbert cubes

Given subset E of natural numbers FS(E) is defined as the collection of all sums of elements of finite subsets of E and any translation of FS(E) is said to be Hilbert cube. We can define the multiplicative analog of Hilbert cube as well. E.G. Strauss proved that for every ε>0 there exists a sequence with density >1−ε which does not contain an infinite Hilbert cube. On the other hand, Nathanson showed that any set of density 1 contains an infinite Hilbert cube. In the present note we estimate the density of Hilbert cubes which can be found avoiding sufficiently sparse (in particular, zero density) sequences. As a consequence we derive a result in which we ensure a dense additive Hilbert cube which avoids a multiplicative one.     

On multiplicative functions on consecutive integers

Let f and g be multiplicative functions of modulus 1. Assume that \( {\lim_{x \to \infty }}\frac{1}{x}\left| {\sum\nolimits_{n \leqslant x} {f(n)} } \right| = A > 0 \) and \( {\lim_{x \to \infty }}\frac{1}{x}\left| {\sum\nolimits_{n \leqslant x} {g(n)} } \right| = 0 \). We prove that, under these conditions,

$ \mathop {\lim }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {f(n)g(n + 1) = 0.}$

Concerning the Liouville function λ, we find an upper estimate for \( \frac{1}{x}\left| {\sum\limits_{n \leqslant x} {\lambda (n)\lambda (n + 1)} } \right| \) under the unproved hypothesis that L(s, χ) have Siegel zeros for an infinite sequence of L-functions.

     

Connection between sets of uniqueness for multiplicative systems of functions and sets of uniqueness for multiplicative transforms

A problem of uniqueness for series over multiplicative systems of functions and for multiplicative transforms is considered. It is shown that each set of uniqueness for a multiplicative transform is specified by a countable collection of sets of uniqueness for series over the corresponding multiplicative system of functions. Each set of uniqueness for a series over a multiplicative system of functions is a portion on [0, 1) of some set of uniqueness for the corresponding multiplicative transform.     

On additive representation functions of finite sets, I (Variation)

If m ∈ ?, ? _m is the additive group of the modulo m residue classes, $\mathcal{A} \subset \mathbb{Z}_m$ and n ∈ ?, ? _m , then let $R\left( {\mathcal{A},n} \right)$ denote the number of solutions of a+a′ = n with $a,a' \in \mathcal{A}$ . The variation $V(\mathcal{A}) = \mathop {\max }\limits_{n \in \mathbb{Z}_m } |R(\mathcal{A},n + 1) - R(\mathcal{A},n)|$ is estimated in terms of the number of a’s with $a - 1 \notin \mathcal{A}$ , $a \in \mathcal{A}$ .     

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.     

On functions behaving like additive functions

Let 0 < 1. In the paper we consider the following inequality: |f(x + y) – f(x) – f(y)| min{|f(x + y)|, |f(x) + f(y)|}, wheref: R R. Solutions and continuous solutions of this inequality are investigated. They have similar properties as additive functions, e.g. if the solution is bounded above (below) on a set of positive inner Lebesgue measure then it is continuous. Some sufficient condition for this inequality is also given.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

On multiplicative functions which are small on average

Let f be a completely multiplicative function that assumes values inside the unit disc. We show that if ${\sum_{n \leq x}f(n)\ll x/(\rm log x)^A}$ ∑ n ≤ x f ( n ) ? x / ( l o g x ) A , ${x \geq 2}$ x ≥ 2 , for some A > 2, then either f(p) is small on average or f pretends to be ${\mu(n)n^{it}}$ μ ( n ) n i t for some t.     

On approximately additive functions

Let C be a convex symmetric subset of a real Banach space F and K be a subgroup of the group (F,+). Let E be a real linear space, h:E→F, and h(x+y)−h(x)−h(y)∈K+C for x,y∈E. We prove that under some additional assumptions h can be represented in the form: h=A+γ+κ with an additive (or linear) A:E→F and some γ:E→C, κ:E→K.