On sets characterizing additive and multiplicative arithmetical functions

Multiplicative functions with bounded seminorm on the set of shifted primes

“…Now, by Lemma 1 the series (9) converges which implies (12). Further, /(2') ^ 0 for at least some t since ||/||i,ip+i > 0, and so by Theorem 3 the estimates (11) p -must hold for all x, and this together with (9) iraplies f € L* (see [11], Lemma 1). Now, the arguments used at the end of the proof of Lemma 1 give /" e Li for every of > 0, whlch proves assertion (i).…”

“…Observe that if (11) holds we obtain by (5) that / e Li(P +1) implies /eil. Further, because of (7) it is only necessary to prove the lower estimate in (11).…”

“…(see[11]) Let f be a non-negative multiplicative function such that < Ar^ for all prime powers p''. If f S L' then either M(f) := lim j /{") = 0 or e for any a > 0.PROOF.…”

Cesàro Means of Additive Functions

“…same proof as in[7] (see(10),(11),(12),(14) and Remark 3 in[7]) with Xj^ instead of x we obtain the convergence of the series (6) and the assertion of (7).For the proof in the other direction (sufficiency) we repeat the arguments used in [7], § 4, word for word and get M(|g] ,x)JJexp lg(p) 1-1 p<x P and 9 ^ L^ ^^ L*. This ends the proof of Theorem 1'.…”

On Sets of Uniqueness for Completely Additive Arithmetic Functions

“…In case G = R and H = {0}, Wolke [8] and, with a different proof, Indlekofer ([5], Theorem 1) showed that for a set A of R-uniqueness every η € Ν must be expressible as a finite product of rational powers of elements of A. Theorem 2 of the article [5] by Indlekofer proves that for Ή = Ζ the sets A of R/Z-uniqueness can be characterized by the property that every η 6 Ν can be expressed as a finite product of integer powers of elements of A. A more Brought to you by | University of Pennsylvania Authenticated Download Date | 6/18/15 7:15 AM The idea of the proof was to consider the multiplicative semigroup Ν as a generator family of the multiplicative group Q+ ·.= {m/n : m,n G N} of positive rationale.…”