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A note on q-multiplicative functions

Authors:

N. L. Bassily I. Kátai

Affiliation:

1.Ain Shams University,Cairo,Egypt;2.Computer Algebra Department,E?tv?s Loránd University,Budapest,Hungary

Abstract:

We prove the following statement. Let $$ q ge 2 $$

and let $t:mathbb{N}_0 to mathbb{R}$ . Suppose that, for all $$ v in mathbb{N} $$

and

, the sequence $eta_{{a_1, a_2 }} left( b right): = tleft( {a_1 + bq^v } right) - tleft( {a_2 + bq^v } right)$ satisfies the relation

$frac{1}{x}sumlimits_{b < x} {eleft( {n_{{a_1, a_2 }} left( b right)} right) to 0quad left( {x to infty } right)}$

where e(u) : = e^2πiu. Then

$mathop {sup }limits_{{g in tilde{mathcal{M}}_q }} left| {frac{1}{x}sumlimits_{n < x} {gleft( n right)eleft( {tleft( n right)} right)} } right| to 0,$

where $tilde{mathcal{M}}_q$ q is the set of q-multiplicative functions g such that $$ left| {gleft( n right)} right| le 1left( {n = 1,2,...} right) $$

Keywords:

q-multiplicative functions theorem of Daboussi

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