A note on q-multiplicative functions

We prove the following statement.

Let \( q \ge 2 \), \( q \in \mathbb{N} \) and let \( t:\mathbb{N}_0 \to \mathbb{R} \). Suppose that, for all \( v \in \mathbb{N} \) and \( 0 \le a_1, a_2 < q^v, a_1 \ne a_2 \), the sequence \( \eta_{{a_1, a_2 }} \left( b \right): = t\left( {a_1 + bq^v } \right) - t\left( {a_2 + bq^v } \right) \) satisfies the relation

where e(u) : = e^2πiu.

Then

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

Authors and Affiliations

1. Ain Shams University, Abbasia, Cairo, Egypt

   N. L. Bassily

2. Computer Algebra Department, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

   I. Kátai

Corresponding author

Correspondence to I. Kátai.

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