A sequence is called $k$-automatic if the $n$’th term in the sequence can be generated by a finite state machine, reading $n$ in base $k$ as input. We show that for many multiplicative functions, the sequence ${\left(f\left(n\right)\text{mod}v\right)}_{n\ge 1}$ is not $k$-automatic. Among these multiplicative functions are ${\gamma }_m\left(n\right),{\sigma }_m\left(n\right),\mu \left(n\right)$ et $\phi \left(n\right)$.

Granville and Soundararajan have recently suggested that a general study of multiplicative functions could form the basis of analytic number theory without zeros of L-functions; this is the so-called pretentious view of analytic number theory. Here we study multiplicative functions which arise from the arithmetic of number fields. For each finite Galois extension K/ℚ, we construct a natural class ${}_K$ of completely multiplicative functions whose values are dictated by Artin symbols, and we show that...

Given a binary recurrence ${u_n}_{n\ge 0}$, we consider the Diophantine equation $u_{n_1}^{x_1}\cdots u_{n_L}^{x_L}=1$ with nonnegative integer unknowns $n_1,...,n_L$, where $n_i\ne n_j$ for 1 ≤ i < j ≤ L, $max|x_i|:1\le i\le L\le K$, and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.

AMS Subj. Classification: MSC2010: 42C10, 43A50, 43A75We perform analysis of certain aspects of approximation in multiplicative systems that appear as duals of ultrametric structures, e.g. in cases of local fields, totally disconnected Abelian groups satisfying the second axiom of countability or more general ultrametric spaces that do not necessarily possess a group structure. Using the fact that the unit sphere of a local field is a Vilenkin group, we introduce a new concept of differentiation in...

We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete 'multiplicative' zero-one law is established akin to the 'simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile 'cross fibering principle'. In a nutshell it enables us to 'lift' zero-one laws to higher...

We consider the Tribonacci sequence $T:={T_n}_{n\ge 0}$ given by T₀ = 0, T₁ = T₂ = 1 and $T_{n+3}=T_{n+2}+T_{n+1}+T_n$ for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

We establish a new multiplicity lemma for solutions of a differential system extending Ramanujan’s classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd positive integers (Nesterenko, 2011).