Multiplicative properties of sequences and digital expansions – MUNUM

The concept of digital expansion is fundamental to various branches of mathematics and computer science. Beside the instances of so-called q-ary numeration systems (binary, hexadecimal etc.), there are plenty of other digital systems which have been studied in recent years. Mention, for instance, digital expansions with respect to a base which satisfies a linear recursion. For a given numeration system there are plenty of digit functionals which are worth studying, both from a theoretical and practical point of view. We mention, for instance, the sum-of-digits function, the digit or block-counting functions, or more generally, so-called q-additive functions. Digital expansions have a variety of applications. For example, the construction of pseudo-random sequences is crucial in cryptography and intimately related to digital expansions.

The aim of this project is to make progress on several fundamental questions which interrelate digital expansions and its functionals (most prominently, the sum-of-digits function and q-additive functions) to classical objects of number theory (integer multiples, prime numbers, polynomials, recurrences). While each of these topics is well investigated, there is still lack of knowledge regarding the interplay between them. The project can be subdivided into three main parts, which are closely connected in terms of the phenomenon of « interaction ».

The first part is concerned with studying distribution properties of digital functionals on polynomials or quasi-polynomials in arithmetic progressions. A famous open problem of A. O. Gelfond (1968) regards the distribution of the sum of digits function of special sparse sequences (primes, polynomials) in arithmetic progressions. The cases of primes and squares have recently effectively been treated by C. Mauduit and J. Rivat, using a wealth of techniques from Fourier analysis, exponential sums and combinatorics. The aim is to pursue further this question for higher degree polynomials, where only lower density estimates are known. Also, we want to treat other open conjectures in this area, where the approach could be useful. The appropriate methodological machinery comes from analytic number theory in conjunction with thorough considerations of the arithmetics behind the notion of « digits ».

The second part is concerned with studying local distribution properties and oscillation phenomena of error terms of digital functions on integer multiples. Newman (1969) showed that the multiples of three have more often an even number of binary ones than an odd number. In recent years, similar « drifting phenomena » have been shown for many other different multiples, but a characterization is still unavailable. We aim for such a result. A related problem is to find asymptotic approximations of the number of n’s whose multiples (with an arbitrary multiplicative factor) have a given sum of digits. J. Schmid (1984) has obtained such a formula in the case of fixed multiplicative factor. It is a an open and demanding problem to provide such formulae for a large range of multiplicative factors.

The third part is concerned with studying the behaviour and distribution of friable integers (« entiers friables »). Friable integers have only small prime divisors and a thorough understanding of their properties is crucial for the quality and complexity of modern factorization algorithms. Recent progress has been made on studying the friable integers among polynomial sequences. The tools to attack such kind of problems of « interaction » stem from probabilistic models. One can refine the notion of friable integers to almost-friable integers, where the friable multiplicative part is large. Another point of interest is to study random multiplicative functions on friable integers, an area of considerable importance in recent years. Finally, it would be interesting to investigate the digital behaviour of friable numbers.