Répartition jointe dans les classes de résidus de la somme des chiffres pour deux représentations d’Ostrowski

Résumé. Pour deux entiers [Formula: see text], nous posons [Formula: see text] et [Formula: see text] (où [Formula: see text]) et nous notons respectivement [Formula: see text] et [Formula: see text] les fonctions sommes des chiffres dans les [Formula: see text] et [Formula: see text]-représentations d’Ostrowski de [Formula: see text]. Soient [Formula: see text] des entiers positifs tels que [Formula: see text] et [Formula: see text], nous obtenons une estimation en [Formula: see text] avec un terme d’erreur [Formula: see text] pour le cardinal de l’ensemble suivant [Formula: see text] pour tous les entiers [Formula: see text] et [Formula: see text] Notre résultat peut être comparé à celui de Bésineau et Kim qui ont traité le cas des [Formula: see text]-représentations dans différentes bases (qui sont premières entre elles). For two distinct integers [Formula: see text], we set [Formula: see text] and [Formula: see text] (where [Formula: see text] is the continued fraction [Formula: see text]) and we let [Formula: see text] and [Formula: see text] denote respectively, the sum of digits functions in the Ostrowski [Formula: see text] and [Formula: see text]-representations of [Formula: see text]. Let [Formula: see text] be positive integers satisfying [Formula: see text] and [Formula: see text], we obtain an estimation [Formula: see text] with an error term [Formula: see text] for the cardinality of the following set [Formula: see text] for all integers [Formula: see text] and [Formula: see text] Our result should be compared to that of Bésineau and Kim who addressed the case of the [Formula: see text]-representations in different bases (that are coprime).

Dirichlet series associated to sum-of-digits functions

We study the Dirichlet series [Formula: see text], where [Formula: see text] is the sum of the base-[Formula: see text] digits of the integer [Formula: see text], and [Formula: see text], where [Formula: see text] is the summatory function of [Formula: see text]. We show that [Formula: see text] and [Formula: see text] have analytic continuations to the plane [Formula: see text] as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions [Formula: see text] and [Formula: see text] to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.

The Properties of the Arithmetic Function as the Consecutive Sum of Digits in Natural Numbers

In this paper defines the consecutive sum of the digits of a natural number, so far as it becomes less than ten, as an arithmetic function called and then introduces some important properties of this function by proving a few theorems in a way that they can be used as a powerful tool in many cases. As an instance, by introducing a test called test, it has been shown that we are able to examine many algebraic equalities in the form of in which and are arithmetic functions and to easily study many of the algebraic and diophantine equations in the domain of whole numbers. The importance of test for algebraic equalities can be considered equivalent to dimensional equation in physics relations and formulas. Additionally, this arithmetic function can also be useful in factorizing the composite odd numbers.

Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue–Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this paper we prove a lower bound for the maximum order complexity of the sum of digits function in Zeckendorf base which is an example of a morphic sequence. We also prove that the polynomial subsequences of this sequence keep large maximum order complexity, such as the Thue–Morse sequence.

Uniform Distribution of the Weighted Sum-of-Digits Functions

The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h 1 sq, γ (n)+h 2 sq,γ (n +1), where h 1 and h 2 are integers such that h 1 + h 2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),s q,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.

A lower bound for Cusick’s conjecture on the digits of n + t

Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density $${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$ T. W. Cusick conjectured that c t > 1/2. We have the elementary bound 0 < c t < 1; however, no bound of the form 0 < α ≤ c t or c t ≤ β < 1, valid for all t, is known. In this paper, we prove that c t > 1/2 – ε as soon as t contains sufficiently many blocks of 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod’ko (2017) and pursued by Emme and Hubert (2018).

Statistical independence in mathematics–the key to a Gaussian law

In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the 1930s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. Among other things, we present the independence of the coefficients in a binary expansion together with a central limit theorem for the sum-of-digits function as well as the independence of divisibility by primes and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $$n\in{\mathbb{N}}$$ n ∈ N . We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.

A NOTE ON THE RELATIVE GROWTH RATE OF THE MAXIMAL DIGITS IN LÜROTH EXPANSIONS

This paper is concerned with the growth rate of the maximal digits relative to the rate of approximation of the number by its convergents, as well as relative to the rate of the sum of digits for the Lüroth expansion of an irrational number. The Hausdorff dimension of the sets of points with a given relative growth rate is proved to be full.