Related papers: The divisor function over integers with a missing …

This paper investigates the randomness properties of a function of the divisor pairs of a natural number. This function, the antecedents of which go to very ancient times, has randomness properties that can find applications in…

We study odd numbers through a straightforward indexing. We focus in particular on odd prime and composite numbers and their distribution. With a counting argument, we calculate the limit of two sums and compare their convergence rate.

In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result…

Let $\tau_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}\tau_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log…

We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: $\kappa_x(n) = n^x + \sum_{d\lfloor n} \kappa_x(d)$, where the sum is over the proper divisors of $n$. We give a geometrical…

In DOI:10.1017/etds.2022.2 the author proved that for each integer $k$ there is an implicit number $M > 0$ such that if $b_1, \cdots , b_k$ are multiplicatively independent integers greater than $M$, there are infinitely many integers whose…

We associate a graph to a possible non-zero zero-divisor in the group algebra of a torsion-free group.

Generating series are crucial in enumerative combinatorics, analytic combinatorics, and combinatorics on words. Though it might seem at first view that generating Dirichlet series are less used in these fields than ordinary and exponential…

We prove the analogue of an identity of Huard, Ou, Spearman and Williams and apply it to evaluate a variety of sums involving divisor functions in two variables. It turns out that these sums count representations of positive integers…

The binary sum-of-digits function $s$ counts the number of ones in the binary expansion of a nonnegative integer. For any nonnegative integer $t$, T.~W.~Cusick defined the asymptotic density $c_t$ of integers $n\geq 0$ such that…

We prove an asymptotic formula with power saving error term for a certain triple divisor sum.

We prove Bombieri-Vinogradov type theorems for primes with a missing digit in their $b$-adic expansion for some large positive integer $b$. The proof is based on the circle method, which relies on the Fourier structure of the integers with…

In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials

An unconditional inequality of the totient function is contributed to the literature. This result is associated with various problems about the distribution of prime numbers.

We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality of separation and covering. We provide analogues for various geometric…

This article studies the zero divisor graphs of the ring of Lipschitz integers modulo $n$. In particular we focus on the number of vertices, the diameter and the girth. We also give some results regarding the domination number of these…

In this article, we try to explain and unify standard divisibility tests found in various books. We then look at recurring decimals, and list a few of their properties. We show how to compute the number of digits in the recurring part of…

We obtain several asymptotic estimates for the sums of the restricted divisor function $$ \tau_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} $$ over short arithmetic progressions, which improve some results of J. Truelsen. Such…

In this paper we introduced an arithmetic graph function which associates with every group G the directed graph whose vertices corresponds to the divisors of |G|. With the help of such functions we introduced arithmetic graphs of classes of…

We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…