Related papers: Explicit formulas for Euler's totient function and…
In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a…
In this paper, we introduce and develop the notion of spanning of integers along functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to a class of problems that requires to determine if the equations of the form…
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…
The purpose of this paper is to give some explicit formulas involving M\"obius functions, which may be known under the generalized Riemann Hypothesis, but unconditional in this paper. Concretely, we prove explicit formulas of partial sums…
For $\mu>0$ we study an asymptotic behavior of the sequence defined as $$T_{n}(\mu)=\frac{max_{1\leq m \leq {n^{\frac{1}{\mu}}}}\{\tau (n + m)\}}{\tau(n)},\ n=1,2,...$$ where $\tau(n)$ denotes the number of natural divisors of the given…
We present some Euler-type recurrences for the partition function $p(n)$.
This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a…
E158 in the Enestrom index. Translation of the Latin original "Observationes analyticae variae de combinationibus" (1741). This paper introduces the problem of partitions, or partitio numerorum (the partition of integers). In the first part…
The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlev\'{e}, etc.) were previously…
We study a weighted divisor function $\mathop{{\sum}'}\limits_{mn\leq x}\cos(2\pi m\theta_1)\sin(2\pi n\theta_2)$, where $\theta_i (0<\theta_i<1)$ is a rational number. By connecting it with the divisor problem with congruence conditions,…
Let $\phi(\cdot)$ and $\sigma(\cdot)$ denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of positive integers $m\le x$ for which the equation $m=n-\phi(n)$ has no…
We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \]…
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.
Euler discovered recurrence for divisor sum functions as a consequence of the pentagonal numbers theorem. With similar idea and also motivated by Ewell's work in 1977, we prove new recurrences for certain divisor sum functions and…
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…
An algorithm is devised for computing $\Phi(n) = \phi(1) + \phi(2) + \cdots + \phi(n)$ in time $\widetilde{\Theta}(n^{2/3})$ and space $\widetilde{\Theta}(n^{1/3})$. The starting point is an existing algorithm based on the Dirichlet…
In this paper, we obtain bounds on the $L^1$ norm of the sum $\sum_{n\le x}\tau(n) e(\alpha n)$ where $\tau(n)$ is the divisor function.
The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading…
Let $f$ be a real polynomial with irrational leading co-efficient. In this article, we derive distribution of $f(n)$ modulo one for all $n$ with at least three divisors and also we study distribution of $f(n)$ for all square-free $n$ with…
In this paper, we develop efficient and accurate algorithms for evaluating $\varphi(A)$ and $\varphi(A)b$, where $A$ is an $N\times N$ matrix, $b$ is an $N$ dimensional vector and $\varphi$ is the function defined by…