Related papers: A divisor function of Wigert and higher degree for…

In this article, we present relations for the Euler totient function $\varphi(n)$ and the number of divisors $\tau(n)$ in terms of finite sums of integer parts of rational numbers or greatest common divisors of pairs of integers. Some of…

A representation of SL(2,Z) by integer matrices acting on the space of analytic ordinary Dirichlet series is constructed, in which the standard unipotent element acts as multiplication by the Riemann zeta function. It is then shown that the…

For functions defined via Dirichlet/generalized Dirichlet series in some half planes of the complex plane, we give a new simple elementary approach to obtain an Approximate Functional Equation(AFE for short) for the product of functions…

Motivated by arithmetic applications on the number of points in a bihomogeneous variety and on moments of Dirichlet $L$-functions, we provide analytic continuation for the series $\mathcal…

We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion…

We study the distribution of divisors of Euler's totient function and Carmichael's function. In particular, we estimate how often the values of these functions have "dense" divisors.

Let $\varphi(n)$ denote the Euler totient function. In this paper, we first establish a new upper bound for $n/\varphi(n)$ involving $K(n)$, the function that counts the number of primorials not exceeding $n$. In particular, this leads to…

Let $d_k(n) = \sum_{n_1 \cdots n_k = n}1$ be the $k$-fold divisor function. We call a function $f:\mathbb{N} \to \mathbb{C}$ a $d_k$-bounded multiplicative function, if $f$ is multiplicative and $|f(n)| \leq d_k(n)$ for every $n \in…

Wigner functions help visualise quantum states and dynamics while supporting quantitative analysis in quantum information. In the discrete setting, many inequivalent constructions coexist for each Hilbert-space dimension. This fragmentation…

A Fej\'er-Dirichlet lift is developed that turns divisor information at the integers into entire interpolants with explicit Dirichlet-series factorizations. For absolutely summable weights the lift interpolates $(a*1)(n)$ at each integer…

We improve unconditional estimates on $\Delta_k(x)$, the remainder term of the generalised divisor function, for large $k$. In particular, we show that $\Delta_k(x) \ll x^{1 - 1.889k^{-2/3}}$ for all sufficiently large fixed $k$.

Let $d(n)$ be the divisor function and it is well known that $\sum_{1\leq n \leq x}d(n) = x\log x+(2\gamma-1)x +\mathcal{O}\left(x^{\theta+\epsilon}\right)$ where $\gamma$ is the Euler constant, $\epsilon>0$ and $1/4<\theta<1/3$. In this…

We show that if $F(s)$ is a nondegenerate ordinary Dirichlet series with nonnegative coefficients and $F(k)$ is a rational number for all large enough positive integers $k$, then the denominators of those rational numbers are unbounded. In…

The Dirichlet divisor problem is used as a model to give a conjecture concerning the conditional convergence of the Dirichlet series of an L-function.

In this paper, we apply the Dirichlet convolution method to \begin{equation*} T_{k}(x)=\sum_{n \leq x} d_{k}(n), \end{equation*} for $k\ge 3$, where $d_{k}(n)$ is the number of ways to represent $n$ as a product of $k$ positive integer…

Let $d(n)$ be the number of divisors of $n$, let $\gamma$ denote Euler's constant and $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote…

The main object of this paper is to give the generalized von mangoldt function using the L-additive function which can help us to make it possible to calculate The Dirichlet series of the arithmetic derivative $\delta$ and Dirichlet series…

N. Minculete has introduced a concept of divisors of order $r$: integer $d=p_1^{b_1}\cdots p_k^{b_k} $ is called a divisor of order $r$ of $n=p_1^{a_1}\cdots p_k^{a_k}$ if $d \mid n$ and $b_j\in\{r, a_j\}$ for $j=1,\ldots,k$. One can…

Several results are obtained concerning the function $\Delta_k(x)$, which represents the error term in the general Dirichlet divisor problem. These include the estimates for the integral of this function, as well as for the corresponding…

Let $$\lambda(s)=\sum_{n=0}^\infty\frac1{(2n+1)^s},$$ $$\beta(s)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(2n+1)^s},$$ and $$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}$$ be the Dirichlet lambda function, its alternating form, and the Dirichlet…