Related papers: Refactorisation of the Dirichlet convolution

The Mordell-Tornheim zeta function and the Herglotz-Zagier function $F(x)$ are two important functions in Mathematics. By generalizing a special case of the former, namely $\Theta(z, x)$, we show that the theories of these functions are…

We prove that in every ring of generalised power series with non-positive real exponents and coefficients in a field of characteristic zero, every series admits a factorisation into finitely many irreducibles of infinite support, the number…

To study a Dirichlet polynomial $f(s)=\frac{a_{m}}{m^{s}}+\cdots +\frac{a_{n}}{n^{s}}$ by regarding it as a multivariate polynomial via the canonical map $\phi$ sending $p_i^{-s}$ to an indeterminate $X_i$, with $p_i$ the $i$th prime…

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq…

We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its…

The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In…

In this article we study some properties of the discrete convolution of Liouville function $S(n):=\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$, which is a Goldbach-type counting function of representations. In…

We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$…

Update: This result was obtained by Milinovich with a better error term. He used $\zeta'(s)$, but we considered $Z'(t)$. We corrected a typo in the main theorem. We consider the sum of the square of the derivative of Hardy's $Z$-function…

Let $x \ge 2$, let $N_x$ be an integer chosen uniformly at random from the set $\mathbb Z \cap [1, x]$, and let $(V_1, V_2, \ldots)$ be a Poisson--Dirichlet process of parameter $1$. We prove that there exists a coupling of these two random…

We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd…

The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions,…

For $f \in H^p(\delta^2)$, $0<p\leq 2$, with Haar expansion $f=\sum f_{I \times J}h_{I\times J}$ we constructively determine the Pietsch measure of the $2$-summing multiplication operator \[\mathcal{M}_f:\ell^{\infty} \rightarrow…

Let $G$ be a profinite group. A strongly admissible smooth representation $\rho$ of $G$ over $\mathbb{C}$ decomposes as a direct sum $\rho \cong \bigoplus_{\pi \in \mathrm{Irr}(G)} m_\pi(\rho) \, \pi$ of irreducible representations with…

We consider the representation growth of split extensions of $SL_2^m(O)$. We prove that the corresponding representation zeta functions factor as a product of the representation zeta function of $SL_2^m(O)$ and the relative representation…

For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive…

We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these…

In 2008, Muneta found explicit evaluation of the multiple zeta star value $\zeta^\star(\{3, 1\}^d)$, and in 2013, Yamamoto proved a sum formula for multiple zeta star values on 3-2-1 indices. In this paper, we provide another way of…

Each irreducible fraction $p/q>0$ corresponds to a primitive vector $(p,q)\in\mathbb Z^2$ with positive coordinates. Such a vector $(p,q)$ can be uniquely written as the sum of two primitive vectors $(a,b),(c,d)\in\mathbb Z_{\geq 0}^2$…

In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a…