The classical Menon's identity [7] states that
\begin{equation*}\label{oldbegin1} \sum_{\substack{a\in\Bbb Z_n^\ast }}\gcd(a -1,n)=φ(n) σ_{0} (n), \end{equation*} where for a positive integer $n$, $\Bbb Z_n^\ast$ is the group of units of the ring $\Bbb Z_n=\Bbb Z/n\Bbb Z$, $\gcd(\ ,\ )$ represents the greatest common divisor, $φ(n)$ is the Euler's totient function and $σ_{k} (n) =\sum_{d|n } d^{k}$ is the divisor function.
In this paper, we generalize Menon's identity with Dirichlet characters in the following way: \begin{equation*}
\sum_{\substack{a\in\Bbb Z_n^\ast b_1, ..., b_k\in\Bbb Z_n}}
\gcd(a-1,b_1, ..., b_k, n)χ(a)=φ(n)σ_k\left(\frac{n}{d}\right), \end{equation*} where $k$ is a non-negative integer and $χ$ is a Dirichlet character modulo $n$ whose conductor is $d$.
Our result can be viewed as an extension of Zhao and Cao's result [16] to $k>0$.
It can also be viewed as an extension of Sury's result [12] to Dirichlet characters.
Menon's identity is a classical identity involving gcd sums and the Euler totient function $ϕ$. A natural generalization of $ϕ$ is the Klee's function $Φ_s$. In this paper we derive a Menon-type identity using Klee's function and a generalization of the gcd function. This identity generalizes an identity given by Lee and Kim in [\textit{J. Number Theory 175, 42--50(2017)}].
Menon's identity is a classical identity involving gcd sums and the Euler totient function $ϕ$. In a recent paper, Zhao and Cao derived the Menon-type identity $\sum\limits_{\substack{k=1}}^{n}(k-1,n)χ(k) = ϕ(n)τ(\frac{n}{d})$, where $χ$ is a Dirichlet character mod $n$ with conductor $d$. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao-Cao identity can be improved if one uses the method we employ here.
Let $χ$ be a Dirichlet character (mod $n$) with conductor $d$. In a quite recent paper Zhao and Cao deduced the identity $\sum_{k=1}^n (k-1,n) χ(k)= φ(n)τ(n/d)$, which reduces to Menon's identity if $χ$ is the principal character (mod $n$). We generalize the above identity by considering even functions (mod $n$), and offer an alternative approach to proof. We also obtain certain related formulas concerning Ramanujan sums.
We define the $k$-dimensional generalized Euler function $φ_k(n)$ as the number of ordered $k$-tuples $(a_1,\ldots,a_k)\in {\Bbb N}^k$ such that $1\le a_1,\ldots,a_k\le n$ and both the product $a_1\cdots a_k$ and the sum $a_1+\cdots +a_k$ are prime to $n$. We investigate some of properties of the function $φ_k(n)$, and obtain a corresponding Menon-type identity.
For every positive integer $n$, Sita Ramaiah's identity states that \medskip \begin{equation*} \sum_{a_1, a_2, a_1+a_2 \in (\mathbb{Z}/n\mathbb{Z})^*} \gcd(a_1+a_2-1,n) = ϕ_2(n)σ_0(n) \; \text{ where } \; ϕ_2(n)= \sum_{a_1, a_2, a_1+a_2 \in (\mathbb{Z}/n\mathbb{Z})^*} 1, \end{equation*} \medskip where $(\mathbb{Z}/n\mathbb{Z})^*$ is the multiplicative group of units of the ring $\mathbb{Z}/n\mathbb{Z}$ and $σ_s(n) = \displaystyle\sum_{d\mid n}d^s$. \smallskip This identity can also be viewed as a generalization of Menon's identity. In this article, we generalize this identity to an algebraic n…
We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary orders. We also point out extensions of Menon's identity in the one variable case, which seems to not appear in the literature.
By considering even functions (mod $n$) we generalize a Menon-type identity by Li and Kim involving additive characters of the group ${\Bbb Z}_n$. We use a different approach, based on certain convolutional identities. Some other applications, including related formulas for Ramanujan sums, are discussed, as well.
The Menon-Sury's identity is as follows: \begin{equation*} \sum_{\substack{1 \leq a, b_1, b_2, \ldots, b_r \leq n\\\mathrm{gcd}(a,n)=1}} \mathrm{gcd}(a-1,b_1, b_2, \ldots, b_r,n)=φ(n) σ_r(n), \end{equation*} where $φ$ is Euler's totient function and $σ_r(n)=\sum_{d\mid n}{d^r}$. Recently, Li, Hu and Kim \cite{L-K} extended the above identity to a multi-variable case with a Dirichlet character, that is, they proved \begin{equation*} \sum_{\substack{a\in\Bbb Z_n^\ast \\ b_1, \ldots, b_r\in\Bbb Z_n}} \mathrm{gcd}(a-1,b_1, b_2, \ldots, b_r,n)χ(a)=φ(n)σ_r{\left(\frac{n}{d}\right)}, \end{equation*} …
We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character free approach for the proof.
We investigate an arithmetic function representing a generalization of the gcd-sum function, considered by Kurokawa and Ochiai in 2009 in connection with the multivariable global Igusa zeta function for a finite cyclic group. We show that the asymptotic properties of this function are closely connected to the Piltz divisor function. A generalization of Menon's identity is also considered.
The $k$-dimensional generalized Euler function $φ_k(n)$ is defined to be the number of ordered $k$-tuples $(a_1,a_2,\ldots, a_k) \in \mathbb{N}^k$ with $1\leq a_1,a_2,\ldots, a_k \leq n$ such that both the product $a_1a_2\cdots a_k$ and the sum $a_1+a_2+\cdots+a_k$ are co-prime to $n$. Tóth proved that the identity \begin{equation*} \sum_{\substack{a_1,a_2,\ldots, a_k=1 \\ \gcd(a_1a_2\cdots a_k,n)=1\\ \gcd(a_1+a_2+\cdots+a_k,n)=1}}^n \gcd(a_1+a_2+\cdots+a_k-1,n) =φ_k(n)σ_0(n), \;\; \text{ where } σ_s(n) = \sum_{d\mid n}d^s \;\; \text{ holds. } \end{equation*} This identity can also be viewed a…
Let $\mathbb{A}=\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$. In this article, we prove a generalization of Tóth identity on $\mathbb{A}$ involving arithmetical functions, multiplicative and additive characters.
Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and collaborators, or the convolution of Bernoulli numbers by Y. Komori and collaborators.
In this paper, we rewrite two forms of an Euler-Ramanujan identity in terms of certain Dirichlet series and derive functional equation of the latter. We also use the Weierstrass-Enneper representation of minimal surfaces to obtain some identities involving these Dirichlet series and one complex parameter.
There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solution $f=f(s,w)$ of the functional equation $L(s-wf)=\exp(f)$, where $L(s)$ is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange-Bürmann formula for power series. The proof is probabilistic in natur…
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 with explicit remainder term. We do this,using a highly generalized identity following Motohashi's approach,wherein he uses a generalisation of Dirichlet's device.We use our method to give an AFE for the product of two Dirichlet L-series corresponding to primitive Dirichlet characters with hitherto unknown estimate for its remainder term.We also consideer AFEs …
In this work we introduce the numerical constant, LOPI, N LOPI is congruent to LOPI mod 18, equal to the lowest odd partition identity in conjunction with the reduced residue system Modulo 18, a complete disjoint covering residue system when considered in its whole set of residues from 0 to 17. By convolution of specific LOPI Dirichlet linear progressions for each LOPI superset, the non-prime elements of the 6 reduced residue congruence classes mod 18 are generated sequentially and uniquely. Through set generation and complete composite number subset generation based on the pattern of sequenti…
We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. A general identity is established. Appearing on one side is an infinite series involving $a(n)$ and modified Bessel functions $K_ν$, wherein on the other side is an infinite series involving $b(n)$ that is an analogue of the Hurwitz zeta function. Seven special cases, including $a(n)=τ(n)$ and $a(n)=r_k(n)$…
We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. Two general identities are established. The first involves the modified Bessel function $K_μ(z)$, and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_μ(z)$, the Bessel functions of imaginary argument…
