arXiv subjects

K Vishnu Namboothiri

Publications and source records attributed to K Vishnu Namboothiri.

17 recordsLinked to original sources

arXiv2025

Two identities involving Cohen-Ramanujan expansions

An arithmetical function $f$ is said to admit a \emph{Cohen-Ramanujan expansion} $f(n) := \sum\limits_{r}\widehat{f}(r)c_r^s(n)$, if the series on the right hand side converges for suitable complex numbers $\widehat{f}(r)$. Here $c_r^s(n)$ denotes the Cohen-Ramanujan sum defined by E. Cohen. We deduce here a Cohen-Ramanujan expansion for the Jordan totient function $J_k(n)$. Further, we give an an asymptotic formula for the sum $\sum\limits_{n \leq N} \frac{J_a(n)}{n^a} \frac{J_b(n+h)}{(n+h)^b}$ using the expansion we derive.

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arXiv2024

Some asymptotic formulae involving Cohen-Ramanujan expansions

Some necessary and sufficient conditions for the existence of Cohen-Ramanujan expansions for arithmetical functions were provided by these authors in [\textit{arXive preprint arXive:2205.08466}, 2022]. Given two arithmetical functions $f$ and $g$ with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for $\sum_{n\leq N}f(n)g(n+h)$ where $h$ is a fixed positive integer. We also provide Cohen-Ramanujan expansions for certain functions to illustrate some of the results we prove consequently.

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arXiv2023

A computable formula for evaluating the mean square sum of $L$-functions

For Dirichlet characters $χ$ mod $k$ where $k\geq 3$, we here give a computable formula for evaluating the mean square sums $\sum\limits_{\substack{χ\text{ mod }k\\χ(-1)=(-1)^r}}|L(r,χ)|^2$ for any positive integer $r\geq 3$. We also give an inductive formula for computing the sum $\sum\limits_{\substack{1\leq m\leq k \\ (m, k)=1}}\frac{1}{\left(\sin\left(\frac{πm}{k}\right)\right)^{2n}}$ where $n$ is a positive integer in terms of Bernoulli numbers and binomial coefficients.

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arXiv2023

On near orthogonality of certain $k$-vectors involving generalized Ramanujan sums

The near orthgonality of certain $k$-vectors involving the Ramanujan sums were studied by E. Alkan in [J. Number Theory, 140:147--168 (2014)]. Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by E. Cohen in [Duke Math. J., 16(2):85--90 (1949)]. We also prove that the weighted average $\frac{1}{k^{s(r+1)}}\sum \limits_{j=1}^{k^s}j^rc_k^{(s)}(j)$ remains positve for all $r\geq 1$. Further, we give a lower bound for $\max\limits_{N}\left|\sum \limits_{j=1}^{N^s}c_k^{(s)}(j) \right|$.

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arXiv2023

On an identity of Delange and its application to Cohen-Ramanujan expansions

Srinivasa Ramanujan provided Fourier series expansions of certain arithmetical functions in terms of the exponential sum defined by $c_q(n)=\sum\limits_{\substack{{m=1}\\(m,q)=1}}^{q}e^{\frac{2 πimn}{q}}$. Later, H. Delange derived the bound $\sum\limits_{q|k}|c_q(n)|\leq n\, 2^{ω(k)}$ and gave a sufficient condition for such expansions to exist. A. Grytczuk gave an exact value for this bound, and derived a converse implication of the absolute convergence stated by H. Delange. We here show that these results have natural generalizations in terms of the Cohen-Ramanujan sum $c_q^{(s)}(n)$ defined by E. Cohen in [\emph{Duke Mathematical Journal, 16(85-90):2, 1949}]. We derive a bound as well as exact value for $\sum\limits_{q|k}|c_q^{(s)}(n)|$ and provide a sufficient condition for the Cohen-Ramanujan expansions to exist.

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arXiv2023

A reciprocity theorem for the Cohen-Ramanujan sums and its application to Cohen-Ramanujan expansions in the second variable

For an arithmetical function $f$, its Ramanujan expansion is a series expansion in the form $f(n)=\sum\limits_{k=1}^{\infty}a(k) c_k(n)$ where $a(k)$ are complex numbers and $c_k(n):= \sum\limits_{\substack{m=1\\(m, k)=1}}^{k}e^{\frac{2πimn}{k}}$ is the Ramanujan sum. Here we prove a reciprocity result on Cohen-Ramanujan sums $c_k^s(n) :=\sum\limits_{\substack{{h=1}\\(h,k^s)_s=1}}^{k^s}e^{\frac{2πi n h}{k^s}}$ to change the position of $k$ and $n$ in a twisted function and use it to prove that for certain arithmetical functions $f$, Cohen-Ramanujan series expansions in the form $\sum\limits_{k=1}^{\infty}a(k) c_k^{(s)}(n)$ exist if and only if expansions in the form $\sum\limits_{k=1}^{\infty}b(k/n) c_n^{(s)}(k)$ exist.

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arXiv2023

On an exponential power sum

Using combinatorial techniques, we derive a recurrence identity that expresses an exponential power sum with negative powers in terms of another exponential power sum with positive powers. Consequently, we derive a formula for the power sum of the first $k$ natural numbers when the power is odd, which when used in combination with Faulhaber's formula for computing power sums helps us to retrieve the Bernoulli numbers in certain cases.

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arXiv2023

A Menon-type Identity derived using Cohen-Ramanujan sum

Menon's identity is a classical identity involving gcd sums and the Euler totient function $ϕ$. We derived the Menon-type identity $\sum\limits_{\substack{m=1\\(m.n^s)_s=1}}^{n^s} (m-1,n^s)_s=Φ_s(n^s)τ_s(n^s)$ in Czechoslovak Math. J., 72(1):165-176 (2022) where $Φ_s$ denotes the Klee's function and $(a,b)_s$ denotes a a generalization of the gcd function. Here we give an alternate method to derive this identity using the concept of Cohen-Ramanujan sum.

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arXiv2023

On a Ramanujan type expansion of arithmetical functions

Srinivasa Ramanujan provided series expansions of certain arithmetical functions in terms of the exponential sums defined by $c_r(n) = \sum\limits_{\substack{{m=1}\\ (m,r)=1}}^{r} e^{\frac{2 πimn}{r}}$ in [Trans. Cambridge Phillos. Soc, 22(13):259-276,1918]. Here we give similar type of expansions in terms of the Cohen-Ramanujan sum defined by E. Cohen in [Duke Mathematical Journal, 16(85-90):2, 1949] as $c_r^s(n)=\sum\limits_{\substack{h=1\\ (h,r^s)_s=1}}^{r^s}e^{\frac{2πi n h}{r^s}}$. We also provide some necessary and sufficient conditions for such expansions to exist.

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arXiv2021

A Menon-type Identity concerning Dirichlet characters and a generalization of the gcd function

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.

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arXiv2021

On two mean square averages of Dirichlet $L$-function

Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $χ$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,χ)$ and $L(4,χ)$ using certain trigonometric sums and Bernoulli polynomials and express them in terms of the Euler totient function $ϕ$ and the Jordan totient function $J_s$.

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arXiv2020

A Menon-type identity using Klee's function

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)}].

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arXiv2018

The Discrete Fourier Transform of $\mathbf{(r,s)-}$even functions

An $(r,s)$-even function is a special type of periodic function mod $r^s$. These functions were defined and studied for the the first time by McCarthy. An important example for such a function is a generalization of Ramanujan sum defined by Cohen. In this paper, we give a detailed analysis of DFT of $(r,s)$-even functions and use it to prove some interesting results including a generalization of the Hölder identity. We also use DFT to give shorter proofs of certain well known results and identities .

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arXiv2017

On the number of solutions of a restricted linear congruence

Consider the linear congruence equation $${a_1^{s}x_1+\ldots+a_k^{s} x_k \equiv b\,(\text{mod } n^s)}\text { where } a_i,b\in\mathbb{Z},s\in\mathbb{N}$$ Denote by $(a,b)_s$ the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$ simultaneously. Given $t_i|n$, we seek solutions $\langle x_1,\ldots,x_k\rangle\in\mathbb{Z}^k$ for this linear congruence with the restrictions $(x_i,n^s)_s=t_i^s$. Bibak et al. [J. Number Theory, 171:128-144, 2017] considered the above linear congruence with $s=1$ and gave a formula for the number of solutions in terms of the Ramanujan sums. In this paper, we derive a formula for the number of solutions of the above congruence for arbitrary $s\in\mathbb{N}$ which involves the generalized Ramanujan sums defined by E. Cohen [Duke Math. J, 16(85-90):2, 1949]

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arXiv2017

On solving a restricted linear congruence using generalized Ramanujan sums

Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\,(\text{mod } n)$ for $b,n\in\mathbb{Z}$. By $(a,b)_s$, we mean the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$ simultaneously. For each $d_j|n$, define $\mathcal{C}_{j,s} = \{1\leq x\leq n^s | (x,n^s)_s = d^s_j\}$. Bibak et al. gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ by proving that for the above linear congruence, the number of solutions is $$\frac{1}{n^s}\sum\limits_{d|n}c_{d,s}(b)\prod\limits_{j=1}^{τ(n)}\left(c_{\frac{n}{d_j},s}(\frac{n^s}{d^s})\right)^{g_j}$$ where $g_j = |\{x_1,\ldots, x_k\}\cap \mathcal{C}_{j,s}|$ for $j=1,\ldots τ(n)$ and $c_{d,s}$ denote the generalized ramanujan sum defined by E. Cohen.

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arXiv2016

Certain weighted averages of generalized Ramanujan sums

We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Toth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose a generalization of Ramanujan sum given by E. Cohen and derive the weighted averages corresponding to the versions of the weighted averages established by Toth.

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arXiv2011

On Counting Twists of a Character Appearing in its Associated Weil Representation

Consider an irreducible, admissible representation $π$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $π_+ + π_-$. If $π=r_θ$, the Weil representation of GL(2,$F$) attached to a character $θ$ of $K^*$ which does not factor through the norm map from $K$ to $F$, then $χ\in \hat{K^*}$ with $(χ>. θ^{-1})|_{F^{*}}=ω_{K/F}$ occurs in ${r_θ}_+$ if and only if $ε(θχ^{-1},ψ_0)=ε(\bar θχ^{-1},ψ_0)=1$ and in ${r_θ}_-$ if and only if both the epsilon factors are -1. But given a conductor $n$, can we say precisely how many such $χ$ will appear in $π$? We calculate the number of such characters at each given conductor $n$ in this work.

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