arXiv subjects

Neha Elizabeth Thomas

Publications and source records attributed to Neha Elizabeth Thomas.

6 recordsLinked to original sources

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.

math.NT↗

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 $\max\limits_{N}\left|\sum \limits_{j=1}^{N^s}c_k^{(s)}(j) \right|$.

math.NT↗

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.

math.NT↗

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.

math.NT↗

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 $J_s$.

math.NT↗

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)}].

math.NT↗