Identities and relations related to the numbers of special words derived from special series with Dirichlet convolution

TL;DR

This paper introduces new number-theoretic functions related to special words and series, deriving identities involving Dirichlet convolutions, zeta functions, Bernoulli numbers, and Eisenstein series.

Contribution

It defines novel functions like necklaces polynomials and special word counts, establishing new identities and relations using Dirichlet convolutions and analytic continuation of zeta functions.

Findings

Derived identities involving Dirichlet series and Lambert series.

Established relations between number-theoretic functions and Eisenstein series.

Connected Bernoulli numbers with special word functions.

Abstract

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, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers. Moreover, we give relations between number-theoretic functions and the Fourier expansion of the Eisenstein series. Finally, we give some observations and remarks on these functions.