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

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.