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The aim of this paper is to define some new number-theoretic functionsincluding necklaces polynomials and the numbers of special words such as Lyndonwords. By using Dirichlet convolution formula with well-known number-theoreticfunctions, we derive some new identities and relations associated withDirichlet series, Lambert series, and also the family of zeta functionsincluding the Riemann zeta functions and polylogarithm functions. By usinganalytic (meromorphic) continuation of zeta functions, we also deriveidentities and formulas including Bernoulli numbers and Apostol-Bernoullinumbers. Moreover, we give relations between number-theoretic functions and theFourier expansion of the Eisenstein series. Finally, we give some observationsand remarks on these functions.

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