  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.
