Evaluation of Dirichlet Series

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

“…On the other hand, Proposition 3.5 can be obtained directly from (2.2) by a straightforward argument consisting of inverting the discrete Fourier transform of m-periodic even sequences (see [3]). The method described in [3] allows one to sum periodic Dirichlet series in general and does not require separate arguments according to the parity of m. However, the approach in [3] does not reveal the regularity of (3.7) and the explicit formula for the inverse of the cosine matrix. In addition, since µ(n) is not periodic, the method does not provide elementary expressions for the x r starting from (2.4), i.e.…”

A generating function for a class of generalized Bernoulli polynomials

“…Our formulas are of such a generality as to allow us to evaluate instances of famous series such as in examples (1) to (4). Notice however that formulas (1) to (3) can be obtained by using results of the first author [3]. This is because the series (1) to (3) are of the form λ j /j n , where {λ j } is a periodic sequence and n ∈ N. Theorem 3 in this note goes beyond the results in [3] by evaluating more general series of the form λ j c j /j n where {λ j } is a periodic sequence, n ∈ N and {c j } is the sequence of Fourier coefficients of a well behaved function.…”

“…Notice however that formulas (1) to (3) can be obtained by using results of the first author [3]. This is because the series (1) to (3) are of the form λ j /j n , where {λ j } is a periodic sequence and n ∈ N. Theorem 3 in this note goes beyond the results in [3] by evaluating more general series of the form λ j c j /j n where {λ j } is a periodic sequence, n ∈ N and {c j } is the sequence of Fourier coefficients of a well behaved function. Here is a final example, with c j = 1/(4π 2 j 2 + 1), The generating function of Theorem 1 has applications in the generalization of the Euler-Maclaurin formula presented in [4].…”

“…Formulas (13) and (15) in Theorem 3 below appear already in [3]. For the proof of formulas (14) and (16) one proceeds as in [3].…”

Generalized Fractional-Order Bernoulli Functions via Riemann-Liouville Operator and Their Applications in the Evaluation of Dirichlet Series

“…For the proof of the following theorems one proceeds as in Balanzario [10], using Theorem 8 and (16).…”