Representations of $$\zeta (2n + 1)$$ and Related Numbers in the Form of Definite Integrals and Rapidly Convergent Series
2020
Let $$\zeta (s)$$
and $$\beta (s)$$
be the Riemann zeta function and the Dirichlet beta function. The formulas for calculating the values of $$\zeta (2m)$$
and $$\beta (2m - 1)$$
(
$$m = 1,\;2,\; \ldots $$
) are classical and well known. Our aim is to represent $$\zeta (2m + 1)$$
, $$\beta (2m)$$
, and related numbers in the form of definite integrals of elementary functions and rapidly converging numerical series containing $$\zeta (2m)$$
. By applying the method of this work, on the one hand, both classical formulas and ones relatively recently obtained by others researchers are proved in a uniform manner, and on the other hand, numerous new results are derived.
- Correction
- Source
- Cite
- Save
- Machine Reading By IdeaReader
8
References
0
Citations
NaN
KQI