Representations of $\zeta(2n+1)$ and related numbers in the form of definite integrals and rapidly convergent series

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 $\beta(2m-1)$ ($m=1,2,\dots$) 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.