On a Ramanujan Identity and Its Generalizations

In the present paper, we propose a new method of derivation of number-theoretic identities which is applied to the proof of the multidimensional analogue of one of the Ramanujan identities. This method allows us to obtain new infinite series representations for the number $\pi$, of the values of the Riemann zeta function, and of the $L$-Dirichlet series at integer points.