The Cotangent Function as an Avatar of the Polylogarithm Function of Order 0 and Ramanujan’s Formula

In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason is that the cotangent function (as a function in the upper half-plane, say) is the polylogarithm function of order 0 (with complex exponential argument), and therefore it shares properties intrinsic to the Lerch zeta-function of order 0. Here we view the Lerch zeta-function defined in the unit circle as a zeta-function in a wider sense, as a function defined in the upper and lower half-planes. As evidence, we give a plausibly most natural proof of Ramanujan’s formula, including the eta transformation formula as a consequence of the modular relation via the cotangent function, speculating the reason why Ramanujan had been led to such a formula. Other evidence includes the pre-Poisson summation formula as the pick-up principle (which in turn is a generalization of the argument principle).