The Riemann zeta function and finite-part integration

Abstract

The derivative of the Riemann zeta function at negative integer arguments was discovered to be related to the finite-part of the divergent integral ∫₀^∞ t^−λ csch(t) dt. The finite-part integral was then expressed using integrals and infinite series to obtain new expressions for the derivative of the zeta function at negative integer arguments. Known expressions were then used to uncover new expressions for the Riemann zeta function at positive odd arguments.