Relations and positivity results for the derivatives of the Riemann ξ function

摘要

We present and evaluate the integer-order derivatives of the Riemann xi function. These derivatives contain logarithmic integrals of powers multiplying a specific Jacobi theta function and as such can be alternatively viewed as certain Mellin transforms at integer argument. We describe how the derivatives at s=0, s=12, and s=1 can be evaluated exactly. We further show, based upon a novel representation, that the even order derivatives at s=12 are all positive, as are all derivatives at s=1. An expression is presented for the derivatives on the critical line, which may be useful in studying the zeros of the function Ξ(t)=ξ(12+it).