The Riemann zeta-function and its derivatives

Abstract

Formulas for higher derivatives of the Riemann zeta-function are developed from Ramanujan’s theory of the ‘constant’ of series. By using the Euler-Maclaurin summation methods, formulas for ζ( n )( s ), ζ( n )(1 – s ) and ζ( n )(0) are obtained. Additional formulas involving the Stieltjes constants are also derived. Analytical expression for error bounds is given in each case. The formulas permit accurate derivative evaluation and the error bounds are shown to be realistic. A table of ζ'( s ) is presented to 20 significant figures for s = –20(0.1)20. For rational arguments, ζ(1/ k ), ζ'(1/ k ) are given for k = –10(1)10. The first ten zeros of ζ'( s ) are also tabulated. Because the Stieltjes constants appear in many formulas, the constants were evaluated freshly for this work. Formulas for the γ n are derived with new error bounds, and a tabulation of the constants is given from n = 0 to 100.