THE RIEMANN ZETA-FUNCTION AND ITS DERIVATIVES

Formulas for higher derivatives of the Riemann zeta-function are devel oped from Ramanujan's theory of the 'constant' of series. By using the Euler-Maclaurin summation methods, formulas for zeta((n))(s), zeta((n ))(1-s) and zeta((n))(0) are obtained. Additional formulas involving t he Stieltjes constants are also derived. Analytical expression for err or bounds is given in each case. The formulas permit accurate derivati ve evaluation and the error bounds are shown to be realistic. A table of zeta'(s) is presented to 20 significant figures for s = -20(0.1)20. For rational arguments, zeta(1/kappa), zeta'(1/k) are given for k=-10 (1)10. The first ten zeros of zeta'(s) are also tabulated. Because the Stieltjes constants appear in many formulas, the constants were evalu ated freshly for this work. Formulas for the gamma(n) are derived with new error bounds, and a tabulation of the constants is given from n = 0 to 100.