TY - JOUR
AU - Turing, A. M.
AB - 180 A. M. TURING [March 16, A METHOD FOE THE CALCULATION OF THE ZETA-FUNCTION By A. M. TUBING. [Received 7 March, 1939.—Read 16 March, 1939.] An asymptotic series for the zeta-function was found by Riemann and has been published by Siegel*, and applied by Titchmarsh f to the calculation of the approximate positions of some of the zeros of the function. It is difficult to obtain satisfactory estimates for the remainders with this asymptotic series, as may be seen from the first of these two papers of Titchmarsh, unless t is very large. In the present paper a method of calculation will be described, which, like the asymptotic formula, is based on the approximate functional equation; it is applicable for all values of 5. It is likely to be most valuable for a range of t where / is neither so small that the Euler-Maclaurin summation method can be used {e.g. t > 30) nor large enough for the Riemann-Siegel asymptotic formula (e.g. t < 1000). Roughly speaking, the method is to use the approximate functional equation for the zeta-function, with the remainder expressed as an integral, f* which for the moment we write as h(x)dz. We approximate to 
TI - A Method for the Calculation of the Zeta‐Function
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s2-48.1.180
DA - 1945-01-01
UR - https://www.deepdyve.com/lp/wiley/a-method-for-the-calculation-of-the-zeta-function-Q2K2k0Ncnh
SP - 180
EP - 197
VL - s2-48
IS - 1
DP - DeepDyve
ER -