C315 Riemann Zeta Function

Function subprograms RRIZET and DRIZET calculate the Riemann zeta function

$\zeta (x)\; =\; \sum$_k=1^&inf; k^-x = {1Γ(x)}∫₀^&inf;{t^x-1e^t-1}dt (x > 1)

for real arguments $x\; \ne 1$ , where $\zeta (x)$ is defined by analytic continuation for x < 1. For x = 1, $\zeta (x)$ has a pole of order one.

On CDC and Cray computers, the double-precision version DRIZET is not available.

Structure:

FUNCTION subprograms
User Entry Names: RRIZET, DRIZET
Files Referenced: Unit 6
External References: GAMMA, DGAMMA (C302), MTLMTR (N002), ABEND (Z035)

Usage:

In any arithmetic expression, RRIZET(X) or DRIZET(X)

has the value $\zeta (X)$ if , and $\zeta (X)-1$ if $X\; >\; 1$ , where RRIZET is of type REAL, DRIZET is of type DOUBLE PRECISION, and where X has the same type as the function name.

Method:

Rational Chebyshev approximation. For the reflection formula for $\zeta (x)$ is used.

Accuracy:

RRIZET (except on CDC and Cray computers) has full single-precision accuracy. For most values of the argument X, DRIZET (and RRIZET on CDC and Cray computers) has an accuracy of approximately one significant digit less than the machine precision.

Error handling:

Error C315.1: $X\; =\; 1$ . The function value is set to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

References:

1. W.J. Cody, K.E. Hillstrom, and H.C. Thather, Jr., Chebyshev approximations for the
   Riemann zeta function, Math. Comp. 25 (1971) 537--547.

C316