zeta

Riemann zeta function of two arguments.

SYNOPSIS:

double x, q, y, zeta();
y = zeta(x, q);

DESCRIPTION:

                 inf.
                  -        -x
   zeta(x,q) =   >   (k+q)
                  -
                 k=0

where x > 1 and q is not a negative integer or zero.
The Euler-Maclaurin summation formula is used to obtain the expansion

                n
                -       -x
zeta(x,q) =   > (k+q)
                -
               k=1

           1-x                 inf. B   x(x+1)...(x+2j)
      (n+q)           1         -     2j
+ --------- - ------- +   >    --------------------
        x-1              x      -                   x+2j+1
                   2(n+q)      j=1       (2j)! (n+q)

where the B2j are Bernoulli numbers.
Note that zeta(x,1) = zetac(x) + 1.
(see zetac)

ACCURACY:

REFERENCE:

Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, Series, and Products, p. 1073; Academic Press, 1980.

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zeta