Riemann Zeta Function

Riemann zeta function.

The Riemann zeta function is the analytic continuation of the infinite series

zeta left-parenthesis s right-parenthesis equals sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts StartFraction 1 Over k Superscript s Baseline EndFraction

where s is a complex variable equal to σ + ti. The series is only convergent when the real part of s, σ, is greater than 1.

Usage

var zeta = require( '@stdlib/math/base/special/riemann-zeta' );

zeta( s )

Evaluates the Riemann zeta function as a function of a real variable s (i.e., t = 0).

var v = zeta( 1.1 );
// returns ~10.584

v = zeta( -4.0 );
// returns 0.0

v = zeta( 70.0 );
// returns 1.0

v = zeta( 0.5 );
// returns ~-1.46

v = zeta( 1.0 ); // pole
// returns NaN

v = zeta( NaN );
// returns NaN

Examples

var linspace = require( '@stdlib/math/utils/linspace' );
var zeta = require( '@stdlib/math/base/special/riemann-zeta' );

var s;
var v;
var i;

s = linspace( -50.0, 50.0, 200 );
for ( i = 0; i < s.length; i++ ) {
    v = zeta( s[ i ] );
    console.log( 's: %d, ζ(s): %d', s[ i ], v );
}