Riemann Zeta Function ζ(x) Calculator

Evaluate the Riemann zeta function ζ(x) for a real argument. For x above 1 it uses a series plus the Euler–Maclaurin formula; for x below 1 it uses the functional equation (reflection formula), with notes on special values, the pole at x=1, and a graph.

Input

Enter a real argument x to evaluate the Riemann zeta function ζ(x). Values with x above 1 use a series with Euler–Maclaurin correction; values below 1 use the reflection formula.

Argument x

Any real number. x = 1 is a pole (diverges); negative even integers give ζ(x) = 0.

Result

Value of ζ(2)

1.6449340668

ζ(2) = π² ÷ 6 (the Basel problem)

Note

Since x is greater than 1, the convergent series was evaluated directly.

Method

Series plus Euler–Maclaurin

Shape of ζ(x)

The blue curve is ζ(x), the red dashed line marks the pole at x = 1, and the orange dot shows the current input. The vertical axis is clamped for readability.

About the Riemann zeta function

ζ(x) is defined for x above 1 as 1/1^x + 1/2^x + 1/3^x + … and is extended to all real x (except x = 1) by the functional equation. It is a central object in analytic number theory, deeply tied to the distribution of primes and the zeros of the zeta function.

How it works

For x greater than 1 the tool sums the leading terms of the series 1/1^x + 1/2^x + … directly and approximates the remaining tail with the Euler–Maclaurin formula, whose correction terms use Bernoulli numbers.
For x less than 1 (including negatives) it applies the functional equation ζ(x) = 2^x π^(x-1) sin(π x ÷ 2) Γ(1-x) ζ(1-x), evaluating 1-x by series and mapping the value back. Γ is computed with the Lanczos approximation.
x = 1 is the only pole of the zeta function, where the value diverges. At negative even integers x = -2, -4, … the value is ζ(x) = 0 (trivial zeros).
Notable special values: ζ(2) = π² ÷ 6 ≈ 1.6449 (the Basel problem), ζ(4) = π⁴ ÷ 90, ζ(0) = -1 ÷ 2, and ζ(-1) = -1 ÷ 12.
Displayed results are floating-point approximations close to double precision, intended for learning and quick checks.

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