\(\zeta\)

You can use the \(\zeta\) function to calculate the Reimann zeta function of a value.

You can use the \zetafn backslash command to insert this function.

The following variants of this function are available:

• \(\text{real } \zeta \left ( \text{<s>} \right )\)

• \(\text{complex } \zeta \left ( \text{<s>} \right )\)

Where \(\text{<s>}\) can be integer, real, or complex.

Note that run-time type conversion allows the result of this function to be assigned to any basic type provided the returned value is compatible with that type.

The \(\zeta\) function is defined by the summation:

\[\zeta \left ( s \right ) = \sum_{ n=1 }^\infty \frac{1}{n^s}\]

Where \(s\) is defined as \(s = \sigma + i t\) and \(\sigma > 1\).

The Reimann zeta function can be further extended over the entire complex plane except where \(\sigma = 1\) and \(\sigma = 0\) by the reflection formula:

\[\zeta \left ( s \right ) = 2 ^ s \pi ^ { s - 1 } \sin \left ( \frac{\pi}{2} s \right ) \Gamma \left ( 1 - s \right ) \zeta \left ( 1 - s \right )\]

Aion also included optimized versions of the Reimann zeta function for integer, real, and complex parameters. Integer and real parameters are evaluated using the Boost library implementation which, for Aion, uses:

• rational approximation for the general case, or

• cached values for select positive odd integer, or

• 0 for negative even integers, or

• Relationship to the Bernoulli numbers for even positive integers or negative odd integers.

For complex values where \(t \ne 0\), values are calculated using convergence of alternating series method.

The examples below show how you can use the \(\zeta\) function:

\[a = \zeta \; 2\]

\[\text{a = 1.64493}\]