Function Repository Resource:

MultipleZetaStar

Source Notebook

Compute the multiple zeta–star special function

Contributed by: Jayanta Phadikar

ResourceFunction["MultipleZetaStar"][{r₁,r₂,…,r_k}]

gives the multiple zeta star function ζ^*(r₁,r₂,…,r_k) defined by the nested sum .

Details

Series definition:

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ResourceFunction["MultipleZetaStar"] is also known as "Multiple Zeta Star". It is similar to MultipleZeta function. The difference is that MultipleZeta uses strictly ordered indices in the series definition, whereas MultipleZetaStar uses weakly ordered indices.

ResourceFunction["MultipleZetaStar"] gives nested sums of reciprocal powers, extending Zeta to multiple weakly ordered indices.

ResourceFunction["MultipleZetaStar"] appears as periods in number theory, as coefficients in ε-expansions of Feynman integrals, and in knot invariants.

MultipleZetaStar is a mathematical function, suitable for both symbolic and numerical manipulation.

The parameters can be arbitrary complex numbers. The defining series converges whenever the real part of each initial partial sum of the arguments exceeds its length, i.e. Re[r₁+⋯+r_m]>m for m=1,…,k.

In ζ^*(r₁,r₂,…,r_k), r₁, r₂,… are the weights, and k is the depth.

ResourceFunction["MultipleZetaStar"][{r}] is same as MultipleZeta[{r}] and Zeta[r].

Examples

Basic Examples (2)

Exact values are generated automatically for small weights:

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Evaluate numerically:

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Scope (14)

Numerical Evaluation (3)

Evaluate with large weight:

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Evaluate to arbitrary precision:

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Evaluate with rational and complex weights:

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Specific Values (5)

For total weight ≤ 7, all possible convergent MultipleZetaStar will autoevaluate in terms of Zeta:

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For total weight > 7, certain MultipleZetaStar usages may not be reducible in terms of Zeta:

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Reduction of a MultipleZetaStar of weight 10:

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MultipleZetaStar starting with 1 in the parameter list is divergent:

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Failure of the convergence criterion can make a MultipleZetaStar divergent:

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Visualization (5)

Log-scale discrete plot of the diagonal depth-2 values MultipleZetaStar[n, n] across n, highlighting rapid decay with increasing weight:

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Side-by-side log-scale traces of MultipleZetaStar[{n,2}] and MultipleZetaStar[{2,n}] that emphasize order asymmetry under index swap:

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Discrete 3-D bars of Log10[MultipleZetaStar[{s1,s2}]] over admissible indices, showing how magnitude varies across the grid:

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Log-scale discrete plot of the diagonal depth-3 values MultipleZetaStar[{n,n,n}] versus n, showing depth-driven suppression with weight:

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Dependence of MultipleZetaStar[{4,4,n}] on the tail index n at fixed head (4, 4):

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Function Representation (1)

Obtain MultipleZetaStar from its series representation:

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$Sum[1/(n1^2 n2^4 n3^1), {n1, 1, Infinity}, {n2, 1, n1}, {n3, 1, n2}]$

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Properties and Relations (2)

At depth-1, MultipleZetaStar, MultipleZeta and Zeta are the same:

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A convergent MultipleZetaStar can be written as a linear combination of MultipleZeta's:

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$Inactive[ ResourceFunction["MultipleZetaStar"]][{2, 6, 8}] == Sum[1/(n1^2 n2^6 n3^8), {n1, 1, Infinity}, {n2, 1, n1}, {n3, 1, n2}]$

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Verify the identity numerically:

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Possible Issues (1)

The argument of MultipleZetaStar must be a list:

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Publisher

Jayanta Kumar Phadikar

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

1.0.0 – 22 August 2026

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License