Multiple Zeta Functions and Their Applications

Multiple zeta functions constitute a family of complex-valued functions defined by nested series of the form ζ(s₁,s₂,…,sₖ)=∑_{n₁>n₂>⋯>nₖ>0} n₁^{-s₁} n₂^{-s₂}⋯nₖ^{-sₖ}, where the arguments sᵢ are complex parameters. As a natural extension of the Riemann zeta function, these objects encode a rich algebraic and analytic structure, revealing deep interrelations between number theory, algebraic geometry and mathematical physics. Analytically, the region of absolute convergence is determined by the real parts of the sᵢ, and analytic continuation techniques extend their domain with isolated singularities. Algebraically, multiple zeta values satisfy families of linear relations—most notably the shuffle and stuffle (double shuffle) relations—which reflect their interpretation as iterated integrals and as nested sums. These relations give rise to conjectures on the dimension of the vector spaces spanned by multiple zeta values of a given weight. In arithmetic geometry, multiple zeta values emerge in the study of mixed Tate motives, linking them to deep questions about periods and Galois actions. In physics, they appear in perturbative quantum field theory, where Feynman integrals often evaluate to combinations of multiple zeta values or their elliptic analogues. Beyond pure theory, numerical and algorithmic advances have facilitated high-precision computation, enabling both verification of conjectured identities and discovery of novel relations. The global significance of multiple zeta functions lies in their unifying role across disciplines, serving simultaneously as probes of prime distribution, invariants of knot and braid groups, and building blocks for special-function hierarchies.

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Technical terms

Multiple zeta function: A nested series generalising the Riemann zeta function, defined by ordered summation over positive integers.

Multiple zeta value: The special value of a multiple zeta function when all arguments are positive integers.

Shuffle (stuffle) relations: Algebraic identities among multiple zeta values arising from concatenation of summation or integral indices.

Mixed Tate motive: An algebro-geometric object whose periods include multiple zeta values and which carries a graded Galois action.

Euler sum: An alternating generalisation of a multiple zeta value, often involving sign factors (–1)ⁿ in the summand.

Generating function: A formal power series whose coefficients encode sums of multiple zeta values subject to linear constraints.

References

1. Euler Sums and Integral Connections. Mathematics (2019).
2. ZETA ELEMENTS IN DEPTH 3 AND THE FUNDAMENTAL LIE ALGEBRA OF THE INFINITESIMAL TATE CURVE. Forum of Mathematics Sigma (2017).
3. A generating function for sums of multiple zeta values and its applications. Proceedings of the American Mathematical Society (2007).