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Double ZetaEdit

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Double zeta refers to the simplest nontrivial case of the broader family of multiple zeta values (MZVs). These objects arise from nested sums and have deep connections across number theory, combinatorics, and mathematical physics. In the standard notation, a double zeta value is written as ζ(a,b), where a and b are positive integers with a > 1 to ensure convergence. The value is defined by the convergent series ζ(a,b) = sum_{m>n>0} 1/(m^a n^b). As a whole, double zeta values live inside the larger algebra of MZVs, which are indexed by a sequence of positive integers (s1, s2, …, sk) with s1 > 1 and each si ≥ 1. The pair (a,b) is the depth-2 case of that structure, and the total weight is a + b.

In common mathematical language, double zeta values are studied as special cases of iterated integrals and Dirichlet-series-type objects. They can be viewed as periods associated with specific algebraic or geometric constructs, and they satisfy a variety of algebraic relations that mirror the way products expand into sums of these values.

Definitions and notation

  • Double zeta value: ζ(a,b) = sum_{m>n>0} (m^(-a))(n^(-b)) with a > 1. This convergence condition guarantees the series makes sense.
  • Weight and depth: The weight is w = a + b, and the depth is 2 for ζ(a,b).
  • Related objects: The broader family of multiple zeta values includes ζ(s1, s2, ..., sk) with s1 > 1 and si ≥ 1 for i > 1.

These values connect to the classic Riemann zeta function, defined by ζ(s) = sum_{n≥1} n^(-s). In particular, certain double zeta values reduce to ordinary zeta values, such as ζ(2,1) = ζ(3). For terminology and broader context, see the Riemann zeta function and Multiple zeta value entries.

Fundamental identities and structure

  • Product relations (stuffle/shuffle): For integers a,b > 1, ζ(a)ζ(b) = ζ(a,b) + ζ(b,a) + ζ(a+b). This relation comes from combining the two sums and rearranging terms and is a prototype of the algebraic structure linking depth-1 and depth-2 zeta values.
  • Sum formula for depth 2: For any integer n ≥ 3, sum_{a=2}^{n-1} ζ(a, n−a) = ζ(n). This identity reflects a balance among all depth-2 decompositions of a fixed weight n.
  • Duality and depth reduction: There are symmetry and duality phenomena for MZVs, which, in the depth-2 case, express certain ζ(a,b) in terms of other depth-2 values or, in special instances, in terms of ζ of a single argument. These relations are part of a broader set of linear relations among MZVs known as the double shuffle relations.

For broader context, these ideas sit alongside the concept of shuffle relations and stuffle relations in the theory of multiple zeta values.

Examples and notable cases

  • ζ(2,1) = ζ(3): A classic and frequently cited identity that illustrates how a depth-2 value can collapse to a single zeta value of higher weight.
  • ζ(3,2) and other weight-5 values: In weight 5, several depth-2 zeta values exist, and their relations are part of the broader web of MZV relations. The exact expressions often involve linear combinations of ζ(5) and products like ζ(2)ζ(3), reflecting the underlying algebraic structure.

The study of double zeta values sits inside the larger program of understanding the vector space generated by all MZVs of a given weight, their algebraic relations, and how those pieces fit together in a coherent picture. For readers seeking a broader context, see Multiple zeta value.

History and development

  • Early roots: The study of reciprocals of powers and nested sums traces back to Euler, who investigated sums of the form sum 1/(n^p m^q) and related expressions.
  • Modern formalism: The modern theory of MZVs, including double zeta values, gained momentum in the late 20th century with work on iterated integrals, associators, and the algebraic relations among zeta values. Key figures include researchers who developed the shuffle and stuffle formalisms and explored the depth and weight structure.
  • Applications in physics and topology: Over the last few decades, double zeta values and their higher-depth cousins have appeared in perturbative expansions in quantum field theory and in the study of knot invariants, situating MZVs at the crossroads of mathematics and physics.

Roles in mathematics and physics

  • Number theory and algebra: Double zeta values are building blocks in the broader algebra of MZVs, contributing to questions about linear independence, basis formation, and the dimension of MZV spaces at fixed weight.
  • Mathematical physics: In perturbative calculations, certain integrals evaluate to combinations of MZVs, including double zeta values, making these constants practically relevant in the computations that underlie physical theories.
  • Knot theory and algebraic geometry: MZVs arise in the study of knot invariants and in periods associated with algebraic structures encountered in geometry and topology.

For related topics, see Knot theory, Feynman diagram (as a representative context in which MZVs appear), and Iterated integral.

See also