Zeta Generating Formulas

• Zeta Generating Formulas (ZGFs) are explicit generating functions that encode infinite families of zeta values using combinatorial, spectral, and algebraic structures.
• They unify classical and generalized zeta identities by linking number theory, special functions, and quantum spectral analysis through precise analytic formulations.
• ZGFs have practical applications in prime number theory, multiple zeta value analysis, and spectral mapping in integrable models.

A Zeta Generating Formula (ZGF) is an explicit functional relation or analytic identity that encodes infinite families of zeta (or zeta-like) values through generating functions or explicit formulae, often linking deep algebraic, spectral, or combinatorial structures. In the context of contemporary mathematics and mathematical physics, ZGFs unify and extend results on the structure, identities, and analytic properties of classical and generalized zeta functions, with constructions spanning number theory, the theory of special functions, the spectral theory of operators, and integrable systems.

1. General Definition and Core Constructions

A ZGF typically manifests as a generating function, explicit product, or functional relation whose expansion yields zeta or multiple zeta values as coefficients or specializations. Classical examples arise from Euler's product for the Riemann zeta function

$$\zeta(s) = \prod_{p~\text{prime}} \left(1 - p^{-s}\right)^{-1}$$

and its subsequent manipulations into explicit or implicit generating forms. In higher order or spectral generalizations, the ZGF encodes entire families of special values, weighted sums, or spectral invariants via generating functions, sometimes in nontrivial combinatorial or algebraic frameworks.

Key types of ZGFs include:

• Ordinary and exponential generating functions for families of zeta or multiple zeta values.
• Functional relations leveraging algebraic structures such as the shuffle, stuffle, or double shuffle relations.
• Spectral ZGFs mapping sets of spectral zeta functions (e.g., spectral traces of quantum Hamiltonians) between sectors in integrable models.

2. ZGFs in Prime Number Theory and Arithmetical Constructions

The ZGF concept is pivotal in analytic number theory both for single zeta values and for understanding prime distributions. Classical foundational results include:

• Euler Prime Product ZGF: The Riemann zeta function for $\Re(s)>1$ is generated via the Euler prime product, and, through analytic manipulations, one obtains new prime product formulas for $|\zeta(s)|^2$ and explicit representations for $\zeta(k)$, the latter often incorporating Bernoulli numbers for even arguments and revealing composite ZGF structures through factorization:

$$\zeta(k)^2 = \zeta(2k)\prod_{n=1}^\infty (1 + p_n^{-k})^{-1}$$

This and related formulas generate, via expansion and specialization, formulae for $\zeta(k)$ up to high orders, including non-integer cases such as $\zeta(3/2)$ (Kawalec, 2019).

• Twin Prime ZGFs: Based on Golomb’s arithmetic formulas, ZGFs for twin primes are constructed via Dirichlet series whose terms encode prime sieving (through the Möbius function and divisor sums) and arithmetic constraints. The resulting ZGF is governed by the analytic structure of the Riemann zeta function and its nontrivial zeros:

$$Z(s) = \sum_{n=2}^\infty \frac{1}{n^s}\sum_{d|n} \mu(d) \log^2 d$$

whose analytic continuation encodes contributions from the nontrivial zeta zeros and generates the asymptotics for twin prime counts (Weber, 2011).

3. ZGFs for Multiple Zeta Values and Combinatorial Identities

ZGFs provide systematic frameworks for organizing relations among multiple zeta values (MZVs), multiple zeta star values (MZSVs), and new variants such as polynomial, Ohno-type, and restricted/weighted Euler sums:

• Generating Functions for MZVs/MZSVs: The canonical approach is to define a generating function, e.g.,

$$G_d(x_1,\dots,x_d) = \sum_{s_i} x_1^{s_1-1}\cdots x_d^{s_d-1} \zeta(s_1,\ldots,s_d)$$

yielding, via coefficient extraction, sum and weighted sum formulas, now generalized to quadruple and polynomial MZVs (Machide, 2012, Machide, 2015, Hirose et al., 2020).

• ZGFs and Double Shuffle Relations: The double shuffle and extended double shuffle relations, central in the algebraic paper of MZVs, admit succinct encoding in ZGF form, providing a uniform method for deriving parameterized, weighted, and restricted sum formulas, e.g. for triple or quadruple zeta values (Yuan et al., 2013, Zhao, 2023).
• Schur Multiple Zeta Values and Polynomial ZGFs: Generalized generating functions involving parameters $(x, y)$ encapsulate both symmetric and polynomial variants of MZVs, with the inclusion of Schur-type combinatorial structures explaining structural resemblances among their sum formulas (Hirose et al., 2020).

Table: Sample ZGF Generating Functions for Multiple Zeta Values

Family	Generating Function (Prototype)	Reference
MZV	$$\Gamma_1(W) = \exp\left(\sum_{k=1}^\infty \frac{\zeta(k)}{k} W^k\right)$$	(Hirose et al., 2020)
MZSV	$\Gamma_1((1+A)W)/\Gamma_1(W)$	(Hirose et al., 2020)
Ohno sum	$O_F(k)$ as a function of index and depth	(Hirose et al., 2019)

4. ZGFs in Spectral Theory and the ODE/IM Correspondence

Recent advances have extended ZGF methodology into quantum spectral problems via the ODE/IM correspondence:

• Spectral ZGFs: For quantum mechanical systems with PT-symmetric or Hermitian potentials (e.g., $V(x)=x^{2K}(ix)^\varepsilon$), ZGFs explicitly map the entire spectrum's spectral zeta functions (SZFs) between different fusion sectors. If $\zeta_K(s) = \sum_\alpha E_{K,\alpha}^{-s}$ is the SZF for sector $K$, the ZGF is a polynomial functional relation:

$$\zeta_K(s) = \sum_{m} c_m \prod_{t=1}^s [\zeta_{K'}(t)]^{m_t}$$

where the multi-indices $m$ are determined by the fusion relations of the underlying T-system, and $c_m$ depend on the combinatorics of Chebyshev polynomials and discrete symmetries of the model (Kamata, 8 Aug 2025).

• Relation to Fusion and Integrable Structures: The construction of ZGFs leverages the exact sum rules and fusion relations among Stokes multipliers in integrable models (A$_{2M-1}$ T-system), with selection rules governed by cyclic symmetry groups $\mathbb{Z}_{2M+2}$ and an explicit mapping between spectral data in different symmetry sectors.

5. Analytic, Hypergeometric, and Symbolic ZGF Methodologies

ZGFs often benefit from analytic and symbolic instrumentality:

• Hypergeometric and WZ Methods: Fast ZGF-based formulas for Hurwitz and related zeta values are proven using the Wilf–Zeilberger (WZ) method, yielding hypergeometric series whose convergence is optimized for computational applications. The structure is controlled by the Pochhammer symbol and telescoping relations (Guillera, 25 Mar 2025, Tauraso, 2018).
• Creative Telescoping and Hypergeometric Reduction: Sophisticated ZGF strategies rely on expressing generating series for interpolated or t-variant multiple zeta values as generalized hypergeometric series. Creative telescoping then reduces recurrences to closed-form (polygamma or zeta) representations, often resolving algebraic structure conjectures for otherwise irreducible families (Au et al., 24 Apr 2024).
• Symbolic and Bernoulli Frameworks: For negative (and more generally, non-standard) arguments, ZGFs are constructed using symbolic methods involving Bernoulli symbols, C-symbols, and analytic continuation, providing explicit closed forms and recursions (Moll et al., 2015, Essouabri et al., 2017).

6. Algebraic Structures, Symmetry, and Global Implications

A recurring feature of ZGF frameworks is the exploitation of algebraic and combinatorial symmetries:

• Underlying Algebraic Symmetry: Many ZGFs reflect hidden structures (e.g., Chebyshev polynomials, $Z_N$ cyclic symmetries, noncommutative polynomial algebras), which control the selection rules and explicit forms of spectral/multiplicative zeta relations (Kamata, 8 Aug 2025, Hirose et al., 2019).
• Global Spectral Data: In its spectral incarnation, ZGFs control the transfer of spectral data between quantum sectors, providing a global, nonrecursively generated view of spectral invariants (Kamata, 8 Aug 2025).

7. Applications and Further Directions

ZGFs unify disparate areas, enabling the derivation and systematization of new identities, weighted and restricted sum formulas, computationally effective series, and deep relations across number theory, special functions, and mathematical physics:

• Accelerated series for Hurwitz and special zeta values with applications in high-precision computation (Guillera, 25 Mar 2025).
• Unified perspectives on MZV algebra and congruence properties (Machide, 2012, Zhao, 2023).
• Novel spectral mappings in integrable models and quantum mechanics (Kamata, 8 Aug 2025).
• New avenues for conjectures and their resolution in the algebraic structure of zeta and multiple zeta families (Au et al., 24 Apr 2024).

Zeta Generating Formulas thus constitute a central tool in analytic number theory, special function theory, and spectral analysis, and their construction and exploitation continue to drive progress in uncovering structural depth and new computational methodologies in these domains.