Koba–Nielsen Local Zeta Functions

• Koba–Nielsen local zeta functions are multivariate complex-power integrals derived from string scattering amplitudes, providing a framework for analytic regularization.
• They employ techniques like embedded resolution and Newton polyhedra to achieve meromorphic continuation and controlled pole structures.
• Defined over local fields such as ℝ, ℂ, ℚₚ, and function fields, these functions bridge string theory, number theory, and statistical-mechanical interpretations.

Searching arXiv for papers on Koba–Nielsen local zeta functions and closely related local-zeta frameworks. I’m checking arXiv records relevant to Koba–Nielsen local zeta functions, including foundational and recent extensions. Koba–Nielsen local zeta functions are multivariate complex-power integrals built from the same singular factors that appear in tree-level string scattering amplitudes. Their basic role is to replace physically defined exponents by independent complex parameters, study the resulting integral as a function of those parameters, and then use its meromorphic continuation as a regularization of the original amplitude. In the formulation emphasized in the local-zeta-function literature, a Koba–Nielsen amplitude is a special instance of a multivariate local zeta function, and these objects can be defined over arbitrary local fields, including $\mathbb R$, $\mathbb C$, $\mathbb Q_p$, and $\mathbb F_q((T))$ (Bocardo-Gaspar et al., 2021).

1. Definitions and basic integral forms

For a local field $K$ of characteristic zero, a general multivariate local zeta function is written as

$$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$

where $f=(f_1,\dots,f_m)$ are nonconstant polynomials and $\varphi$ is a test function (Bocardo-Gaspar et al., 2021). Koba–Nielsen local zeta functions arise by taking the $f_i$ to be the linear factors that occur after fixing three insertion points in tree-level string amplitudes, namely $x_i$, $\mathbb C$0, and $\mathbb C$1 (Bocardo-Gaspar et al., 2021).

For the $\mathbb C$2-adic open string $\mathbb C$3-point tree amplitude, the Koba–Nielsen-type integral is

$\mathbb C$4

with

$\mathbb C$5

(Bocardo-Gaspar et al., 2016). The associated multivariate local zeta function depends on independent complex parameters $\mathbb C$6, $\mathbb C$7, and is defined by integrating the same Koba–Nielsen factors with exponents $\mathbb C$8 instead of the kinematic invariants (Bocardo-Gaspar et al., 2016).

For $\mathbb C$9, the open-string amplitude reduces to the Veneziano integral

$\mathbb Q_p$0

(Bocardo-Gaspar et al., 2021). For closed strings over $\mathbb Q_p$1, the analogous amplitude is essentially of the same type, with a polynomial factor and with the exponents scaled as in the standard complex formulation (Bocardo-Gaspar et al., 2021). This places open and closed tree amplitudes inside the same algebro-analytic framework.

The central identification is therefore literal rather than metaphorical: the amplitudes are not merely analogous to local zeta functions; they are local zeta functions with a distinguished hyperplane-arrangement divisor coming from the Koba–Nielsen factors (Bocardo-Gaspar et al., 2016).

2. Convergence and the status of the amplitudes as bona fide integrals

A recurring theme in the literature is that Koba–Nielsen amplitudes are bona fide integrals. In the $\mathbb Q_p$2-adic setting, the multivariate zeta function

$\mathbb Q_p$3

is integrated over $\mathbb Q_p$4, where

$\mathbb Q_p$5

and the initial convergence domain is

$\mathbb Q_p$6

(Bocardo-Gaspar et al., 2016). The exclusion of the divisor $\mathbb Q_p$7 ensures that the complex powers are well-defined away from their zeros (Bocardo-Gaspar et al., 2016).

Over local fields of characteristic zero, the same basic conclusion holds: the Koba–Nielsen local zeta functions are genuinely convergent on a nonempty open region in parameter space, and then extend meromorphically (Bocardo-Gaspar et al., 2019). Explicit low-point cases illustrate the structure of the convergence domain. For $\mathbb Q_p$8, the convergence region is exactly

$\mathbb Q_p$9

(Bocardo-Gaspar et al., 2019). For $\mathbb F_q((T))$0, the region is given by inequalities such as

$\mathbb F_q((T))$1

together with

$\mathbb F_q((T))$2

$\mathbb F_q((T))$3

and

$\mathbb F_q((T))$4

(Bocardo-Gaspar et al., 2019).

The same work states that these $\mathbb F_q((T))$5 conditions are sharper than a previously claimed domain in the literature, and gives a counterexample showing that the larger domain is incorrect (Bocardo-Gaspar et al., 2019). This is one of the clearer corrections in the subject: convergence is subtle and depends on the precise geometry of the divisor after blow-up.

In the more recent convex-domain generalization, the Euclidean Koba–Nielsen local zeta function on $\mathbb F_q((T))$6 is shown to have a nonempty convergence region, with a concrete box

$\mathbb F_q((T))$7

(Veys et al., 10 Jul 2025). This extends the convergence theory beyond the original full-space setting.

3. Meromorphic continuation and regularization

The principal analytic statement is that Koba–Nielsen local zeta functions admit meromorphic continuation in the complex parameters. In the $\mathbb F_q((T))$8-adic case, the associated multivariate zeta functions continue to all of $\mathbb F_q((T))$9 as rational functions in the variables $K$0 (Bocardo-Gaspar et al., 2016). This is the standard Igusa phenomenon in the Koba–Nielsen setting and gives a canonical regularized value whenever the original integral diverges (Bocardo-Gaspar et al., 2016).

In the characteristic-zero local-field setting, the continuation is obtained uniformly by embedded resolution of singularities. If

$K$1

then Hironaka’s theorem provides an embedded resolution

$K$2

such that the pullback divisor has normal crossings and each polynomial factor locally becomes a unit times a monomial (Bocardo-Gaspar et al., 2021). The integral is then reduced to finite sums of monomial integrals, from which one reads off convergence and poles (Bocardo-Gaspar et al., 2021).

For Koba–Nielsen amplitudes over any local field of characteristic zero, the continuation theorem is stated in the form that the amplitude $K$3 converges on a nonempty open set, extends meromorphically to all of the momentum space, and has poles lying in the inverse image of the zeta-function polar hyperplanes under the map $K$4 (Bocardo-Gaspar et al., 2019). In the $K$5-adic case, the resulting amplitude is rational in the variables $K$6 (Bocardo-Gaspar et al., 2019).

The same analytic mechanism is also used as a regularization procedure. In the $K$7-adic literature, the physical amplitude is defined by specialization,

$K$8

so that when the original Koba–Nielsen integral diverges, the continued zeta function still provides a canonical regularized value (Bocardo-Gaspar et al., 2016). This regularization is explicitly compared with the use of meromorphic continuation for Feynman-type amplitudes (Bocardo-Gaspar et al., 2016).

A specific contrast with the Archimedean case is recorded in the $K$9-adic regularization paper: “As far as we know, there is no a similar result for the Archimedean Koba-Nielsen amplitudes” (Bocardo-Gaspar et al., 2016). In context, this refers to the explicit multivariate rational continuation and regularization result proved there for the $$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$0-adic case.

4. Pole structure, embedded resolution, and Newton-polyhedral control

The polar locus of Koba–Nielsen local zeta functions is governed by the geometry of the singular divisor. After embedded resolution, the poles lie on affine hyperplanes determined by the numerical data of exceptional divisors (Bocardo-Gaspar et al., 2021). In the characteristic-zero Koba–Nielsen framework, these hyperplanes have the form

$$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$1

up to the standard shifts appearing in the local theory (Bocardo-Gaspar et al., 2021). In the real and complex cases, the continuation can be expressed schematically as a finite sum of gamma factors evaluated at linear combinations of the kinematic variables, with holomorphic coefficients (Bocardo-Gaspar et al., 2019).

For the Euclidean full-space integral, the convex-domain extension sharpens this picture. The relevant arrangement is

$$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$2

and the paper uses an explicit embedded resolution obtained by blowing up only the dense intersections

$$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$3

in the ranges stated there, together with the analogous centers at infinity for the compactified arrangement (Veys et al., 10 Jul 2025). Its main criterion is that a subspace $$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$4 contributes to the polar locus if and only if

$$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$5

(Veys et al., 10 Jul 2025). This gives a domain-sensitive pole criterion for bounded and unbounded convex subsets.

The same paper states that the meromorphic continuation can be reinterpreted as weighted sums of Gamma functions, evaluated at linear combinations of the complex parameters, where the weights are holomorphic functions (Veys et al., 10 Jul 2025). In the Euclidean case, it further proves that the full list of convergence inequalities is independent, so that no inequality is redundant (Veys et al., 10 Jul 2025).

On the Archimedean side, the Newton-polyhedron method provides a complementary description for local zeta functions attached to non-degenerate polynomials. If $$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$6 is a primitive inward normal vector to a facet of the Newton polyhedron, then the candidate poles lie in

$$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$7

together with the universal family

$$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$8

(Aroca et al., 2018). The significance of that result for Koba–Nielsen-type Archimedean integrals is that, under a non-degeneracy hypothesis, the pole structure is controlled by the facet normals of the Newton polyhedron, and fake candidate poles from extra rays in a regular subdivision can be discarded (Aroca et al., 2018).

5. Non-Archimedean generalizations and neighboring local-zeta frameworks

Several nearby non-Archimedean theories illuminate the analytic type of Koba–Nielsen local zeta functions. One direction replaces compactly supported Bruhat–Schwartz test functions by functions in a nuclear Sobolev-type space $$Z_{\varphi}(f,s)=\int_{K^n}\varphi(x)\prod_{i=1}^m |f_i(x)|_K^{s_i}\,dx,\qquad \Re(s_i)>0,$$9, leading to integrals of the form

$f=(f_1,\dots,f_m)$0

and, more generally,

$f=(f_1,\dots,f_m)$1

(Zúñiga-Galindo, 2017). That work explicitly states that it does not define Koba–Nielsen local zeta functions as a formal class, but also states that there is a clear conceptual connection that can be inferred: the integrals are of the same analytic type, namely complex powers of norms of polynomial expressions integrated against a suitable weight (Zúñiga-Galindo, 2017). A plausible implication is that operator-theoretic generalizations of local zeta functions can be relevant for Koba–Nielsen-style regularization problems.

Another direction studies local zeta functions attached to rational functions $f=(f_1,\dots,f_m)$2. In that setting, the zeta function

$f=(f_1,\dots,f_m)$3

admits meromorphic continuation as a rational function in $f=(f_1,\dots,f_m)$4, and in contrast with the classical polynomial case, poles can have both positive and negative real parts (Bocardo-Gaspar et al., 2017). A twisted version with multiplicative characters,

$f=(f_1,\dots,f_m)$5

is analyzed by a multivariate $f=(f_1,\dots,f_m)$6-adic stationary phase formula and Newton polyhedra, with candidate poles determined by primitive normal vectors (Bocardo-Gaspar, 2020). These papers are not about Koba–Nielsen amplitudes specifically, but they provide explicit templates for local rational integrals with meromorphic continuation, Newton-polyhedral pole control, and character twists.

From the viewpoint of the subject, these neighboring theories show that Koba–Nielsen local zeta functions sit inside a broader class of multivariate local-zeta constructions in which resolution data, Newton polyhedra, stationary phase, and test-function spaces all materially affect the continuation and pole structure.

6. Limits, deformations, and further interpretations

A major development is the $f=(f_1,\dots,f_m)$7 program. The open $f=(f_1,\dots,f_m)$8-adic string amplitudes are first regularized by their multivariate local zeta functions, and then the Denef–Loeser limit produces topological zeta functions and corresponding amplitudes (Bocardo-Gaspar et al., 2017). In this framework one passes to unramified extensions $f=(f_1,\dots,f_m)$9 and takes the limit $\varphi$0, which is the standard way to model $\varphi$1 (Bocardo-Gaspar et al., 2017). The resulting amplitudes are called Denef–Loeser amplitudes (Bocardo-Gaspar et al., 2021).

In the four-point case, the topological amplitude is explicitly

$\varphi$2

(Bocardo-Gaspar et al., 2021). For $\varphi$3, these topological amplitudes are stated to coincide with the tree-level Feynman amplitudes obtained from the $\varphi$4 limit of the Gerasimov–Shatashvili effective Lagrangian with logarithmic potential (Bocardo-Gaspar et al., 2021). The same comparison is made in the detailed $\varphi$5 analysis, which states that the Denef–Loeser amplitudes exactly match the tree-level Feynman amplitudes of the Gerasimov–Shatashvili theory (Bocardo-Gaspar et al., 2017).

The local-zeta-function formalism also accommodates twisted amplitudes. With Chan–Paton factors and a constant $\varphi$6-field, one is led to twisted multivariate local zeta functions

$\varphi$7

and embedded resolution again gives meromorphic continuation, with poles on hyperplanes determined by the resolution data (Bocardo-Gaspar et al., 2021). A parallel formulation appears in the general characteristic-zero Koba–Nielsen theory, where the oscillatory and sign factors are expanded into linear combinations of twisted Koba–Nielsen zeta functions (Bocardo-Gaspar et al., 2019).

Another interpretation is statistical-mechanical. The Koba–Nielsen local zeta functions can be viewed as partition functions of one-dimensional logarithmic Coulomb gases with fixed endpoints $\varphi$8 and $\varphi$9, Hamiltonian

$f_i$0

and partition function

$f_i$1

(Bocardo-Gaspar et al., 2019). This identifies the same local-zeta object simultaneously with a regularized string amplitude and a log-Coulomb partition function.

A recent extension broadens the integration domain from the full Euclidean space to bounded or unbounded convex subsets and then to arbitrary hyperplane arrangements (Veys et al., 10 Jul 2025). In that setting, generalized Selberg-Mehta-Macdonald integrals and Dotsenko-Fateev-like integrals occur as particular cases of the new Koba–Nielsen local zeta functions, or of a further generalization to arbitrary hyperplane arrangements (Veys et al., 10 Jul 2025). This places the subject at the intersection of local zeta theory, string amplitudes, hyperplane-arrangement geometry, and special-function integrals.