---
title: Koba–Nielsen Local Zeta Functions
url: https://www.emergentmind.com/topics/koba-nielsen-local-zeta-functions
type: topic
---

# Koba–Nielsen Local Zeta Functions

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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 \(\mathbb F_q((T))\) [2105.00298].

## 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 [2105.00298]. 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 \(x_i\), \(1-x_i\), and \(x_i-x_j\) [2105.00298].

For the \(p\)-adic open string \(N\)-point tree amplitude, the Koba–Nielsen-type integral is
\[
A^{(N)}(k) = \int_{\mathbb{Q}_p^{N-3}} \prod_{i=2}^{N-2} |x_i|_p^{\,k_i\cdot k_i} \,|1-x_i|_p^{\,k_{N-1}\cdot k_i} \prod_{2\le i<j\le N-2}|x_i-x_j|_p^{\,k_i\cdot k_j} \, d x_2\cdots d x_{N-2},
\]
with
\[
\sum_{i=1}^N k_i=0,\qquad k_i\cdot k_i=2
\]
[1611.03807]. The associated multivariate local zeta function depends on independent complex parameters \(s=(s_{ij})\in\mathbb C^D\), \(D=\frac{N(N-3)}{2}\), and is defined by integrating the same Koba–Nielsen factors with exponents \(s_{ij}\) instead of the kinematic invariants [1611.03807].

For \(N=4\), the open-string amplitude reduces to the Veneziano integral
\[
A^{(4)}(k)=\int_K |x|_K^{k_1\cdot k_2}|1-x|_K^{k_2\cdot k_3}\,dx
\]
[2105.00298]. For closed strings over \(\mathbb C\), the analogous amplitude is essentially of the same type, with a polynomial factor and with the exponents scaled as in the standard complex formulation [2105.00298]. 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 [1611.03807].

## 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 \(p\)-adic setting, the multivariate zeta function
\[
Z^{(N)}(s)=\int_{\mathbb{Q}_p^{N-3}\setminus A} F(s,x;N)\,\prod_{i=2}^{N-2}dx_i
\]
is integrated over \(\mathbb{Q}_p^{N-3}\setminus A\), where
\[
A= \left\{ x\in \mathbb{Q}_p^{N-3} : \prod_{i=2}^{N-2} x_i(1-x_i)\prod_{2\le i<j\le N-2}(x_i-x_j)=0 \right\},
\]
and the initial convergence domain is
\[
\Re(s_{ij})>0 \quad\text{for all } i,j
\]
[1611.03807]. The exclusion of the divisor \(A\) ensures that the complex powers are well-defined away from their zeros [1611.03807].

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 [1905.10879]. Explicit low-point cases illustrate the structure of the convergence domain. For \(N=4\), the convergence region is exactly
\[
\Re(s_{12})>-1,\qquad \Re(s_{32})>-1,\qquad \Re(s_{12})+\Re(s_{32})<-1
\]
[1905.10879]. For \(N=5\), the region is given by inequalities such as
\[
\Re(s_{ij})>-1,
\]
together with
\[
\Re(s_{12})+\Re(s_{13})+\Re(s_{23})>-2,\qquad
\Re(s_{42})+\Re(s_{43})+\Re(s_{23})>-2,
\]
\[
\Re(s_{12})+\Re(s_{42})+\Re(s_{23})<-1,\qquad
\Re(s_{13})+\Re(s_{43})+\Re(s_{23})<-1,
\]
and
\[
\Re(s_{12})+\Re(s_{13})+\Re(s_{42})+\Re(s_{43})+\Re(s_{23})<-2
\]
[1905.10879].

The same work states that these \(N=5\) conditions are sharper than a previously claimed domain in the literature, and gives a counterexample showing that the larger domain is incorrect [1905.10879]. 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 \(\mathbb R^N\) is shown to have a nonempty convergence region, with a concrete box
\[
\frac{-2}{N+1}<\operatorname{Re}(s_{ij})<\frac{-2}{N+3}
\qquad\text{for all }ij
\]
[2507.08120]. 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 \(p\)-adic case, the associated multivariate zeta functions continue to all of \(\mathbb C^D\) as rational functions in the variables \(p^{-s_{ij}}\) [1611.03807]. This is the standard Igusa phenomenon in the Koba–Nielsen setting and gives a canonical regularized value whenever the original integral diverges [1611.03807].

In the characteristic-zero local-field setting, the continuation is obtained uniformly by embedded resolution of singularities. If
\[
D=\{x_2\cdots x_{N-2}(1-x_2)\cdots(1-x_{N-2})\prod_{i<j}(x_i-x_j)=0\},
\]
then Hironaka’s theorem provides an embedded resolution
\[
\pi:X\to K^{N-3}
\]
such that the pullback divisor has normal crossings and each polynomial factor locally becomes a unit times a monomial [2105.00298]. The integral is then reduced to finite sums of monomial integrals, from which one reads off convergence and poles [2105.00298].

For Koba–Nielsen amplitudes over any local field of characteristic zero, the continuation theorem is stated in the form that the amplitude \(A_K^{(N)}(k)\) 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 \(k\mapsto (k_i\cdot k_j)\) [1905.10879]. In the \(p\)-adic case, the resulting amplitude is rational in the variables \(q^{-k_i\cdot k_j}\) [1905.10879].

The same analytic mechanism is also used as a regularization procedure. In the \(p\)-adic literature, the physical amplitude is defined by specialization,
\[
A^{(N)}(k)=Z^{(N)}(s)\big|_{s_{ij}=k_i\cdot k_j},
\]
so that when the original Koba–Nielsen integral diverges, the continued zeta function still provides a canonical regularized value [1611.03807]. This regularization is explicitly compared with the use of meromorphic continuation for Feynman-type amplitudes [1611.03807].

A specific contrast with the Archimedean case is recorded in the \(p\)-adic regularization paper: “As far as we know, there is no a similar result for the Archimedean Koba-Nielsen amplitudes” [1611.03807]. In context, this refers to the explicit multivariate rational continuation and regularization result proved there for the \(p\)-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 [2105.00298]. In the characteristic-zero Koba–Nielsen framework, these hyperplanes have the form
\[
\sum_j N_{ij}s_j+\nu_i=0
\]
up to the standard shifts appearing in the local theory [2105.00298]. 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 [1905.10879].

For the Euclidean full-space integral, the convex-domain extension sharpens this picture. The relevant arrangement is
\[
A_N=f_N^{-1}(0), \qquad
f_N(x)= \prod_{i=1}^N x_i \prod_{i=1}^N(1-x_i) \prod_{1\le i<j\le N}(x_i-x_j),
\]
and the paper uses an explicit embedded resolution obtained by blowing up only the dense intersections
\[
x_1=\cdots=x_r=0,\qquad
x_1=\cdots=x_r=1,\qquad
x_1=x_2=\cdots=x_r
\]
in the ranges stated there, together with the analogous centers at infinity for the compactified arrangement [2507.08120]. Its main criterion is that a subspace \(Z_j\) contributes to the polar locus if and only if
\[
\dim(Z_j\cap D)=\dim(Z_j)
\]
[2507.08120]. 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 [2507.08120]. In the Euclidean case, it further proves that the full list of convergence inequalities is independent, so that no inequality is redundant [2507.08120].

On the Archimedean side, the Newton-polyhedron method provides a complementary description for local zeta functions attached to non-degenerate polynomials. If \(u\) is a primitive inward normal vector to a facet of the Newton polyhedron, then the candidate poles lie in
\[
P(u)=\left\{-\frac{|u|+k}{2\nu_u(f)}: k\in\mathbb N\right\},
\]
together with the universal family
\[
-\frac{1+\mathbb N}{2}
\]
[1812.05514]. 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 [1812.05514].

## 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 \(H^\infty(K^n)\), leading to integrals of the form
\[
Z_g(s,f)=\int_{K^n\setminus f^{-1}(0)} |f(x)|_K^s\, g(x)\,|d^n x|_K,
\qquad \Re(s)>0,\quad g\in H^\infty(K^n),
\]
and, more generally,
\[
I_g(\alpha,b)=\int_{K^n}\prod_{i=1}^r |b_i(x)|_K^{\alpha_i} g(x)\,|d^n x|_K
\]
[1704.07965]. 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 [1704.07965]. 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 \(f/g\). In that setting, the zeta function
\[
Z(s,f/g) = \int_{K^n\setminus D_K}\left|\frac{f(x)}{g(x)}\right|_K^s\,|dx|_K
\]
admits meromorphic continuation as a rational function in \(q^{-s}\), and in contrast with the classical polynomial case, poles can have both positive and negative real parts [1702.06938]. A twisted version with multiplicative characters,
\[
Z(s,\chi,f/g) = \int_{\mathcal O_K^n\setminus D_K} \chi\!\left(\operatorname{ac}\!\left(\frac{f(x)}{g(x)}\right)\right) \left|\frac{f(x)}{g(x)}\right|_K^{\,s} \,|dx|_K,
\]
is analyzed by a multivariate \(\pi\)-adic stationary phase formula and Newton polyhedra, with candidate poles determined by primitive normal vectors [2003.06569]. 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 \(p\to1\) program. The open \(p\)-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 [1712.08725]. In this framework one passes to unramified extensions \(K_e/\mathbb Q_p\) and takes the limit \(e\to0\), which is the standard way to model \(p\to1\) [1712.08725]. The resulting amplitudes are called Denef–Loeser amplitudes [2105.00298].

In the four-point case, the topological amplitude is explicitly
\[
A^{(4)}_{\text{top}}(k) = -1 + \frac{1}{1+k_1\cdot k_2} + \frac{1}{1+k_2\cdot k_3} + \frac{1}{1+k_2\cdot k_4}
\]
[2105.00298]. For \(N=4,5\), these topological amplitudes are stated to coincide with the tree-level Feynman amplitudes obtained from the \(p\to1\) limit of the Gerasimov–Shatashvili effective Lagrangian with logarithmic potential [2105.00298]. The same comparison is made in the detailed \(p\to1\) analysis, which states that the Denef–Loeser amplitudes exactly match the tree-level Feynman amplitudes of the Gerasimov–Shatashvili theory [1712.08725].

The local-zeta-function formalism also accommodates twisted amplitudes. With Chan–Paton factors and a constant \(B\)-field, one is led to twisted multivariate local zeta functions
\[
Z_{\varphi}(s,\chi,f) = \int \varphi(x)\prod_i \chi_i(\mathrm{ac}(f_i(x)))|f_i(x)|^{s_i}\,dx,
\]
and embedded resolution again gives meromorphic continuation, with poles on hyperplanes determined by the resolution data [2105.00298]. 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 [1905.10879].

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 \(x_1=0\) and \(x_{N-1}=1\), Hamiltonian
\[
H_K(x_2,\dots,x_{N-2};e) = -\sum_{i=2}^{N-2} e_i e_1 \log|x_i|_K -\sum_{i=2}^{N-2} e_i e_{N-1}\log|1-x_i|_K -\sum_{2\le i<j\le N-2} e_i e_j \log|x_i-x_j|_K,
\]
and partition function
\[
Z_{K,N,e}(\beta) = \int \prod_{i=2}^{N-2}|x_i|_K^{\beta e_ie_1} |1-x_i|_K^{\beta e_ie_{N-1}} \prod_{2\le i<j\le N-2}|x_i-x_j|_K^{\beta e_ie_j} \,dx_2\cdots dx_{N-2}
\]
[1905.10879]. 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 [2507.08120]. 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 [2507.08120]. This places the subject at the intersection of local zeta theory, string amplitudes, hyperplane-arrangement geometry, and special-function integrals.

Source: https://www.emergentmind.com/topics/koba-nielsen-local-zeta-functions