Goss Zeta Functions in Positive Characteristic

• Goss zeta functions are function-field analogues of the Riemann zeta function, defined over finite fields with non-Archimedean values and Euler product expansions.
• They incorporate analytic methods over the Goss plane and local theories, enabling explicit evaluations at integer points and the study of v-adic and P-adic variants.
• Their deep connections with Drinfeld modules, Anderson t-modules, and shtuka functions yield significant insights into special values, transcendence, and arithmetic rigidity.

Searching arXiv for recent and foundational work on Goss zeta functions. Search query: Goss zeta functions arXiv recent foundational papers Goss zeta functions are the function-field analogues of the Riemann zeta function and classical Dirichlet $L$-functions, formulated in the arithmetic of global function fields over finite fields and taking values in non-Archimedean completions rather than in $\mathbb C$. In the rational case one works with $A=\mathbb F_q[T]$ or $A=\mathbb F_q[\theta]$, its fraction field $K=\mathbb F_q(T)$ or $K=\mathbb F_q(\theta)$, and the completion at the infinite place; more generally one fixes a smooth projective curve $X/\mathbb F_q$ with a distinguished place $\infty$, and $A$ is the ring of functions regular away from $\infty$. The theory developed by Goss from 1979 onward includes ideal-theoretic Euler products, non-Archimedean parameter spaces such as $\mathbb C$0 or $\mathbb C$1, special values at integer points, $\mathbb C$2-adic and $\mathbb C$3-adic variants, and extensive interactions with Drinfeld modules, Anderson $\mathbb C$4-modules, shtuka functions, and Tate algebras (Pellarin et al., 7 Jun 2026, Gezmiş et al., 2021).

1. Function-field framework and basic definitions

A standard starting point fixes a finite field $\mathbb C$5, the rational function field $\mathbb C$6, and $\mathbb C$7. For a finite geometric extension $\mathbb C$8 with integral closure $\mathbb C$9, the ideal-theoretic Goss zeta function is defined by

$A=\mathbb F_q[T]$0

where the sum is over nonzero ideals $A=\mathbb F_q[T]$1, the product is over nonzero prime ideals $A=\mathbb F_q[T]$2, and the variable $A=\mathbb F_q[T]$3 lies in the Goss plane $A=\mathbb F_q[T]$4 (Phagan, 17 Sep 2025). In the simplest rational case, with $A=\mathbb F_q[T]$5 denoting the monic polynomials in $A=\mathbb F_q[T]$6, one writes

$A=\mathbb F_q[T]$7

and the values at positive integers are the Carlitz zeta values (Gezmiş et al., 2021).

Another formulation emphasizes the infinite place. If $A=\mathbb F_q[T]$8, every $A=\mathbb F_q[T]$9 has a Teichmüller-like decomposition

$A=\mathbb F_q[\theta]$0

and the natural parameter space is

$A=\mathbb F_q[\theta]$1

For $A=\mathbb F_q[\theta]$2, one defines

$A=\mathbb F_q[\theta]$3

This is the positive-characteristic analogue of $A=\mathbb F_q[\theta]$4, and it underlies the usual Goss series over monic polynomials of fixed degree and then over all degrees (Hu et al., 2021).

In the broader curve-theoretic setting, one fixes a smooth projective geometrically connected curve $A=\mathbb F_q[\theta]$5, a rational point $A=\mathbb F_q[\theta]$6, and

$A=\mathbb F_q[\theta]$7

The completion at $A=\mathbb F_q[\theta]$8 is a local field $A=\mathbb F_q[\theta]$9, one chooses a sign function $K=\mathbb F_q(T)$0, and defines “monic” elements by $K=\mathbb F_q(T)$1. Goss’s parameter space is then described as

$K=\mathbb F_q(T)$2

with integer points embedded by $K=\mathbb F_q(T)$3, and ideal exponentiation is defined so that $K=\mathbb F_q(T)$4 for $K=\mathbb F_q(T)$5 (Pellarin et al., 7 Jun 2026).

2. Analytic structure, special values, and trivial zeros

A basic analytic property is that Goss zeta functions are entire in the sense of Goss. For fixed $K=\mathbb F_q(T)$6, the degree-$K=\mathbb F_q(T)$7 pieces

$K=\mathbb F_q(T)$8

form an entire power series in $K=\mathbb F_q(T)$9, uniformly convergent on bounded subsets of $K=\mathbb F_q(\theta)$0 (Hu et al., 2021). In the curve-based formulation, partial zeta functions $K=\mathbb F_q(\theta)$1 extend from the half-plane $K=\mathbb F_q(\theta)$2 to continuous functions on all of $K=\mathbb F_q(\theta)$3, and the Dedekind-style functions $K=\mathbb F_q(\theta)$4 attached to finite extensions extend to entire functions on $K=\mathbb F_q(\theta)$5 as well (Pellarin et al., 7 Jun 2026).

At arithmetic points, the theory exhibits strong algebraicity. Carlitz showed that for $K=\mathbb F_q(\theta)$6 divisible by $K=\mathbb F_q(\theta)$7,

$K=\mathbb F_q(\theta)$8

with $K=\mathbb F_q(\theta)$9, and Goss generalized the same pattern to arbitrary $X/\mathbb F_q$0: for $X/\mathbb F_q$1 with $X/\mathbb F_q$2,

$X/\mathbb F_q$3

with $X/\mathbb F_q$4, where $X/\mathbb F_q$5 is generated by ideal-exponentiation values and $X/\mathbb F_q$6 is the Hilbert class field (Pellarin et al., 7 Jun 2026).

The theory also has a precise notion of trivial zeros. In the polynomial case, Goss’s special polynomials $X/\mathbb F_q$7 satisfy: $X/\mathbb F_q$8 and in that case the zero at $X/\mathbb F_q$9 is simple (Perkins, 2014). Böckle and Sheats proved the exact degree formula

$\infty$0

where $\infty$1 is the sum of the base-$\infty$2 digits of $\infty$3 (Perkins, 2014). Perkins generalized this to multivariate special polynomials

$\infty$4

with exact degree

$\infty$5

and vanishing criterion

$\infty$6

again with a simple zero (Perkins, 2014).

Bernoulli–Goss polynomials encode a related special-value structure. For

$\infty$7

the Bernoulli–Goss polynomials are

$\infty$8

and they control divisibility of zeta polynomials of cyclotomic function fields (Shiomi, 2018).

3. Local theories: $\infty$9-adic, $A$0-adic, and $A$1-adic variants

Goss also introduced local zeta functions at finite places. For a prime $A$2 of $A$3, the $A$4-adic Goss zeta function is defined on

$A$5

by

$A$6

with

$A$7

When $A$8 has degree one, one may assume $A$9, and for fixed $\infty$0 the function $\infty$1 is actually a polynomial in $\infty$2 (Diaz-Vargas et al., 2014). In that case the paper “Riemann Hypothesis for Goss $\infty$3-adic Zeta Function” proves that all zeros of $\infty$4 are simple and lie in

$\infty$5

a statement established by explicit control of the valuations

$\infty$6

and the Newton polygon slopes (Diaz-Vargas et al., 2014).

At the infinite place and at finite places $\infty$7, one also has $\infty$8-adic zeta values attached to Anderson $\infty$9-modules. For the tensor power $\mathbb C$00 of the Carlitz module,

$\mathbb C$01

where

$\mathbb C$02

The $\mathbb C$03-twisted form is

$\mathbb C$04

If $\mathbb C$05, then

$\mathbb C$06

The same simple-zero phenomenon holds for $\mathbb C$07-adic Pellarin $\mathbb C$08-series when $\mathbb C$09, and for almost all $\mathbb C$10-adic Dirichlet–Goss $\mathbb C$11-series of type $\mathbb C$12, with the stronger statement that if $\mathbb C$13, it holds for all Dirichlet characters of type $\mathbb C$14 (Calvo, 6 Jun 2026).

A common misconception is to treat these local theories as direct copies of the classical $\mathbb C$15-adic theory. The cited work instead emphasizes that the local variables, target fields, and vanishing criteria are specific to positive characteristic: “even” weights mean multiples of $\mathbb C$16, and the order of vanishing is controlled by $\mathbb C$17-module unit modules, class modules, and $\mathbb C$18-adic logarithms rather than by classical archimedean gamma factors (Calvo, 6 Jun 2026).

4. Hurwitz-type refinements, twisted variants, and zeta functions over curves

One major refinement is the Hurwitz-type refinement of the Goss zeta function. For

$\mathbb C$19

Hu defines

$\mathbb C$20

For fixed $\mathbb C$21, the map $\mathbb C$22 is entire on $\mathbb C$23, and there is a convergent infinite-order linear difference operator

$\mathbb C$24

such that

$\mathbb C$25

This is presented as the positive-characteristic analogue of infinite-order differential equations for the Hurwitz zeta and $\mathbb C$26-adic Hurwitz-type Euler zeta; unlike the complex case, the positive-characteristic equation is convergent (Hu et al., 2021).

A second line of generalization replaces the basic norm term by admissible maps on ideals. If $\mathbb C$27 is an admissible map, $\mathbb C$28 are $\mathbb C$29-algebra homomorphisms, $\mathbb C$30 are continuous homomorphisms, and $\mathbb C$31, then the twisted characteristic-$\mathbb C$32 zeta functions of Anglès, Ngo Dac, and Tavares Ribeiro are several-variable series of the form

$\mathbb C$33

and these series converge both in the $\mathbb C$34-adic and $\mathbb C$35-adic settings (Anglès et al., 2016). The same framework contains twisted Carlitz–Goss zeta functions, Pellarin $\mathbb C$36-series, and several-variable deformations over Tate algebras.

A third generalization replaces the Goss plane by the curve itself. For a smooth projective curve $\mathbb C$37 with rational point $\mathbb C$38, Pellarin-type partial zeta functions attached to an ideal $\mathbb C$39 are

$\mathbb C$40

These are rigid analytic functions on $\mathbb C$41, and in arbitrary genus they satisfy functional identities involving a normalized rank-one Drinfeld module $\mathbb C$42, its period lattice $\mathbb C$43, and an adjoint shtuka function $\mathbb C$44. One of the central identities is

$\mathbb C$45

equivalently

$\mathbb C$46

The same work proves that the space of special functions $\mathbb C$47 satisfies

$\mathbb C$48

generalizing genus $\mathbb C$49 and genus $\mathbb C$50 identities of Pellarin and of Green–Papanikolas (Ferraro, 2022).

The survey “Zeta functions over curves” stresses that these functions over curves interact with Goss’s original zeta functions but remain fundamentally different: their domain is the rigid analytic curve $\mathbb C$51, not the Goss plane $\mathbb C$52, and Ferraro’s rationality theorem produces a function that serves as a partial analogue of Riemann’s $\mathbb C$53-function in this geometric setting (Pellarin et al., 7 Jun 2026).

5. Drinfeld modules, $\mathbb C$54-motives, and special-value arithmetic

Goss zeta functions are embedded in a broader $\mathbb C$55-function theory attached to Drinfeld modules and Anderson–Goss $\mathbb C$56-motives. For an abelian $\mathbb C$57-motive $\mathbb C$58 over $\mathbb C$59, with local Euler factors

$\mathbb C$60

the Goss $\mathbb C$61-function at a positive integer $\mathbb C$62 is

$\mathbb C$63

For the Carlitz module $\mathbb C$64, the associated motive $\mathbb C$65 satisfies

$\mathbb C$66

so the Carlitz zeta values are the simplest Goss $\mathbb C$67-values (Gezmiş et al., 2021).

This motivic viewpoint leads to strong arithmetic statements. Gezmiş and Namoijam prove that if $\mathbb C$68 is a Drinfeld $\mathbb C$69-module of rank $\mathbb C$70 over $\mathbb C$71, then for every positive integer $\mathbb C$72, the special value $\mathbb C$73 is transcendental over $\mathbb C$74; if $\mathbb C$75 has everywhere good reduction, then

$\mathbb C$76

is also transcendental for every positive integer $\mathbb C$77 (Gezmiş et al., 2021). Their method interprets Goss $\mathbb C$78-values in terms of Taelman $\mathbb C$79-values of explicitly constructed abelian $\mathbb C$80-modules, then applies transcendence results for logarithms.

For tensor powers of rank-one sign-normalized Drinfeld modules over the coordinate ring of an elliptic curve, explicit zeta values appear as coordinates of logarithms. If $\mathbb C$81 is such a Drinfeld module and

$\mathbb C$82

then for $\mathbb C$83 there exists a vector

$\mathbb C$84

such that

$\mathbb C$85

with $\mathbb C$86 the Hilbert class field. This yields transcendence of the values $\mathbb C$87 and of certain ratios with periods (Green, 2017).

Tensor products, symmetric squares, and alternating squares of Drinfeld modules also admit Rankin–Selberg-type convolution formulas. For Drinfeld modules $\mathbb C$88 of equal rank $\mathbb C$89, the tensor-product $\mathbb C$90-function satisfies

$\mathbb C$91

where $\mathbb C$92 is defined by Schur-polynomial-weighted coefficients built from the Frobenius polynomials of $\mathbb C$93 and $\mathbb C$94 (Huang, 2023). At $\mathbb C$95, Fang’s class module formula expresses these convolution values in terms of regulators and class modules of the tensor product.

6. Arithmetic rigidity, multiplicity one, and current directions

Goss zeta functions exhibit strong rigidity with respect to their Euler factors. For a finite geometric extension $\mathbb C$96 of $\mathbb C$97, the local factors of $\mathbb C$98 encode the splitting type of primes of the base field. Phagan proves a strong multiplicity one theorem: if two Goss zeta functions share cofinitely many Euler factors, then the full zeta functions coincide,

$\mathbb C$99

The same statement holds for the Teichmüller lifts

$A=\mathbb F_q[T]$00

where $A=\mathbb F_q[T]$01 is the Teichmüller character into Witt vectors (Phagan, 17 Sep 2025). The proofs use splitting types, Chebotarev density, and Gassmann theory rather than any functional equation.

This last point addresses a recurrent issue in the subject. Unlike the classical Riemann and Dedekind zeta functions, Goss zeta functions do not presently come with a generally accepted functional equation. The absence of such an equation is explicitly identified as an obstacle for formulating a fully satisfactory Riemann Hypothesis in the characteristic-$A=\mathbb F_q[T]$02 setting (Phagan, 17 Sep 2025). At the same time, the literature shows that several deep structural properties persist without it: strong multiplicity one, convergent infinite-order difference equations, explicit $A=\mathbb F_q[T]$03-adic order-of-vanishing results, and rigidity phenomena over curves.

Current directions recorded in the cited works include several distinct programs. One concerns relative zeta functions over curves: for a finite extension $A=\mathbb F_q[T]$04, the functions

$A=\mathbb F_q[T]$05

extend to entire rigid analytic functions on $A=\mathbb F_q[T]$06, and Ferraro conjectures that in a ring class field case the order of vanishing at the canonical point $A=\mathbb F_q[T]$07 is $A=\mathbb F_q[T]$08, with numerical evidence (Pellarin et al., 7 Jun 2026). Another concerns Dirichlet–Goss $A=\mathbb F_q[T]$09-series: the available $A=\mathbb F_q[T]$10-adic theory proves simple zeros at $A=\mathbb F_q[T]$11 for almost all characters of a fixed type, and asks whether this holds for all characters in general (Calvo, 6 Jun 2026). A third concerns higher-rank Drinfeld modules, where transcendence is now established for many special $A=\mathbb F_q[T]$12-values, but a full description of algebraic relations analogous to the Carlitz case remains open (Gezmiş et al., 2021).

Taken together, these developments show that “Goss zeta functions” no longer denote a single isolated analogue of $A=\mathbb F_q[T]$13. The term now covers an interconnected family of positive-characteristic zeta and $A=\mathbb F_q[T]$14-functions: ideal-theoretic Goss zetas on the Goss plane, $A=\mathbb F_q[T]$15-adic and $A=\mathbb F_q[T]$16-adic variants, Hurwitz-type and twisted refinements, Pellarin-type functions over curves, and motivic $A=\mathbb F_q[T]$17-series attached to Drinfeld modules and $A=\mathbb F_q[T]$18-motives. Their common structure is the replacement of classical complex analysis by Frobenius-twisted, rigid analytic, and $A=\mathbb F_q[T]$19-motivic methods.