--- title: Goss Zeta Functions in Positive Characteristic url: https://www.emergentmind.com/topics/goss-zeta-functions type: topic --- # Goss Zeta Functions in Positive Characteristic 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 \(K_\infty^\times\times \mathbb Z_p\) or \(\mathbb C_\infty^\times\times \mathbb Z_p\), special values at integer points, \(v\)-adic and \(P\)-adic variants, and extensive interactions with Drinfeld modules, Anderson \(t\)-modules, shtuka functions, and Tate algebras [2606.08848, 2110.02569]. ## 1. Function-field framework and basic definitions A standard starting point fixes a finite field \(\mathbb F_q\), the rational function field \(F=\mathbb F_q(T)\), and \(A=\mathbb F_q[T]\). For a finite geometric extension \(K/F\) with integral closure \(\mathcal O_K\), the ideal-theoretic Goss zeta function is defined by \[ \zeta_K^{[p]}(s)=\sum_I \frac{1}{N(I)^s} =\prod_{\mathfrak P}\frac{1}{1-N(\mathfrak P)^{-s}}, \] where the sum is over nonzero ideals \(I\subset \mathcal O_K\), the product is over nonzero prime ideals \(\mathfrak P\subset \mathcal O_K\), and the variable \(s\) lies in the Goss plane \(C_F^*\times \mathbb Z_p\) [2509.14083]. In the simplest rational case, with \(A_+\) denoting the monic polynomials in \(A\), one writes \[ \zeta_A(s)=\sum_{a\in A_+} a^{-s}, \] and the values at positive integers are the Carlitz zeta values [2110.02569]. Another formulation emphasizes the infinite place. If \(K_\infty=\mathbb F_q((1/T))\), every \(a\in K_\infty^\times\) has a Teichmüller-like decomposition \[ a=\omega_\infty(a)\,\langle a\rangle, \qquad \omega_\infty(a)=a_mT^m,\qquad \langle a\rangle=1+u,\ |u|_\infty<1, \] and the natural parameter space is \[ S=K_\infty^\times\times \mathbb Z_p. \] For \(w=(s_0,s)\in S\), one defines \[ a^w=s_0^{v_\infty(a)}\langle a\rangle^s. \] This is the positive-characteristic analogue of \(n^s\), and it underlies the usual Goss series over monic polynomials of fixed degree and then over all degrees [2111.11911]. In the broader curve-theoretic setting, one fixes a smooth projective geometrically connected curve \(X/\mathbb F_q\), a rational point \(\infty\), and \[ A=H^0(X\setminus\{\infty\},\mathcal O_X). \] The completion at \(\infty\) is a local field \(K_\infty\), one chooses a sign function \(sgn:K_\infty^\times\to \mathbb F_q^\times\), and defines “monic” elements by \(A^+=\{a\in A:sgn(a)=1\}\). Goss’s parameter space is then described as \[ \mathbb S_\infty \cong \mathbb C_\infty^\times\times \mathbb Z_p, \] with integer points embedded by \(n\mapsto s_n=(\pi^{-n},n)\), and ideal exponentiation is defined so that \(a^{s_n}=a^n\) for \(a\in A^+\) [2606.08848]. ## 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 \(s\in\mathbb Z_p\), the degree-\(l\) pieces \[ \sum_{\substack{a\in R_1\\ \deg(a)=l}} a^{-w} \] form an entire power series in \(s_0\), uniformly convergent on bounded subsets of \(K_\infty\) [2111.11911]. In the curve-based formulation, partial zeta functions \(Z_I:\mathbb S_\infty\to \mathbb C_\infty\) extend from the half-plane \(|x|>1\) to continuous functions on all of \(\mathbb S_\infty\), and the Dedekind-style functions \(\zeta_{E,A}\) attached to finite extensions extend to entire functions on \(\mathbb S_\infty\) as well [2606.08848]. At arithmetic points, the theory exhibits strong algebraicity. Carlitz showed that for \(n\) divisible by \(q-1\), \[ \zeta(n)=\alpha_n\,\widetilde{\pi}^n \] with \(\alpha_n\in \mathbb F_q(\theta)\), and Goss generalized the same pattern to arbitrary \(A\): for \(n>0\) with \(q-1\mid n\), \[ \zeta_A(n)=\alpha_n\,\widetilde{\pi}^n \] with \(\alpha_n\in VH\), where \(V\) is generated by ideal-exponentiation values and \(H\) is the Hilbert class field [2606.08848]. The theory also has a precise notion of trivial zeros. In the polynomial case, Goss’s special polynomials \(z(B,t_0)\) satisfy: \[ z(B,1)=0 \quad \Longleftrightarrow \quad B>0 \text{ and } B\equiv 0 \pmod{q-1}, \] and in that case the zero at \(t_0=1\) is simple [1402.4000]. Böckle and Sheats proved the exact degree formula \[ \deg_{t_0} z(B,t_0)=\min_{i\ge 0}\left\lfloor \frac{l(p^iB)}{q-1}\right\rfloor, \] where \(l(B)\) is the sum of the base-\(q\) digits of \(B\) [1402.4000]. Perkins generalized this to multivariate special polynomials \[ z(B_1,\dots,B_s,t_0)=\sum_{d\ge 0} t_0^d\sum_{a\in A^+(d)} X_1(a)^{B_1}\cdots X_s(a)^{B_s}, \] with exact degree \[ \deg_{t_0} z(B_1,\dots,B_s,t_0) = \min_{i\ge 0} \left\lfloor \frac{l(p^iB_1)+\cdots+l(p^iB_s)}{q-1} \right\rfloor, \] and vanishing criterion \[ z(B_1,\dots,B_s,1)=0 \quad\Longleftrightarrow\quad B_1+\cdots+B_s\equiv 0 \pmod{q-1}, \] again with a simple zero [1402.4000]. Bernoulli–Goss polynomials encode a related special-value structure. For \[ S_i(n)=\sum_{\substack{a\in A_+\\ \deg a=i}} a^n, \qquad C_n(u)=\sum_{i=0}^\infty S_i(n)u^i, \] the Bernoulli–Goss polynomials are \[ B_n(u)= \begin{cases} C_n(u),& n\not\equiv 0 \pmod{q-1},\\[0.4em] \dfrac{C_n(u)}{1-u},& n\equiv 0 \pmod{q-1}, \end{cases} \] and they control divisibility of zeta polynomials of cyclotomic function fields [1808.06782]. ## 3. Local theories: \(v\)-adic, \(t\)-adic, and \(P\)-adic variants Goss also introduced local zeta functions at finite places. For a prime \(v\) of \(A=\mathbb F_q[t]\), the \(v\)-adic Goss zeta function is defined on \[ S_v:=C_v^\sharp \times \varprojlim \mathbb Z/(q^{\deg v}-1)p^n\mathbb Z \] by \[ \zeta_v(s)=\sum_{d=0}^\infty S_{d,v}(y)\,x^d, \qquad s=(x,y), \] with \[ S_{d,v}(y)=\sum_{\substack{a\in \mathbb F_q[t]\ \mathrm{monic}\\ \deg a=d,\ (v,a)=1}} a^y. \] When \(v\) has degree one, one may assume \(v=t\), and for fixed \(y\) the function \(\zeta_t(x,y)\) is actually a polynomial in \(x\) [1408.1111]. In that case the paper “Riemann Hypothesis for Goss \(t\)-adic Zeta Function” proves that all zeros of \(\zeta_v(x,y)\) are simple and lie in \[ K_v=\mathbb F_q((t))\subset C_v, \] a statement established by explicit control of the valuations \[ v_d(y)=\operatorname{val}_v(S_{d,v}(y)) \] and the Newton polygon slopes [1408.1111]. At the infinite place and at finite places \(P\), one also has \(P\)-adic zeta values attached to Anderson \(t\)-modules. For the tensor power \(C^{\otimes n}\) of the Carlitz module, \[ L(C^{\otimes n}/A)=\zeta(n), \qquad L_P(C^{\otimes n}/A)=\zeta_P(n), \] where \[ \zeta_P(n)=\sum_{d\ge 0}\sum_{\substack{a\in A_{+,d}\\ P\nmid a}} \frac{1}{a^n}. \] The \(z\)-twisted form is \[ \zeta_P(n,z)=\sum_{d\ge 0}\sum_{\substack{a\in A_{+,d}\\ P\nmid a}}\frac{1}{a^n}z^d. \] If \(q-1\mid n\), then \[ \operatorname{ord}_{z=1}\zeta_P(n,z)=1. \] The same simple-zero phenomenon holds for \(P\)-adic Pellarin \(L\)-series when \(n\equiv s\pmod{q-1}\), and for almost all \(P\)-adic Dirichlet–Goss \(L\)-series of type \(s\), with the stronger statement that if \(s1\}, \qquad v_\infty(a)=-m, \] Hu defines \[ \zeta_\infty(s_0,s,a,n) = \sum_{l=0}^\infty s_0^{-(m+1+l)n} \sum_{\substack{k\in \mathbb F_q[1/T]\\ \deg_{1/T}(k)\le l}} (k+a)^{-s}. \] For fixed \((a,n)\), the map \((s_0,s)\mapsto \zeta_\infty(s_0,s,a,n)\) is entire on \(S\), and there is a convergent infinite-order linear difference operator \[ L=\operatorname{id}+\sum_{\substack{j\ge 1\\ (q-1)\mid j}}\frac{1}{-s}(1+\Delta(s,n))^j, \qquad \Delta(s,n)f=f(s+1,n+1)-f(s,n), \] such that \[ L\Big[\zeta_\infty\big(1/T,s,a,0\big)\Big] = \sum_{\gamma\in\mathbb F_q}(a+\gamma)^{-s}. \] This is presented as the positive-characteristic analogue of infinite-order differential equations for the Hurwitz zeta and \(p\)-adic Hurwitz-type Euler zeta; unlike the complex case, the positive-characteristic equation is convergent [2111.11911]. A second line of generalization replaces the basic norm term by admissible maps on ideals. If \(\eta:I(A)\to K_\infty^\times\) is an admissible map, \(\rho_i\) are \(F_p\)-algebra homomorphisms, \(\sigma_j\) are continuous homomorphisms, and \(u\in C_\infty^\times\times \mathbb Z_p^n\), then the twisted characteristic-\(p\) zeta functions of Anglès, Ngo Dac, and Tavares Ribeiro are several-variable series of the form \[ \zeta_{S,O_E}(\rho,\psi;\sigma,\eta;\chi;u) = \sum_{m\ge 0} \sum_{\substack{I\in I_S(B),\ I\subset O_E\\ \deg N_{E/K}(I)=m}} \chi(I)\,\rho_1(\psi_1(N_{E/K}(I)))\cdots \rho_s(\psi_s(N_{E/K}(I)))\,N_{E/K}(I)_{\sigma,\eta}^{-u}, \] and these series converge both in the \(\infty\)-adic and \(v\)-adic settings [1603.04076]. The same framework contains twisted Carlitz–Goss zeta functions, Pellarin \(L\)-series, and several-variable deformations over Tate algebras. A third generalization replaces the Goss plane by the curve itself. For a smooth projective curve \(X/\mathbb F_q\) with rational point \(\infty\), Pellarin-type partial zeta functions attached to an ideal \(I\subset A\) are \[ \zeta_I = \sum_{a\in I\setminus\{0\}} a^{-1}\otimes a \in K_\infty\widehat{\otimes}A. \] These are rigid analytic functions on \(X_{\mathbb C_\infty}\setminus\{\infty\}\), and in arbitrary genus they satisfy functional identities involving a normalized rank-one Drinfeld module \(\phi\), its period lattice \(\rho_I I\), and an adjoint shtuka function \(f_*\). One of the central identities is \[ \zeta_I = -(a_I^{-1}\otimes a_I) \prod_{i\ge 0} \bigl((\rho_I a_I\otimes 1)^{1-q}f_*^{(1)}\bigr)^{(i)}, \] equivalently \[ \frac{\left((\rho_I^{-1}\otimes1)\zeta_I\right)^{(-1)}}{(\rho_I^{-1}\otimes1)\zeta_I}=f_*. \] The same work proves that the space of special functions \(\Sf(\phi)\) satisfies \[ \Sf(\phi)=\frac{(\rho_I\otimes 1)h}{\zeta_I}(\mathbb F_q\otimes IJ), \] generalizing genus \(0\) and genus \(1\) identities of Pellarin and of Green–Papanikolas [2212.07823]. 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 \(X_{\mathbb C_\infty}\setminus\{\infty\}\), not the Goss plane \(\mathbb S_\infty\), and Ferraro’s rationality theorem produces a function that serves as a partial analogue of Riemann’s \(\xi\)-function in this geometric setting [2606.08848]. ## 5. Drinfeld modules, \(t\)-motives, and special-value arithmetic Goss zeta functions are embedded in a broader \(L\)-function theory attached to Drinfeld modules and Anderson–Goss \(t\)-motives. For an abelian \(t\)-motive \(M\) over \(K\), with local Euler factors \[ P_M(v,X)=\det\bigl(1-X\,\mathrm{Frob}_v\mid H(M)_v^{I_v}\bigr), \] the Goss \(L\)-function at a positive integer \(n\) is \[ L(M,n)=\prod_{v\notin S} P_M(v,v^{-n})^{-1}. \] For the Carlitz module \(C\), the associated motive \(M_C\) satisfies \[ L(M_C,n)=\sum_{a\in A_+} a^{-(n-1)} \qquad (n\ge 2), \] so the Carlitz zeta values are the simplest Goss \(L\)-values [2110.02569]. This motivic viewpoint leads to strong arithmetic statements. Gezmiş and Namoijam prove that if \(\phi\) is a Drinfeld \(A\)-module of rank \(r\ge 2\) over \(K\), then for every positive integer \(n\), the special value \(L(M_\phi,n)\) is transcendental over \(K\); if \(\phi\) has everywhere good reduction, then \[ L(\wedge^{r-1}M_\phi,n) \] is also transcendental for every positive integer \(n\) [2110.02569]. Their method interprets Goss \(L\)-values in terms of Taelman \(L\)-values of explicitly constructed abelian \(t\)-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 \(\phi\) is such a Drinfeld module and \[ S_\phi(b;s)=\sum_{\mathfrak a\subset A} b^{\sigma_{\mathfrak a}}\,\mathfrak a^{-1}\,d(\phi_{\mathfrak a})^{-s}, \] then for \(1\le n\le q-1\) there exists a vector \[ z=(*,\dots,*,CS_\phi(b;n))^\top\in \mathbb C_\infty^n \] such that \[ \mathrm{Exp}_{\phi^{\otimes n}}(z)\in H^n, \] with \(H\) the Hilbert class field. This yields transcendence of the values \(S_\phi(b;n)\) and of certain ratios with periods [1706.06048]. Tensor products, symmetric squares, and alternating squares of Drinfeld modules also admit Rankin–Selberg-type convolution formulas. For Drinfeld modules \(\phi,\psi\) of equal rank \(r\ge 2\), the tensor-product \(L\)-function satisfies \[ L((\phi\otimes\psi)^\vee,s) = L(A,\chi_\phi\chi_\psi,rs+2)\cdot L({}_\phi\times{}_\psi,s), \] where \(L({}_\phi\times{}_\psi,s)\) is defined by Schur-polynomial-weighted coefficients built from the Frobenius polynomials of \(\phi\) and \(\psi\) [2308.06340]. At \(s=0\), 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 \(K/F\) of \(F=\mathbb F_q(T)\), the local factors of \(\zeta_K^{[p]}\) 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, \[ \zeta_F^{[p]}=\zeta_K^{[p]}. \] The same statement holds for the Teichmüller lifts \[ \zeta_K^{[0]}(s)=\sum_I \frac{1}{\chi(N(I)^s)} = \prod_{\mathfrak P}\frac{1}{1-\chi(N(\mathfrak P)^{-s})}, \] where \(\chi:C_F\to W\) is the Teichmüller character into Witt vectors [2509.14083]. 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-\(p\) setting [2509.14083]. At the same time, the literature shows that several deep structural properties persist without it: strong multiplicity one, convergent infinite-order difference equations, explicit \(P\)-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 \(E/H/K\), the functions \[ \zeta_{E,A}(\chi)=\sum_{\mathfrak I\subset B}\frac{[B/\mathfrak I]_A^+\otimes 1}{1\otimes [B/\mathfrak I]_A^+} \] extend to entire rigid analytic functions on \(X_{\mathbb C_\infty}\setminus\{\infty\}\), and Ferraro conjectures that in a ring class field case the order of vanishing at the canonical point \(\Xi\) is \(h_R-1\), with numerical evidence [2606.08848]. Another concerns Dirichlet–Goss \(L\)-series: the available \(P\)-adic theory proves simple zeros at \(z=1\) for almost all characters of a fixed type, and asks whether this holds for all characters in general [2606.08085]. A third concerns higher-rank Drinfeld modules, where transcendence is now established for many special \(L\)-values, but a full description of algebraic relations analogous to the Carlitz case remains open [2110.02569]. Taken together, these developments show that “Goss zeta functions” no longer denote a single isolated analogue of \(\zeta(s)\). The term now covers an interconnected family of positive-characteristic zeta and \(L\)-functions: ideal-theoretic Goss zetas on the Goss plane, \(v\)-adic and \(P\)-adic variants, Hurwitz-type and twisted refinements, Pellarin-type functions over curves, and motivic \(L\)-series attached to Drinfeld modules and \(t\)-motives. Their common structure is the replacement of classical complex analysis by Frobenius-twisted, rigid analytic, and \(t\)-motivic methods. Source: https://www.emergentmind.com/topics/goss-zeta-functions