What is a zeta function?

Question

In my readings, I've come across a wide variety of objects called zeta functions. For example, the Ihara zeta function, Igusa local zeta function, Hasse-Weil zeta function, etc. My question is simple: What makes something a zeta function? There are a couple things that the zeta functions I've seen have in common. For example: they're usually defined as some sort of infinite sum, have an Euler product and a functional equation. Also, rationality of certain zeta functions seems to be an important idea. Is there any way to understand this from a "big picture" point of view?

Answer 1

To a first approximation, we might agree that zeta functions are generating functions that encode arithmetic data. Usually the idea of generating functions is that they encode information about structures that combine and interact in ways that is mirrored or shadowed by basic polynomial operations. For more information on this topic, one may look into analytic combinatorics and combinatorial species. The latter invokes category theory nicely for understanding structures.

However in the context of zeta functions, our functions encode data about objects which combine and interact in ways that is mirrored or shadowed by basic arithmetic operations. For example, multiplying power series corresponds to convolution of the coefficients under the operation of addition, whereas multiplying Dirichlet series corresponds to multiplicative convolution. In both contexts factorizations suggest or encode relationships $-$ example, $\prod(1-x^n)^{-1}=\sum p(n)x^n$ connects counting numbers with integer partitions, as $\prod(1-p^{-s})^{-1}=\sum n^{-s}$ connects primes with integers, encoding the fundamental theorem of arithmetic. These equalities are analogous.

Another recurring feature of zeta functions is that they're formed using established "internal pieces" or "points of interest" of various associated objects. Dedekind uses ideals of number fields. Hasse-Weil and Igusa use points on curves and varieties. Ihara uses prime walks on graphs, Selberg uses closed geodesics. Sautoy uses finite-index subgroups. And probably more I don't know of. Since the "pieces" or "points" of an object can be considered "data," this is the same observation as the last paragraph to some extent, only made more precise.

In many cases, a "prototypical" form of a zeta function becomes twisted by some kind of harmonic data encoded by characters or more generally representations. This is apparent in the L-functions of Dirichlet, Hecke and Artin, as well as Igusa's. Arguably the numbery zeta functions are harmonic in nature even without any extra twisting: this is spiritually corollary to the adelic reformulation of functional equations seen in Iwasawa-Tate theory.

With the Riemann zeta function, the local Euler factors are Mellin transforms of Gaussians, or in other words multiplicative Fourier transforms of unique fixed points of additive Fourier transforms, as is the mysterious gamma factor $\pi^{-s/2}\Gamma(s/2)$ appearing in the functional equation (only, taken in the archimedean local place). When the local pieces are patched into a global construct with the correct topologization (adeles/ideles), the functional equation is a direct manifestation of Poisson.

The factorizations of zeta functions (particularly into rational functions) presumably derive from the fact that they're designed to encode arithmetic data $-$ if the associated objects and their internal organs are structured and interacting nicely enough then it's reasonable to speculate the associated "generating functions" will also be structured and decompose into pieces nicely. This is clear with number field zeta functions for example; the Euler product corresponds to prime ideal factorization, and it isn't so surprising with zetas associated to nilpotent groups (since they decompose as direct products of Sylows) (more generally rationality is conjecturally related to Frattini index).

And the functional equations, again at least with the numbery zetas, seem to derive from the fact that zeta functions belie deeper harmonic-analytic natures. I don't have a good idea of where critical line phenomena $-$ the Riemann hypothesis and analogues, Voronin universality, spectral properties channeling GUEs and quasicrystals $-$ or $s=1$ behavior $-$ Birch-Swinnerton-Dyer and nonvanishing of $L(1,\chi)$ $-$ come from: at this stage these sorts of features are largely conjectural and so maybe not good to understand as definitional aspects of zeta functions.

Special values of Riemann zeta (i.e. at even naturals) though do seem to come from somewhere identifiable: namely the Minakshisundaram-Pleijel zeta function. The MP zeta associated to the Laplacian of a circle is $\zeta(2s)$, so the facts (a) that the values $\zeta(2n)$ involve $\pi$ and (b) closed-forms only are known at even naturals instead of odd ones are both less mysterious in light of the fact we're looking at a circle and traces of operators (which have nice formulas for natural number powers, but no such expectation otherwise). This broaches the topic of operator trace zetas.

Zetas defined as traces of operator powers still somewhat fit the picture we've painted so far: they are "generating functions" that encode important data about some object. And their analytic continuations are meaningful $-$ they allow regularized operator determinants $-$ but their lack of factorization and the flexibility in how an operator spectrum can look suggests that these are broader than our scope of interest and do not comport with the "moral design" of the other great examples of modern zeta functions exhibited thus far.

Many other things called zeta functions also do not fit this mold. Herwitz zeta, Shintani cone zetas and multiple zetas come to mind as well. Though this is only my personal impression, it seems these are outgrowths that jut from the core paradigm and thus share many of the same facets of the programs exploring the main territory yet are growing in different directions. I do not feel in any way qualified to comment on if the (conjectural) Selberg class as a whole deserves to be wholly under the umbrella of zeta functions or if it breaks free into a broader special functions crest.

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Like most interesting concepts to humans, the concept of zeta function might ultimately be a bit of a radial category, like the concept of "number": something that has been developed historically over time with many different features patched together that have become laden with a complex web of configurations of necessary/sufficient conditions to actually be in that category. Qiaochu put it succinctly in his "inductive definition." This is a decidedly unsatisfying view but an importantly realistic perspective to keep in mind at any rate.

Without being too specific and choosing what's necessary or sufficient and when, we can still list out the conditions we feel comprise the ultimate "moral design" of zeta functions. These are:

• formed from important pieces of an associated object
• factorizations tell how pieces relate to 'atomic' pieces
• amenable to being enriched or twisted with extra data
• functional equations derive from inherent harmonic nature
• analytic behavior speaks to quantitative behavior of object (??)

While we may view pure mathematics as brute fact not subject to the arbitrariness of nature in our particular universe, it may still yet be a useful aesthetic to understand math as another part of nature ultimately. In this way we can expect some messy classifications like the periodic table in chemistry or the taxonomy of life in biology that don't fit into neat singular packages and are inevitably organized partly by human subjectivity in our effort to tame what's out there.