4. Algebraic Geometry and Selberg Zeta Analogies Path

This path draws analogies from algebraic geometry and the theory of zeta functions of varieties, as well as Selberg’s zeta function for automorphic forms. The idea is to mimic the proofs of RH analogues in those domains for the classical RH. These analogies inspired major approaches (Weil’s proof in function fields, Deligne’s work, etc.).

Base Camp 1: The Weil Conjectures and Function Field RH – Learn how RH is proved in algebraic geometry. André Weil conjectured (and proved for curves) that for any proper algebraic variety over a finite field, the eigenvalues of Frobenius on $i$th $\ell$-adic cohomology have absolute value $q^{i/2}$, which is an analogue of RH[46]. The crucial case of curves gives an analogue of RH for zeta functions of curves (a special case of the Weil conjectures, later proved fully by Deligne). Key resources:

“Dix Exposés sur la Cohomologie des Schémas (SGA 5)” –* specifically the chapter by J-P. Serre on Zeta and $L$-functions. This advanced reference introduces the zeta function of a variety and sketches the cohomological proof strategy (though Deligne’s full proof is deep). It shows concretely how zeros lie on “critical lines” ($\Re(s)=i/2$) because they come from eigenvalues of Frobenius with weight $i$[46]. While heavy, it’s authoritative for the function field analogy.

“Algebraic Number Theory” by Jürgen Neukirch – Chapter VII discusses zeta functions of number fields vs. function fields, and includes a section on the analogy with curves over $\mathbf{F}_q$. It explains how the Dedekind zeta of a number field is an analogue of the zeta of a curve (where RH is true by Weil’s theorem). This helps build intuition that Spec($\mathbb{Z}$) (the spectrum of integers) might play the role of a “curve over $\mathbf{F}_1$ (field of one element)” in a grand analogy, suggesting a geometric approach to RH[47]. Neukirch provides the necessary algebraic number theory background to appreciate this analogy.

“L-functions and Galois Representations” (London Math. Soc. Lecture Note Series 320) – Contains an article by K. Rubin and A. Silverberg, “Riemann Hypothesis in Characteristic $p$”, which is a digestible account of why the RH holds for function field zeta functions (Weil/Deligne) and how concepts like Hodge structures and monodromy enforce it. It’s a bridge between the highbrow cohomology proofs and a number-theorist’s understanding.

Base Camp 2: Selberg Trace Formula and Zeta for Modular Forms – Study the analogy for automorphic $L$-functions. A triumph analogous to RH was Atle Selberg’s proof that for certain modular forms, all nontrivial eigenvalues of the Laplacian lie in $[1/4,\infty)$, which translates to the Selberg zeta having zeros on $\Re(s)=1/2$. This is essentially an RH analogue for Selberg zeta[28]. Important references:

“Automorphic Forms and the Selberg Trace Formula” by A. Knapp – Introduces Selberg’s spectral theory on $GL(2)$ and derives the Selberg zeta function. Knapp shows how the Selberg trace formula serves as an explicit formula relating lengths of closed geodesics (analogous to primes) and the spectrum of the Laplacian (analogous to zeros). This is directly parallel to the explicit formula in prime number theory[48][49] and Weil’s criterion, but here it leads to an actual proof of an RH-like statement for compact surfaces.

“Eigenvalues of the Laplacian and Modular Forms” by Henryk Iwaniec – Iwaniec, a leading expert, provides intuition on why the Laplacian eigenvalues $\lambda$ on a congruence quotient of the upper half-plane should satisfy $\lambda \ge 1/4$ (which is the critical line $s=1/2$ in the $s$-plane). He explains partial results and the relation to the Selberg eigenvalue conjecture (proved for special cases). This connects to known Selberg zeta zeros lying on the line and hints at approaches for classical RH by constructing analogous trace formulas[48][49].

“Explicit Formulae” by Anton Deitmar – A monograph on explicit formulae (in the style of Weil’s and Selberg’s formula). It treats side by side the explicit formula for number fields and the Selberg trace formula for surfaces. Studying this helps one see exactly what ingredient is present in the Selberg case (the existence of a suitable Hilbert space and Hermitian operator) that is missing in the number field case. Deitmar’s comparative approach can inspire ideas to import tools from the Selberg world into the Riemann zeta world.

Base Camp 3: Deninger’s Cohomological Approach – Investigate bold attempts to create a “cohomology of Spec(Z)” (the dream of many). Deninger (1990s) proposed a hypothetical cohomology theory for arithmetic schemes that would produce the right spectral behavior for zeta:

“Some Analogies between Number Theory and Dynamical Systems” by Christopher Deninger (1998) – Deninger’s paper (often cited in RH contexts[47]) outlines how one might interpret $\log \zeta(s)$ as a “determinant” of an infinite-dimensional operator, akin to how zeta functions of varieties are expressed as determinants on cohomology[50]. He introduced the idea of a “mystery cohomology” whose eigenvalues would be $e^{i t_n}$ for $1/2 + it_n$ zeros. This paper, though conjectural, is widely respected for its depth and has inspired a lot of follow-up research.

“Noncommutative Geometry, Dynamical Systems and $\zeta$” by Alain Connes – In this work (Chapter 5 of “Noncommutative Geometry” or various papers), Connes builds on Deninger’s ideas by using tools from noncommutative geometry (e.g. adeles classes) to derive the explicit formula and interpret it as a trace formula[41]. While not yet achieving a proof of RH, Connes’ approach ties together the Hilbert–Pólya spectral idea with the need for a cohomological framework. It’s useful to study for those aiming to extend algebraic geometry methods to $\Spec(\mathbb{Z})$.

“Equivalents of the Riemann Hypothesis, Vol. 2: Analytic Equivalences” by K. Broughan (2017) – This comprehensive work lists many criteria equivalent to RH and often discusses their geometric or spectral meaning. It includes Weil’s positivity criterion and others that might hint at underlying structure[51]. Broughan’s books (Vol.1 and Vol.2) are a well-respected catalog of RH-related statements, some of which emerged from the algebraic geometry analogy (e.g. the positivity of certain bilinear forms à la Weil[52]). It’s an excellent reference to ensure one hasn’t missed any algebraic-geometric insight that translates into an RH criterion.

By following this path, one sees RH not just as an isolated conjecture, but as part of a grand tapestry including the Weil conjectures and Selberg’s eigenvalue theorem. The challenge and hope is to import the powerful tools from those domains (cohomology, trace formulas) into the number field case. The resources above show the state of these efforts.