Of course, the truth of the Riemann hypothesis is a central question in analytic number theory. Does its truth/falsehood have important consequences in purely algebraic number theory as well? Moreover, are there any known methods of studying or "attacking" the Riemann hypothesis that are more algebraic than analytic?

  • $\begingroup$ For understand why the RH for $\zeta(s)$ might be true, you have to look at all the similar functions, and the necessary and sufficient condition for a Dirichlet series having a RH. Once you understand all those examples and counter-examples, you are in front of a philosophical question : what is so special with the prime numbers, compared to all other (similar) sequences of integers ? $\endgroup$ – reuns Oct 16 '16 at 0:22
  • $\begingroup$ IMO, knowing many examples and counter-examples of L-functions is helpful essentially for proving that a particular attack to the RH cannot work. $\endgroup$ – reuns Oct 16 '16 at 0:25

You can find here http://www.claymath.org/sites/default/files/sarnak_rh_0.pdf
a short but very enlightening account on RH and GRH by P. Sarnak , « Problem of the Millennium : The Riemann hypothesis », 2004 (building on E. Bombieri’s official presentation of the problem). Your question , as well as many others on this subject, could be placed under the common headline « What is so interesting about the zeroes of the Riemann Zeta function ? » (as termed by Karmal, April 24). I can see that a large majority of the answers concentrate on applications to the distribution of primes, which is natural since Riemann himself started the subject, but one has the right to marvel at e. g. how GRH can lead to an information on the arithmetic of elliptic curves (Serre’s result recalled by Sarnak op. cit.). Even more wonderful is the parallel with the Zeta function of a curve (Weil’s theorem recalled by Jake) and more generally, of a smooth projective variety (Weil’s conjectures, proved by Deligne and others) over a finite field. This is not just an analogy, but a manifestation of the unity of mathematics : « What was really eye-catching [in the Weil conjectures], from the point of view of other mathematical areas, was the proposed connection with algebraic topology. Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil (based on working out some examples) was striking and novel. It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on. The analogy with topology suggested that a new homological theory [étale cohomology] be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre » (Wikipedia).

Modifying slightly your question, let us now concentrate on the “significance of the zeroes of the Zeta function to algebraic number theory”. Away from RH, the deep arithmetic properties surrounding the so called trivial zeroes are all the more astonishing in view of the opposition discrete vs. analytic. Looking at the special values $\zeta (n)$, $n \in \mathbf Z$ and using the functional equation, one can easily show that $\zeta (0) = 1/2$, and thus restrict to the integers $n$ of the form $-2m$ or $1 – 2m$, $m$ > 0. Then:

1) The special value $\zeta (1 - 2m)$ = - $B_{2m}$ /$2m$, where $B_k$ is the $k$-th Bernouilli number, is a plain rational value. But when a number theorist falls upon a rational number, he inevitably asks whether the numerator and the denominator could not be the orders of some finite groups. Astonishingly, this is the case, see below.
2) $\zeta (- 2m) = 0$ . These trivial zeroes are simple and the special values $\zeta (- 2m)^* $ are by definition the first non zero terms in the Taylor expansion at $s = \zeta (- 2m)$ .

The general formula for all the special values then reads: for all $n \ge 1$, $\zeta (- n)^* $ is equal, up to a sign and a power of 2, to $R_n (\mathbf Q)$. # $K_{2n}(\mathbf Z)$ / # torsion $K_{2n+1}(\mathbf Z)$. Here the $K_i (\mathbf Z)$ are the Quillen K-groups associated to the ring $\mathbf Z$ and $R_n(\mathbf Q)$ is the $n$-th Borel regulator. This was the so called Lichtenbaum conjecture (around 1973), now a theorem.

Comments: Quillen’s definition of the K-groups of a ring comes from algebraic topology – a further manifestation of the unity of mathematics. Note that the K- groups of $\mathbf Z$ are in no way easy to compute. For any number field $F$, one has the K-groups of the ring of integers $O_F$; one can define also the Borel regulators of $F$, higher analogues of the Dedekind regulator which appears in the classical expression of $\zeta_F (0)^* $; and one can extend the Lichtenbaum conjecture to $F$ to get a higher analogue of the class number formula. The conjecture has been proved for abelian fields. Its depth can be judged by simply looking at the list of tools which come into play: the values of $\zeta_F(s)$ at positive integers expressed in terms of polylog functions; the Main Conjecture of Iwasawa theory on p-adic L-functions proved by Mazur and Wiles; the so called Milnor-Bloch-Kato conjecture relating the K-theory of fields to Galois cohomology, proved by Voevodsky; the Quillen-Lichtenbaum conjecture relating the the K-theory of rings of integers to étale cohomology, proved by Vv. and others. Note that a far reaching generalization of the Lichtenbaum conjecture has been proposed by Bloch-Kato concerning the special values of “motivic” L-functions – another strong indication, if still needed, of the unity of mathematics. A convenient introductory book is “The Bloch-Kato conjecture for the Riemann Zeta function”, Coates-Raghuram-Saikia-Sujatha ed., London Math. Soc. LN 418, 2015 .

  • $\begingroup$ Honestly, this could be formatted a whole lot better; as it stands, it's a wall of text that makes an interesting subject look uninteresting. $\endgroup$ – J. M. is a poor mathematician Jul 25 '16 at 13:01
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    $\begingroup$ In my opinion this is actually an incredible answer. I'm only just seeing it. Never knew any of this-- why is it that I've only heard the most basic (prime number theorem error term related) theory surrounding the zeta function? $\endgroup$ – Vik78 Oct 16 '16 at 2:27

This is only somewhat related but I hope you find this useful.

The Extended Riemann Hypothesis is a generalization of the Riemann Hypothesis to number fields (Just replace the Riemann Zeta function with the Dedekind Zeta function of a number field). It's truth implies an effective error term for the Chebotarev Density Theorem which has implications for the distribution of ideals in Ideal Classes, Ray Classes etc.

Another algebraic analogue of the Riemann Zeta function is the Zeta function of a Curve over a finite field. Here, the analogue of the Riemann Hypothesis has the following important consequence (among a lot of other interesting information): If $C$ is a smooth curve of genus $g$ over the finite field $\mathbb F_q$, denoting the number of points of $C/\mathbb F_q$ as $N_q$, $$|N_q-q-1|\leq 2g\sqrt q $$

This is actually a theorem!

  • $\begingroup$ A quotation in Neukirch's ANT asserts that "the Zeta function knows everything about algebraic number fields, the only problem is to make it speak" (attributed to Harder). May I suggest that you take a quick glimpse at the book “The Bloch-Kato conjecture for the Riemann Zeta function” which I cited ? You'll get an idea of the depth at which Zeta intervenes in the arithmetic of number fields. $\endgroup$ – nguyen quang do Oct 25 '16 at 15:45

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