# Resources and outstanding conjectures about the Epstein zeta function

I am looking for a reference to the Epstein zeta function. For the Riemann zeta function, there is Titchmarsh's treatment. However, I do not know of any references regarding the Epstein zeta function and I'll be thankful to anyone who points me to a seminal paper/book that talks about properties of the Epstein zeta function and why it is an interesting object of study. This question is probably both broad and imprecise, but I hope it will attract an interesting set of answers.

In addition, I was wondering if there are some "interesting" conjectures that drive research regarding the Epstein zeta function (or where the Epstein zeta-function naturally occurs).

I would recommend Harold Stark's papers on the Epstein Zeta function, such as "L-functions and character sums for quadratic forms, I", Acta Arith. vol 14 (1967/68), PP. 35–50, and "On the zeros of Epstein's zeta function", Mathematika vol 14 (1967), pp. 47–55.

These contributed to the understanding of the Deuring–Heilbronn phenomenon, and the solution of class number problems for class number 1, 2, and 3.

An earlier reference (in German) is Deuring's "Zetafunktionen quadratischer Formen", J. Reine Angew. Math., vol 172, (1934), pp. 226-252.

As to open problems, here's a suggestion, with some number theory background for context. There is a very elegant theorem about discriminants of binary quadratic forms with one class per genus, which motivates the desire to classify them. The congruence class of a prime number $$p$$ modulo $$d$$ determines which form of discriminant $$-d<0$$ represents $$p$$ if and only if there is one class per genus \cite{Cox}.

Such discriminants which are congruent to $$0$$ modulo $$4$$ are of course Euler's numeri idonei or idoneal numbers. Euler expected there would be infinitely many such discriminants, and was surprised to be unable to find more. It was Gauss who conjectured that the only such discriminants are the 65 examples (not necessarily fundamental) known to Euler. There are also 65 known fundamental discriminants (not necessarily even) with one class per genus. The existence of a 66th is still an open problem. By genus theory we know that for discriminants with one class per genus, the class group satisfies $$\mathcal C(-d)\cong \left(\mathbb Z/2\right)^{g-1},$$ where $$g$$ is the number of prime divisors of $$d$$. Obviously $$d$$ is bigger than the absolute value of the smallest fundamental discriminant with $$g$$ prime divisors, $$d_g\overset{\text{def.}}=3\cdot4\cdot5\cdot7\dots \cdot p_g.$$ From lower bounds on the size of $$p_g$$, the $$g$$th prime and on $$\theta(x)=\sum_{p, one can show that $$d_g>g^g.$$ Since $$2^{g-1} \ll \sqrt{g^g},$$ lower bounds for the class number which we expect to be true rule out the possibility for one class per genus for large $$g$$. Chowla used this idea to show that the number of classes per genus tends to infinity with $$d$$.

In 1973, Peter Weinberger showed that on GRH, no fundamental discriminant $$-d<-5460$$ has one class per genus, and unconditionally there is at most one more such $$d$$. Weinberger used Tatuzawa's version of Siegel's Theorem to deduce there is at most one such $$d$$ bigger than $$d_{11}=401120980260\approx 4\times 10^{11}$$, and sieving to eliminate the $$d.

In contrast, Oesterle explicitly observed that the lower bound due to Goldfeld-Gross-Zagier is not strong enough to finish the classification of discriminants with one class per genus: $$\log(g^g)$$ is $$\ll 2^{g-1}$$. Iwaniec and Kowalski observed that even the full strength of the Birch Swinnerton-Dyer conjecture, "the best effective lower bounds which current technology allows us to hope for" would not suffice, as $$\log(g^g)^r$$ is $$\ll 2^{g-1}$$ for any $$r$$. In fact, the outlook is still more bleak: Watkins observed that if the discriminant $$-d$$ is divisible by all the primes up to $$(\log\log d)^3$$ (as $$d_g$$ certainly is), the product over primes dividing $$d$$ in the Goldfeld-Gross-Zagier lower bound is so small the resulting bound is worse than the trivial bound.

The Deuring-Heilbronn phenomenon says that the existence of a Landau-Siegel zero for a Dirichlet $$L$$-function $$L(s,\chi_{-d})$$ would affect the position of zeros of other $$L$$-functions. In this context, the low lying complex zeros of $$\zeta(s)L(s,\chi_{-d})$$ for class number 1 are shown to be on the critical line in the papers of Deuring and Stark above. This is the local version of 'modified GRH'. Moreover the location of the zeros on the critical line is constrained; they are forced to be nearly periodic.

An interesting application of this phenomenon can be seen in Stark's PhD thesis. Stark used explicit values for the zeros of $$\zeta(s)$$ to show no discriminant $$-d$$ had class number $$1$$ in the range $$163.

It would be interesting to adapt the methods of Stark, (and later improvements of Montgomery-Weinberger, and of Watkins) to the study of discriminants with one class per genus. Traditionally looks at binary quadratic forms $$Q(x,y)=ax^2+bxy+cy^2$$ which are reduced; in this case $$a$$ is the minimum nonzero value represented by $$Q$$. However if there is one class per genus, it is more convenient to look at forms with $$a|d$$, $$a<\sqrt{d}$$; the classes are in one to one correspondence with such divisors. These are the so-called 'ambiguous' forms. Most such forms are reduced, and those which are not are still close enough to being reduced that we get good estimates.

So here's a suggestion for an open problem on the Epstein zeta function: adapt the methods of Stark's thesis (for the Epstein zeta function attached to the. principal form, i.e. $$h=1$$) to Epstein zeta functions attached to ambiguous forms, to be able to use zeros of $$\zeta(s)$$ to eliminate the possibility of one class per genus, up to some very large value of $$d$$.

I am not suggesting this is straightforward - I spent some time thinking about this and got nowhere.

• Thanks, this is the sort of answer I am looking for. Do you know of some conjectures that involve the Epstein zeta-function? Feb 3 at 15:14
• @user859588 see edits to answer. Feb 10 at 18:24
• Thank you so much for the detailed account! That's very kind of you. :) Feb 12 at 14:03