# For which Functions is the (generalized) Riemann hypothesis known?

In [1], Lin Weng shows that the Riemann hypothesis (RH) holds for certain linear combinations of shifted completed Riemann zetas. Further, Deligne's result on the Riemann hypothesis for function fields gives a RH for the latter. I would like to know whether these are all examples for which the RH is known.

To be precise, we say that a meromorphic function on $\mathbb C$ satisfies the generalized Riemann hyporthesis (genRH), if all of its poles and zeroes lie in the union of a finite number of vertical lines $c+i\mathbb R$ and the real line. Are there any known examples, other than those in [1], or those derived from Deligne's work, for which the genRH has been proven? Of particular interest would be a function which can be written as a Dirichlet series for $\Re(s)>>0$.

[1] Symmetries and the Riemann hypothesis. Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), 173–223, Adv. Stud. Pure Math., 58, Math. Soc. Japan, Tokyo, 2010.

• Weil's and Deligne's proofs of the function field Riemann hypothesis can be expressed in this form (where Deligne's is more general). However in this case $a_n$ is only nonvanishing for $n$ a power of some fixed prime $p$. Nov 28 '17 at 8:26
• @reuns: what about this paper: M. Suzuki, A proof of the Riemann Hypothesis for the Weng zeta function of rank 3 for the rationals, in Conference on L-Functions, 175-200, World Sci. (2007)?
– user1688
Nov 28 '17 at 9:22
• @reuns: yes sure, do you know any examples?
– user1688
Nov 28 '17 at 13:00
• Please reformulate your question correctly from this paper, not this isn't about proving the RH for any Dirichlet series. Nov 28 '17 at 13:30

Not an answer, but an extended comment on the ambiguity in the OP's question. Resolving that ambiguity makes the 'Riemann Hypothesis' aspect less interesting than it first appears.

The OP says

In 1, Lin Weng shows that the Riemann hypothesis (RH) holds for certain linear combinations of shifted Riemann zetas.

Not exactly. Let $\Lambda(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ so $\Lambda(s)=\Lambda(1-s)$. The simplest example seems to be in the paper of Lagarias and Suzuki that reuns links in the comments above: $$\mathbb Z_{2,\mathbb Q}(s)=\frac{\Lambda(2s)}{1-s}-\frac{\Lambda(2-2s)}{s}$$ But $\Lambda(s)$ is not the Riemann zeta function $\zeta(s)$. It seems like I'm being pedantic but this leads to the confusion already in the title of the question:

To be precise, we say that a Dirichlet series $$D(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ satisfies the generalized Riemann hyporthesis (genRH), if...

$\mathbb Z_{2,\mathbb Q}(s)$ is not a Dirichlet series, nor even a Dirichlet series completed with Gamma factors.

Lagarias and Suzuki recount the history and discuss several proofs that the zeros lie on a line. Here's a sketch of an idea they don't mention: $$\mathbb Z_{2,\mathbb Q}(s)=0\Leftrightarrow \frac{\Lambda(2s)}{1-s}=\frac{\Lambda(2-2s)}{s}.$$ Stirling's formula tells you the magnitude of the Gamma factors on the two sides above, and with some work one should be able to show that off the critical line, these two functions can not have the same magnitude. The zeta function can't cause too much trouble because it's evaluated at $2s$, so we're in the region of absolute convergence of the underlying Dirichlet series.

Once we know the zeros are on the critical line, the next question is statistics of the spacing between them. Upon renormalizing so the average gap is $1$, zeros of many number theoretic $L$-functions are conjectured to have the same statistics as eigenvalues of random unitary matrices, the so-called GUE distribution. Partial results in this direction were proved by Montgomery for $\zeta(s)$, and impressive numerical evidence was obtained by Odlyzko. And, by the work of Katz and Sarnak, this is a theorem in the function field case, just as the Riemann Hypothesis is.

The distribution of the zeros of $\mathbb Z_{2,\mathbb Q}(s)$ is very different. This is worked out by Lagarias for a slightly different function in "Zero Spacing Distributions for Differenced $L$-Functions"; this case is very much the same.

With $s=1/2+it$, the two terms which make up the function are complex conjugates. Their difference is $0$ if and only if the imaginary part of each is $0$, or the argument is $\pm\pi/2 \bmod \pi$. Stirling's formula for $\log(\Gamma(s))$ gives very precise information about the argument of that term. The argument of $\zeta(1+2it)$ again can be controlled, since Titchmarsh (5.17.4) gives $$\frac{\zeta^\prime(1+it)}{\zeta(1+it)}=O\left(\frac{\log(t)}{\log\log(t)}\right),$$ the Fundamental Theorem of Calculus recovers the argument with the same bound. (NB the $\log\log(t)$ saving in the denominator will be crucial below.). One finds the number of zeros to height $T$ is the same as for $\Lambda(2s)$ up to an error of (I think) $O\left(\log(T)\right)$. And, the precise estimate given by Stirling's formula means that the zeros are very regular: the normalized gap between the zeros at height $T$ is $$\delta=1+O\left(\frac{1}{\log\log(T)}\right)$$

The graphic below shows the argument of $\mathbb Z_{2,\mathbb Q}(s)$ interpreted as a color, for $0<\sigma<1$ and $10^6<t<10^6+1$.

Up to the scale, the picture looks that same at any height.

• I think you exaggerate the work of Katz and Sarnak slightly. What they show is that, e.g., for most quadratic extensions of $\mathbb F_q(t)$, with $q$ sufficiently large with respect to the discriminant, the zeroes have the expected spacing. Equally they show that for some quadratic extensions, the spacing is very bad. The conjecture that for fixed $q$, for all quadratic extension of $\mathbb F_q(t)$ with sufficiently large discriminant, the spacing is as expected, remains open. Dec 4 '17 at 7:48

See also (full disclosure: my student) Kim Klinger-Logan's preprint arXiv 1706.08552, which gives a very general (though not RH-proving) argument for all zeros of ... something... being on-the-line.

The answer is that there's trivially many, even if you want it to be defined by a Dirichlet series. Dirichlet polynomials (such as $1+2^{-s}$) will work. But that's cheap because these are too small (bounded on vertical lines). Starting with any Dirichlet series $f(s)=\sum a_n n^{-s}$ that analytically continues to an entire function, you can take the function $\exp f(s)$, which can also be defined as a Dirichlet series and is nowhere zero. But that's cheap because these are too big (not order 1 functions) and hence cannot be defined as a product over their zeros.

If you wanted functions that are neither too big nor too small then conjecturally all L-functions satisfy this, but there isn't a single function that you can prove this for, and for all we know no such function exists. There is an earlier mathoverflow question that defines this precisely and asks for a function satisfying the weaker property of having a "zero-free region", which is provided in the answer.

• Are you aware of some work towards analyzing the properties of the equation $g(x) = g(\lfloor x \rfloor) = \sum_\beta \frac{\pm x^\beta}{\beta}$, in order to "construct" (from their zeros/poles $\beta$) all the Dirichlet series $\frac{F'(s)}{F(s)} = s \int_1^\infty g(x) x^{-s-1}dx$ for which $F(s)$ has a meromorphic continuation and functional equation ? Dec 5 '17 at 14:31
• It's a fascinating question. If you just want the asymptotics to look correct to the far right and the far left then that likely corresponds to some bounds on the density function of the zeros, but it's much more difficult if you want it to end up being a Dirichlet series. I unfortunately am not aware of any progress in this direction, or any natural conditions on the zeros that would make it a Dirichlet series, so far all we know there's a fundamental analytic obstruction to all Dirichlet series satisfying a Riemann hypothesis. Dec 6 '17 at 6:36