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The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences of imaginary parts of zeta zeros show an unusual tendency not to be close to imaginary parts of zeta zeros. "Riemann zeros repel their deltas."

The people I have talked to have said that this simple observation appears to be completely new and potentially groundbreaking. Here are my questions:

1) Has anyone independently verified these statistics? (See my histograms below, and also the excellent graphs in Jonathan Bober's answer.)

2) The author discusses the "eñe product" as a source of a mathematical explanation for his observation, and he refers to his own unpublished manuscripts in which he introduces this product. Are these manuscripts available? Does anyone have any reference which defines the eñe product?

Finally, any insights into Marco's results would be much appreciated.

The histograms (produced in Mathematica) are of the deltas of the first 10000 zeros of $\zeta(s)$ falling in $[100, 110]$ and $[190, 200]$, respectively. The purple bars show the locations of the zeros.

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Pierre Cartier told me last summer about those results. He thought as you that they were groundbreaking, and his opinion was they were sound. – Joël Dec 9 '11 at 4:03
The paper has 0 theorems and I do not understand from an initial reading how the author's multiplication "explains" the numerical data. – KConrad Dec 9 '11 at 6:38
The phenomenon itself should not be too surprising. Everyone (?) believes the conjecture that the imaginary parts of the zeta zeros should be linearly independent over the rationals. Consequently, the equation $\gamma_1-\gamma_2=\gamma_3$ should have no solutions in zeta zeros $\frac{1}{2}+i\gamma_i$. Numerically, therefore, there will be a repulsion effect for imaginary parts away from differences -- what would be interesting is to see an explanation for this repulsion. If one believes all of this, then there ought to be similar repulsion for other linear combination equations. (ctd) – Trevor Wooley Dec 9 '11 at 12:37
(ctd) ... For example, it would be interesting to compute differences of the shape $2\gamma_1-\gamma_2$ with $\gamma_1\neq \gamma_2$. Since the equation $2\gamma_1-\gamma_2=\gamma_3$ should have only the trivial solutions in imaginary parts of zeta zeros, one might also expect to see $2\gamma_1-\gamma_2$ avoid imaginary parts of zeta zeros. Similarly for other linear combinations. Would be good to see some more computations ... and of course prove the linear independence hypothesis(!) – Trevor Wooley Dec 9 '11 at 12:42
I dunno, Trevor, if I choose random points on the line via Poisson process or something, these are linearly independent with probability 1. but I don't see that the locations of the points should repel distances in this scenario. – JSE Dec 9 '11 at 15:02

4 Answers 4

up vote 38 down vote accepted

I have not looked at the paper terribly closely, but my impression is that

1.) The (experimental/statistical) observation is the article is correct, but not really so new.

2.) The author's explanation might not be quite correct. (There doesn't seem to be a great amount of explanation, but what little there is points to the zeros of the zeta function, when the actual explanation seems to be in "shadows" of the zeros on the line Real(s) = 1.)

More details:

The tendency for gaps between zeros of the zeta function to stay away from the zeros of the zeta function has been observed elsewhere. One place that I know of where this is nicely described is in "Riemann zeros and random matrix theory" by Nina Snaith, Milan Journal of Math, Volume 78, Number 1. There is a nice graph in that paper, which I feel I have seen elsewhere, but can't seem to find anywhere else right now. I have posted a copy of the graph at

Snaith, Figure 3

(There ought to me some nice lectures about this stuff available from MSRI, but after more than 10 months, the videos still seem to be in post-production.)

This sort of behavior seems to be explained by the "L-function ratios conjecture" of Conrey, Farmer, and Zirnbauer ( ), which might be better called the "L-function ratios recipe for making conjectures." This explanation is not really satisfactory, in that it is a conjecture, but it is not really only a conjecture; it is more like a somewhat plausible sounding "hand-waving" argument that happens to make astoundingly accurate predictions in many cases.

Anyway, when you use the ratios conjecture/recipe to compute the "two-point correlation" of the zeros of the zeta function, terms will come up that involve the zeta function on the 1-line. And from the prediction you will expect there to be less zero-gaps of size comparable to the minima of the absolute value of $\zeta(1 + it)$. Since the minima of $\zeta(1 + it)$ tend to be close to the zeros of $\zeta(1/2 + it)$, you can "see" the zeros (or shadows of the zeros, as I called them above) in these statistics. (This is why I say that the author's explanation might not be correct.)

The ratios prediction (now I am using yet another word) and the phenomenon of zeros influencing gaps, has been seen/tested in other cases. For a somewhat random sample of the unreasonable influence of the zeros of $\zeta(s)$ in other places, see:


Huynh, Keating, Snaith

Hiary and Odlyzko

(and, of course, the explanatory text around each of those pictures).

As to whether there is anything to this "eñe product," I don't know. It would be nice if there were, but there are rather few details in Marco's paper.

For the specific case of the zeta function, it should not be too hard to get longer range histograms of the zero-spacings. Going out to one thousand or so should be enough to distinguish between minima of $\zeta(1 + it)$ and zeros of $\zeta(1/2 + it)$.


Update number 2: I happen to have access to lots of zeros of the zeta function computed to high precision by Dave Platt. (These will be more publicly available sometime soon, hopefully, when certain computers have their disk storage upgraded.) I've been wanting to look at them for a while, and I wanted to see a better picture, so I made one.

The following picture is a histogram of the differences of the first two billion zeros of the zeta function, restricted to zero spacings between 1000000 and 1000010, along with the Bogomolny-Keating/Conrey-Farmer-Zirnbauer/Conrey-Snaith prediction ("right click", or whatever, to view the image by itself, for higher resolution):


Some notes:

  • The histogram has 1024 steps, so a box size of 10/1024. It takes a lot of zeros to make a smooth picture with such a small box size. (There are 57,465,000,000 pairs of zeros "contained" in this histogram.)

  • I haven't been too careful in all of my computations. For example, I don't know exactly which zero was the last one in the histogram (which affects the prediction a little bit), and the formula for the prediction is a little complicated, and I haven't haven't checked it carefully. So I don't know if the prediction is a little larger than the actual values, or if this is my error.

  • The histogram is not normalized! I like it like this, but it hides the fact that the error in the prediction here is generally less than 0.1%, even if I computed it wrong. (The histogram varies by around 4.5%, for comparison.)

It can be seen that the zeros (red dots) still have some influence in this range, but it is nowhere near as clear-cut as it is for small spacing size.

Specifically, the green line is (or is supposed to be)

$$ \frac{10}{1024} \cdot \frac{1}{(2\pi)^2}\Re\Bigg[ 2T\left(\frac{\zeta'}{\zeta}\right)'(1 + iX) - 2T \cdot B(iX) + T\left(\log \frac{T}{2\pi}\right)^2 - 2T\log\frac{T}{2\pi} + 2T $$ $$ + \frac{2 \zeta(1 - iX)\zeta(1 + iX)A(iX)}{(2\pi)^{iX}}\left(\frac{T^{1 - iX} - 1}{1 - iX}\right)\Bigg], $$ where $$ A(s) = \prod_p\left(1 - \frac{1}{p^{1+s}}\right)\left(1 - \frac{2}{p} + \frac{1}{p^{1 + s}}\right)\left(1 - \frac{1}{p}\right)^{-2}, $$ $$ B(s) = \sum_p \left(\frac{\log p}{p^{1 + s} - 1}\right)^2, $$ $T = 732565723.921443$ (approximately the end of the range of zeros considered) and $X$ runs from 1000000 to 1000010.

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Someone named "reference" attempted to add a comment as follows: you say in your answer that Pérez Marco's results are "not really so new". Could you give any references to the numerical analysis of sections 4 to 8? – S. Carnahan Dec 13 '11 at 5:15
For the pair correlation of zeros of an L-function other than the zeta function, I don't know offhand of any other experiments, but the ratios conjecture provides a way to make a prediction for what the answer should be. The computations are a little messy, but I think it shouldn't be too hard to replicate what is done in Section 4 of Conrey and Snaith's "Applications of the L-functions ratios conjectures." (See .) I'm less certain about the experiments with two different L-functions at the same time, but maybe they can be predicted as well. – Jonathan Bober Dec 13 '11 at 11:49
The line of text "zeros-histogram-..." is meant to be the image… : is there any way to recreate or find this image? I guess William Stein might be one to contact... – David Roberts Sep 23 at 1:12

My student, Brad Rodgers, has just posted a paper on the arXiv at which proves a partial result towards the repulsion effect (that differences of two imaginary parts of Riemann zeroes tend to avoid another Riemann zero), in the spirit of Montgomery's partial result towards his pair correlation conjecture (i.e. this repulsion effect can be detected when tested against sufficiently band-limited test functions, assuming RH).

Ultimately, the reason for this repulsion lies in the obvious approximate formula

$$ |\Lambda(n)|^2 \approx \Lambda(n) \log n$$

where $\Lambda$ is the von Mangoldt function. If one compares this with the explicit formula, which is formally of the form

$$ \Lambda(n) = 1 - \sum_\rho n^{\rho-1} + \ldots$$

one begins to see the negative correlation between differences of imaginary parts of zeroes $\rho$ (which show up in the expansion of $|\Lambda|^2$) and in the imaginary parts of zeroes themselves. (Making this intuition rigorous, though, is somewhat non-trivial, requiring manipulations similar to those in Montgomery's original paper to deal with the fact that the explicit formula as given above is only convergent in a very weak sense.)

EDIT: it is likely that a similar analysis would also explain why Riemann zeroes correlate with differences of zeroes of other functions. For instance, starting from $|\Lambda(n) \chi(n)|^2 \approx \Lambda(n) \log n$, one can predict that differences of imaginary parts of zeroes of a Dirichlet L-function should repel away from the imaginary part of zeroes of zeta.

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Dear Terry, this phenomenon looks like it should be universal. The building blocks of all L-functions are, conjecturally, the cuspidal automorphic representations $\pi$ of $\mathrm{GL}_n(\mathbf{A_Q})$. If $\Lambda_{\pi}(n)$ denotes the $n$th Dirichlet coefficient of $-L'(s,\pi)/L(s,\pi)$, a Tauberian calculation together with an analysis of Rankin-Selberg L-functions gives $\sum_{n \sim X} |\Lambda_{\pi}(n)|^2\sim X\log{X}$, so $|\Lambda_{\pi}(n)|^2 \approx \Lambda(n) \log{n}$ on average. – David Hansen Mar 19 '12 at 16:07
Is it possible to give a similar heuristic which explains why $\zeta^\prime/\zeta$ (v. its derivative) appears in the 1 level density of the zeros of quadratic Dirichlet $L$-functions? – Stopple Mar 21 '12 at 18:11

I'm taking this from pages 15-16 from the linked article. The ene product $f \star g$ (not correct notation, I know) of two polynomials seems to be the polynomial which has roots the products $\alpha\beta$ for $\alpha$ a root of $f$ and $\beta$ a root of $g$. Then the ene product of a pair of Euler products

$$ F(s) \bar{\star} G(s) = \prod_p F_p(p^{-s}) \bar{\star} \prod_p G_p(p^{-s})$$

where each of the $F_p$ and $G_p$ are polynomials indexed by primes $p$, is $$ F\;\bar{\star}G(s) = \prod_p F_p \star G_p (p^{-s}) $$

This is clearly merely the definition, and doesn't even ask what it might do. All I can glean is that the set of Dirichlet $L$-functions becomes a ring under this product with the Riemann zeta (suitably normalised) becoming the multiplicative unit.

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How is this different from the tensor product of two L-functions (given as Euler products). There again, to form it you take all the products of roots, no? – Junkie Dec 9 '11 at 6:46
I don't know! I was just copying from the article. Exactly which roots does one take products of for the tensor $F\otimes G$? – David Roberts Dec 9 '11 at 7:36
I don't think it is supposed to be the tensor product (it is defined on polynomials, not inverses of polynomials). Look at the calculation on p. 16: $\zeta(s)\tilde\star\zeta(s)=\zeta(s+1/2)^{-1}$. Not that I understand what's going on either... – B R Dec 9 '11 at 8:02
(oops, the second $\zeta$ should have a bar over it). – B R Dec 9 '11 at 16:10

To my understanding the "eñe product" is the same as the usual product of Dirichlet series considered as elements in the big ring of Witt vectors over the complex numbers say. So as a reference you might want to take a look at Hazewinkels encyclopedic article .

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