Watching Over the Zeroes

How confident should we be that the Riemann Hypothesis is true?

Image may be NSFW. Clik here to view.

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Andrew Odlyzko and Herman te Riele, in a 1985 paper, refuted a once widely believed conjecture from 1885 that implies the Riemann Hypothesis. The belief began a U-turn in the 1940s, and by the late 1970s the community was convinced its refutation would come from algorithmic advances to bring the needed computation into a feasible range. Odlyzko and te Riele duly credit advances in algorithms—not mere computing power—for their refutation.

Today we consider reasons for and against a belief in the Riemann Hypothesis (RH), contrast them with the situation for Image may be NSFW.
Clik here to view., and point out that there are numerous conjectures weaker than RH but wide open which an RH claimant might try to crack first.

The conjecture in question is named for Franz Mertens but was first made by Thomas Stieltjes in a letter to Charles Hermite. We have talked about it in some detail here and here. Shorn of detail, the Mertens conjecture has the form:

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Clik here to view.

where Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. are computable—indeed Image may be NSFW.
Clik here to view.. If any conjecture of this kind is false, it can be falsified by a finite computation that gives an Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view..

Recall the RH says that all the zeroes of the analytically-continued complex function Image may be NSFW.
Clik here to view., other than “trivial” ones Image may be NSFW.
Clik here to view. for integers Image may be NSFW.
Clik here to view., have the form Image may be NSFW.
Clik here to view. for some real Image may be NSFW.
Clik here to view.. There are statements of the same kind that are equivalent to RH, including this by Jeff Lagarias:

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Clik here to view.

Enrico Bombieri, in his essay on RH for the Clay Mathematical Institute, details how RH (plus the condition that all the zeroes are simple) can be verified directly for Image may be NSFW.
Clik here to view. up to any fixed Image may be NSFW.
Clik here to view. by a finite computation. Odlyzko and te Riele and others have computed many zeroes to high precision, not only up to certain heights Image may be NSFW.
Clik here to view. but in regions around select much higher Image may be NSFW.
Clik here to view. values. Odlyzko and Arnold Schönhage designed an algorithm that enabled higher computations including Xavier Gourdon’s 2004 verification of RH for the first Image may be NSFW.
Clik here to view. zeroes and some higher patches.

Some Lessons

This work yields some further lessons that we feel are relevant to the current discussion over and what bearing Sir Michael Atiyah recent ideas may have on it.

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Clik here to view. Know the data. The key data in Odlyzko and te Riele’s computation came from their own extensive tabulation of zeroes of Image may be NSFW.
Clik here to view.. Odlyzko’s tables are online.

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Clik here to view. Initial data may mislead. This is an old lesson but always bears repeating. The Mertens conjecture was initially believed based on its holding for incredibly many Image may be NSFW.
Clik here to view.. Now we know that all Image may be NSFW.
Clik here to view. up to Image may be NSFW.
Clik here to view. obey Image may be NSFW.
Clik here to view. and there are strong reasons for believing this continues through Image may be NSFW.
Clik here to view. and maybe Image may be NSFW.
Clik here to view..

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Clik here to view. Augment computation by proof. Odlyzko and te Riele did not simply search for a counterexample Image may be NSFW.
Clik here to view.. Instead their finite computation used a portion of their known zeroes, new lattice reduction techniques, and methods of complex analysis to prove that Image may be NSFW.
Clik here to view. must somewhere rise above Image may be NSFW.
Clik here to view.. In fact, no concrete Image may be NSFW.
Clik here to view. is yet known—only an upper bound of about Image may be NSFW.
Clik here to view. in a 2006 paper by te Riele with Tadej Kotnik.

This situation also still holds with regard to John Littlewood’s refutation of Bernhard Riemann’s further quasi-conjecture that the number Image may be NSFW.
Clik here to view. of primes up to Image may be NSFW.
Clik here to view. is strictly bounded above by the log integral Image may be NSFW.
Clik here to view., which would imply the RH. He proved that Image may be NSFW.
Clik here to view. giving Image may be NSFW.
Clik here to view. must exist, and his student Stanley Skewes at Cambridge extracted an astronomical but finite upper bound for Image may be NSFW.
Clik here to view. from the analytical techniques. The challenge of improving the bounds and maybe finding an Image may be NSFW.
Clik here to view. drew Alan Turing into drafting computational devices and methods for RH. See Odlyzko’s 2012 Turing Centennial slides for a blueprint of Turing’s “zeta machine” and his 2012 paper with Dennis Hejhal for much more.

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Clik here to view. Know the neighbors. Odlyzko and te Riele address the weaker form of the Mertens conjecture that asserts Image may be NSFW.
Clik here to view.. Stieltjes thought he had proved this. They give reasons to expect that expanding their methods and their zeta dataset will falsify this too by replacing the constant Image may be NSFW.
Clik here to view. to an arbitrarily high one. For a case in point, the paper with Kotnik also raised the constant to Image may be NSFW.
Clik here to view..

The RH is, however, equivalent to Image may be NSFW.
Clik here to view.. Similarly, it is equivalent to

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Clik here to view.

This leads to a further remark. If there is a zero Image may be NSFW.
Clik here to view. with Image may be NSFW.
Clik here to view. then Image may be NSFW.
Clik here to view. sometimes exceeds Image may be NSFW.
Clik here to view. and vice-versa. It has not even been proved that there are no zeroes with real part Image may be NSFW.
Clik here to view. Such a partial result ought to be more tractable, for reasons including that the possible densities of zeroes Image may be NSFW.
Clik here to view. are known to dwindle as Image may be NSFW.
Clik here to view. approaches Image may be NSFW.
Clik here to view. (or Image may be NSFW.
Clik here to view. by symmetry), while at least 40% of the zeroes are known to be exactly on the line. Hence bounding Image may be NSFW.
Clik here to view. strictly away from Image may be NSFW.
Clik here to view. ought to be a first step for an RH claimant that may yield easier verification—and even if that is the only thing proved it would still be a monumental advance.

Beliefs About the Riemann Hypothesis

The great RH may be solved or may not. But it is interesting to see that not all believe that the RH is true. Most papers on the RH are cautious about whether it is true or not. The original paper of Riemann called it “very probable.” Littewood famously stated his belief that the RH is false; Paul Turán disbelieved it and for most of his life so did Turing. The last main chapter of John Derbyshire’s 2003 book Prime Obsession is titled with a quote given him by Odlyzko:

Either it’s true, or it isn’t.

We in computer theory have our own Clay Problem, the Image may be NSFW.
Clik here to view. problem. While most seem to believe that Image may be NSFW.
Clik here to view. is not equal to Image may be NSFW.
Clik here to view., some do believe that it could be the case that Image may be NSFW.
Clik here to view.. One important difference from the RH is that neither side is known to be falsifiable or verifiable by a finite computation. We note the near-perfect balance between claims of Image may be NSFW.
Clik here to view. and claims of Image may be NSFW.
Clik here to view. on Gerhard Woeginger’s claims page. On the other hand, we noted above that a refutation of the RH that can be feasibly executed and checked might need to do argumentation rather than just a finite computation.
This suggests that it may be useful to treat the problems similarly and look at the reasons behind beliefs both for and against the RH. So let’s look at the usual arguments—we follow Wikipedia and a 2003 survey by Aleksandar Ivić on reasons to doubt the RH—with a view to how they are tending.

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Clik here to view. Several analogues of the RH over finite fields have already been proved. This is emphasized in nice detail by Bombieri’s essay and we defer to it.

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Clik here to view. Numerical evidence goes beyond verifying that the first 10 trillion zeroes obey the RH. Turing’s negative belief was not only that the RH is false but that past a certain point Image may be NSFW.
Clik here to view., counterexamples would occur with non-negligible frequency. If so, then one could refute the RH by doing relatively less computation but starting at much higher values of Image may be NSFW.
Clik here to view.. Odlyzko and others have sampled such high regions and found many zeroes there. All of them obey the RH.

To be sure, there are reasons for believing that critical magnitudes for testing the RH have not yet been reached. And of course there are other conjectures in analytic number theory besides those mentioned above that have been supported by large amounts of numerical evidence yet have turned out to be false. So watch out. We note also this paper about numerical RH experiments with beautiful plots.

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Clik here to view. The probabilistic argument for the RH based on the behavior of the Möbius function. If it behaves roughly randomly, then the Image may be NSFW.
Clik here to view. bound mentioned above should hold and make the RH true. See, however, these remarks by Eric Bach and Jeffrey Shallit.

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Clik here to view. Odlyzko in 1987 opened a new avenue by deep statistical testing of a conjecture by Hugh Montgomery about correlations between pairs of zeroes of Image may be NSFW.
Clik here to view.. This enhanced an observation by Freeman Dyson that the correlations are the same as for random Hermitian matrices from the Gaussian unitary ensemble (GUE) often used in physics. David Hilbert and George Pólya had earlier conjectured that the zeroes correspond to eigenvalues of some positive linear operator in a way that implies the RH. This led to a spurt of optimism, represented by a 2000 paper by John Brian Conrey on the “GUE Conjecture,” for finding such an operator. This 2015 survey by Marek Wolf gives more recent news. It is titled “Will a physicist prove the Riemann Hypothesis?” and its later sections may verge on the intuitions accompanying Sir Michael Atiyah’s claims about the RH.

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Clik here to view. A flip side about these correlations, however, starts with Derrick Lehmer’s 1956 observation of a phenomenon where two zeros are sometimes very close. This leads to a possible argument that the RH is false. In January of this year, Brad Rogers and Terence Tao turned up the heat on this connection by proving that the GUE conjecture entails the existence of infinitely many Lehmer pairs. Further, they showed that an equivalent form of the RH in terms that an asymptotically-defined numerical constant Image may be NSFW.
Clik here to view. being non-positive can in fact hold only if Image may be NSFW.
Clik here to view. is exactly zero. Intuitively this reduces the “margin of error” for the RH to hold—see also discussion in this earlier survey.

This last point brings us right back to our question about proving partial results toward the RH. A PolyMath project on bounding Image may be NSFW.
Clik here to view. from above has achieved Image may be NSFW.
Clik here to view., and a bound of Image may be NSFW.
Clik here to view. is known to follow if and when the RH is verified for Image may be NSFW.
Clik here to view. up to Image may be NSFW.
Clik here to view.. There was previously a lower bound Image may be NSFW.
Clik here to view., a hair-breadth away from the critical value Image may be NSFW.
Clik here to view.. A partial result on the way to Image may be NSFW.
Clik here to view. would be proving that Image may be NSFW.
Clik here to view. is not in time Image may be NSFW.
Clik here to view.. Proving such a just-above-linear lower bound for Image may be NSFW.
Clik here to view. wouldn’t even be placing it outside of Image may be NSFW.
Clik here to view. yet it would be monumental, just as would proving that no zero has real part above Image may be NSFW.
Clik here to view.. Claimers of Image may be NSFW.
Clik here to view. never seem to address such simple milestones, and the same seems to be true about RH claims that we have perused.

Open Problems

If we plotted community belief in the RH over time on a scale from 0% likely to 100% likely, what would the graph look like? Is it still ascending, or are there signs of its rounding over as with the (big-Image may be NSFW.
Clik here to view. form of the) Mertens conjecture? Odlyzko’s quote prompts us to wonder, has it ever touched the line of being 50-50?

Another issue: why do claimers always claim “it all”—why do they never claim a big partial result? Is this some deep psychological human trait?

[added Bach-Shallit reference for random argument]