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Riemann Hypothesis—Why So Hard?

February 21, 2021


“If I were to awaken after having slept a thousand years, my first question would be: has the Riemann Hypothesis been proven?” — David Hilbert

Steklov Institute memorial page

Sergei Voronin was an expert in number theory, who studied the Riemann zeta function, but who sadly died young over twenty years ago. We discussed his amazing 1975 result about the Riemann zeta function here. Others call the result the amazing theorem. I (Dick) am getting old—I almost forgot that we did a post on his theorem again over four years ago.

Today I thought we would recall his theorem, sketch why the theorem is true, and then discuss some extensions.

Starting with Alan Turing we have been interested in universal objects. Turing famously proved that there are universal machines: these can simulate any other machine on any input. Martin Davis has an entire book on this subject.

Universal objects are basic to complexity theory. Besides Turing’s notion, a universal property is key to the definition of NP-complete. A set {S} in NP is NP-complete provided all other sets in NP can be reduced to {S} in polynomial time. Michael Nielsen once began a discussion of universality in this amusing fashion:

Imagine you’re shopping for a new car, and the salesperson says, “Did you know, this car doesn’t just drive on the road.” “Oh?” you reply. “Yeah, you can also use it to do other things. For instance, it folds up to make a pretty good bicycle. And it folds out to make a first-rate airplane. Oh, and when submerged it works as a submarine. And it’s a spaceship too!”

Voronin’s Insight

In 1975 Voronin had the brilliant insight that the Riemann zeta {\zeta(s)} function has an interesting universality property. Roughly speaking, it says that a wide class of analytic functions can be approximated by shifts {\zeta(s+it)} with real {t}. Recall

\displaystyle  \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}

for {\Re(s) >1}, and it has an analytic extension for all other values but {s=1}.

The intense interest in the {\zeta(s)} function started in 1859 with Bernhard Riemann’s breakthrough article. This was the first statement of what we call the Riemann Hypothesis (RH).

In over a century of research on RH before Voronin’s theorem, many identities, many results, many theorems were proved about the zeta function. But none saw that the {\zeta(s)} function was universal before Voronin. Given the zeta function’s importance in understanding the structure of prime numbers this seems to be surprising.

Before we define the universal property I thought it might be useful to state a related property that the {\zeta(s)} function has:

Theorem 1 Suppose that {P} is a polynomial so that for all {s},

\displaystyle  P\left(\zeta(s), \zeta{'}(s),\dots,\zeta^{(m)}(s) \right) = 0.

Then {P} is identically zero.

Since {s} is a single variable, this says that {\zeta(s)} and its derivatives {\zeta'(s)} and {\zeta''(s) \dots } do not satisfy any polynomial relationship. This means intuitively that {\zeta(s)} must be hypertranscendental. Let’s now make this formal.

Voronin’s Theorem

Here is his theorem:

Theorem 2 Let {0<r<1/4}. Let {f(s)} be an analytic function that never is zero for {|s| \le r}. Then for any {\epsilon>0} there is a real {t} so that

\displaystyle  \max_{\left | s \right | \leq r} \left | \zeta(s + \frac{3}{4} + i t) - f(s) \right | < \epsilon.

See the paper “Zeroes of the Riemann zeta-function and its universality,” by Ramunas Garunkstis, Antanas Laurincikas, and Renata Macaitiene, for a detailed modern discussion of his theorem.


Note that the theorem is not constructive. However, the values of {t} that work have a positive density—there are lots of them. Also note the restriction that {f(s)} is never zero is critical. Otherwise one would be able to show that the Riemann Hypothesis is false. In 2003, Garunkstis et al. did prove a constructive version, in a paper titled, “Effective Uniform Approximation By The Riemann Zeta-Function.”

Voronin’s Proof

The key insight is to combine two properties of the zeta {\zeta(s)} function: The usual definition with the Euler product. Recall the Riemann zeta-function has an Euler product expression

\displaystyle  \zeta(s) = \prod_p \frac{1}{1-p^{-s}}.

where {p} runs over prime numbers. This is valid only in the region {\Re(s) > 1}, but it makes sense in a approximate sense in the critical strip:

\displaystyle  1/2 < \Re(s) < 1.

Then take logarithms and since {\log(p)} are linearly independent over {Q}, we can apply the Kronecker approximation theorem to obtain that any target function {f(s)} can be approximated by the above finite truncation. This is the basic structure of the proof.

Open Problems

Voronin’s insight was immediately interesting to number theorists. Many found new methods for proving universality and for extending it to other functions. Some methods work for all zeta-functions defined by Euler products. See this survey by Kohji Matsumoto and a recent paper
by Hafedh Herichi and Michel Lapidus, the latter titled “Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality.”

Perhaps the most interesting question is:

Can universality be used to finally unravel the RH?

See Paul Gauthier’s 2014 IAS talk, “Universality and the Riemann Hypothesis,” for some ideas.

[fixed missing line at end]

7 Comments leave one →
  1. Frank Vega permalink
    February 21, 2021 11:35 pm

    I’m really proud of have been working with you in a paper about the Riemann Hypothesis. Unfortunately, we have stopped the communication, but I hope some day we can finally continue on this. From our interchange, I learned a lot, principally in how to organize better the ideas (I need to learn more on that). You prefer to be only mentioned in the acknowledgements, that’s why I have to remove all your contributions and only pick some tiny things from our collaboration (for just to be honest with you). I wish that you could know that my invitation to work together on this is still open: this work can be improved much more until make it a perfect and verified proof (this has not been validated yet). This is all I have moved on these few past weeks:

    https://doi.org/10.5281/zenodo.3648511

    What do you think?

  2. Anonymous permalink
    February 22, 2021 2:11 am

    Nice introductory article for the general reader in the theory of universality of the Riemann zeta function. One comment I have is that on the last paragraph on “Open Problems” the last phrase is not typed properly, I guess you meant to say: “ See Khoji’s Matsumoto’s survey and also the work done by Hafedh Herichi and Michel Lapidus on quantizing the universality theorem of Voronin”. I suggest to fix this phrase, since in his survey, K. Matsumoto does not discuss the research work of Herichi and Lapidus on a quantum analog of the universality theorem of Voronin involving using truncated spectral operators. Overall a very nice art introductory article !

    • February 22, 2021 10:37 am

      Thanks! I suspect a line of text was lost owing to a technical glitch.

  3. Ivan Gusev permalink
    February 22, 2021 4:13 am

    There is universality. It is “topological quantization” where “critical line” is a boundary of certain manifold and pole at 1 is Arnold tongue. I think I’m 3 theorem away from RH. First — If a submanifold is constructable by local zeta function(“emergent acting potential” so to say) and has equalized action to outside and inside to zero, then meeting “super-secret” condition results in existence of self-adjoint unbounded operator that constructs this manifold out of nothingness, it emerges, so to say, and this operator is linear. Also, it cannot be wandering. Second — For any topological cycle on “topological quanta” there does exist algebraic cycle that has certain rotation number(zero maybe? I’m a bit struggling with this condition) if a set of topological quanta is closed for particular topological cycle. Using the fact that “critical line” is a boundary of a manifold we show that critical line is “fixable”(we can repair it) and bring zeroes to its place by moving zeroes in relation to each other, it is like if RH is false then we can come up with better function that does the same and has its all zeroes on the line, this criteria is super easy. Third — there does exist cycle on “topological quanta” of a form p + 2 = p (twin prime conjecture) and it occurs infinitely often. Assuming that it is false we show that in this case Arnold tongue would make pole at 1 finite value and therefore strip RZF of its analytic continuation. And then we state that pole at 1 implies that our set of “topological quanta” is a self-boundary of certain manifold that gets created by RZF in an act of “self-emergence” and that if any such cycle P+n1=n2 would fail it would imply that n1 or n2 is not in N. I call it “criteria of incremental stability”. You can imagine natural line to be a topology based vector(with of quanta 1 and repeat it infinitely often for all numbers on the vector) and prime numbers to be a directional quanta of this vector in all possible directions and for each P there does exist similar rerepresentaion on quanta of this quantized vector and it all unites in continuous “thing”. It is all going to be super-confusing, I know, but imagine zero-knowledge proofs of action in dynamical systems. You give me DS and claim that you have seen action in it. How do I verify without obtaining knowledge about your proof that it is right? I’m just checking that “our topological quanta” coincide that’s it. And PvsNP is harder then this because it requires measuring “density of cycles” in all possible ways, I think I’m going to give up on P!=NP because of this. By the end of this year I will, maybe, post my paper or go to prison for criticizing Putin. I just have no formal education and don’t think that anybody will be interested in a reading of my fabrication, still the process of making one is quite interesting!

  4. February 22, 2021 3:50 pm

    Zeta universality does have the proof of P vs NP. However, writing it out in detail is NP hard!

  5. Why is P vs NP so important? permalink
    February 25, 2021 4:35 pm

    If NP complete languages are subset of Indexed languages which are subset of Context sensitive languages which are subset of Turing complete languages which humans are capable of why is P vs NP important?

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