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The recent post("Long-standing conjectures in analysis ... often turn out to be false") prompted me to think about a question which I have not given much though before: to what extent the Riemann hypothesis (RH) may be regarded as a problem in analysis. It may actually be not as silly as it sounds.

The particular side of it I am curious about is the following. The

Theorem : $\zeta(s)\neq 0$ for $\Re s>1$.

may be considered a very weak consequence of RH. However, this statement is only trivial in the context of number theory. One may ask, is it possible to give it a "purely analytic" proof, without using Euler product and other stuff related to the primes? Apparently, it is not possible to formulate this as a precise mathematical question because all the theories used to formalize analysis contain a good deal of arithmetic, but it should be precise enough for practical purposes. (You know number theory when you see it.)

Most likely such a proof does not exist yet, but some readers may know more about the relevant things then I do. I am honestly curious because while it definitely looks tough, it may be not entirely implausible. If such a proof is found, it might give us a fresh look on the old problem.

Alex Gavrilov
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    Since $\zeta(s)$ is a Dirichlet series for ${\text Re}(s) > 1$ with constant term $1$, we have $\zeta(s)\rightarrow 1$ as the real part of $s$ tends to infinity. Thus it is "trivial" that $\zeta(s) \not= 0$ when the real part of $s$ is sufficiently large. Bringing this fact all the way back to ${\text Re}(s) > 1$ needs some more input about the zeta-function. For example, the finite sums $\sum_{n=1}^{N} 1/n^s$ have zeros with real part greater than $1$ when $N$ is big enough (I think $N > 30$ or $40$ might be enough). – KConrad Aug 02 '18 at 10:23
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    I remember once attending a talk by Serre on the history of the Riemann Hypothesis, where he explained (IIRC) that RH was once considered mainly a problem in analysis rather than number theory, and that [some famous mathematician whose name I now can't remember] had been assigned, for his doctorate, by [some other famous mathematician] the problem of proving RH, as a problem in analysis. (Evidently, this did not go well. Sadly, I don't remember who were the people involved, and I may have misremembered the whole story anyway.) – Gro-Tsen Aug 02 '18 at 11:46
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    @Gro, it was Barnes who proposed that to the student, Littlewood. – Gerry Myerson Aug 02 '18 at 12:27
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    a dissenting opinion, by J. Brian Conrey: It is my belief that RH is a genuinely arithmetic question that likely will not succumb to methods of analysis. There is a growing body of evidence indicating that one needs to consider families of L-functions in order to make progress on this difficult question. If so, then number theorists are on the right track to an eventual proof of RH, but we are still lacking many of the tools.

    http://www.ams.org/notices/200303/fea-conrey-web.pdf

     – Carlo Beenakker Aug 02 '18 at 12:36
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    It should be noted that the assignment of RH directly as a thesis problem was in the early 20th century, when its difficulty (particularly when British mathematics was somewhat isolated) was not fully appreciated. Similarly, in the 1930s Hasse assigned to a graduate student the meromorphic continuation and functional equation of the zeta-function of elliptic curves over $\mathbf Q$ as a thesis topic at a time when Weil was not convinced such a result was plausible. See Weil's comments in his collected works (Vol. 2, p. 529) on his paper about Jacobi sums as "Grossencharaktere". – KConrad Aug 02 '18 at 12:44
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    You may wish to look at this recent paper by Banks: https://arxiv.org/pdf/1709.08551.pdf – Lucia Aug 02 '18 at 21:03
  • Thank you, this proof of Banks is interesting. (Nobody would call it "purely analytic" though.) – Alex Gavrilov Aug 03 '18 at 06:02
  • @KConrad Re the trivial argument that "$\zeta(s)\ne0$ when the real part of $s$ is sufficiently large", in fact $\sigma\ge 7/4$ suffices. – Stopple Aug 09 '18 at 22:04
  • @AlexGavrilov: You may wish to look at this recent preprint (https://arxiv.org/abs/1706.08868). It concerns the zeros of the entire functions as Fourier transforms. The basic complex analysis method used there is based on the approach of Polya and Hurwitz. It is purely analytic because primes do not appear in the preprint. – mike Aug 30 '19 at 00:45

2 Answers2

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First, let me say that a proof just of $\zeta(s) \not= 0$ for ${\rm Re}(s) > 1$ that does not in any straightforward way also yield a larger nonvanishing half-plane should not be regarded as a "fresh look" at anything as deep as RH. In fact, since the nonvanishing of $\zeta(s)$ on the line ${\rm Re}(s) = 1$ is equivalent to the Prime Number Theorem, it is not realistic that an approach to nonvanishing on ${\rm Re}(s) > 1$ alone is going to get you anything more since you don't just stumble into a new proof of the Prime Number Theorem.

With that in mind, consider the following integral representation of $\zeta(s)$ for ${\rm Re}(s) > 1$, which follows from partial summation on the Dirichlet series for $\zeta(s)$ (it is in Serre's Course in Arithmetic, for example): $$ \zeta(s) = s\int_1^\infty \frac{[x]}{x^{s+1}}\,dx = \frac{s}{s-1} + s\int_1^\infty \frac{\{x\}}{x^{s+1}}\,dx. $$

If $\zeta(s) = 0$ at an $s$ with ${\rm Re}(s) > 1$, then by the above formula $$ \frac{1}{1-s} = \int_1^\infty \frac{\{x\}}{x^{s+1}}\,dx. $$ Taking absolute values of both sides and using $|\{x\}|<1$ and writing $s=\sigma + it$, $$ \frac{1}{|1-s|} < \frac{1}{\sigma}, $$ so $\sigma < |1-s|$. Squaring both sides, $\sigma^2 < (1-\sigma)^2 + t^2$, which is the same as $\sigma < (1+t^2)/2$. Combining this with the restriction $1 < \sigma$ puts all $s$ with ${\rm Re}(s) > 1$ and $\zeta(s) = 0$ in a restricted range in the half-plane ${\rm Re}(s) > 1$, but not into the empty set. Perhaps by not using something as simple as $|\{x\}| < 1$ in the estimate on the integral someone can restrict the range of possible $s$ even further.

KConrad
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  • I think we can prove the result for the whole half plane $\Re(s) > 1$ by noting that $\int_1^{\infty} \frac{{x }}{x^{s+1}} {\rm d} x < \frac{1}{2} \cdot \int_1^{\infty} \frac{1}{x^{s+1}} {\rm d} x$. To see this, write $\int_n^{n+1} \frac{{ x}}{x^{s+1}} {\rm d} x = \int_{0}^{1/2}\Big( \frac{u}{(n+u)^{s+1}} + \frac{1-u}{(n+1-u)^{s+1}} \Big) {\rm d} u \= \int_{0}^{1/2}\Big( \frac{1}{(n+1-u)^{s+1}} + u \cdot \Big( \frac{1}{(n+u)^{s+1}} - \frac{1}{(n+1-u)^{s+1}} \Big)\Big) {\rm d} u < \frac{1}{2}\int_{0}^{1/2}\Big(\frac{1}{(n+1-u)^{s+1}} + \frac{1}{(n+u)^{s+1}} \Big) {\rm d} u$. – Noah Stephens-Davidowitz Aug 02 '18 at 23:37
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    When $s$ is complex it does not make sense to have an inequality of such integrals. – KConrad Aug 03 '18 at 04:15
  • Oh, of course. Sorry for the stupidity. – Noah Stephens-Davidowitz Aug 03 '18 at 05:23
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    Replacing the bound $|{ x }|<1$ with $|{ x } - 1/2 | < 1/2$ pushes the zero free region out further, but still doesn't do it. Does anyone want to try some higher Bernoulli polynomials and see what happens? – David E Speyer Aug 08 '18 at 16:49
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Andrew Booker is of the opinion that it is the nonvanishing of zeta (or an L-function) to the right of the critical strip which is more fundamental than the Euler product. See Slide 10 of his recent talk https://heilbronn.ac.uk/wp-content/uploads/2018/07/Booker-talk.pdf

Terry Tao
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    I did enjoy the uses of xkcd 927 and xkcd 483, but (being a non-specialist, and hence rather lost towards the end) I half expected to see xkcd 349 ... – Yemon Choi Aug 08 '18 at 16:31
  • This $L$-datum business seems like an impossible way to define anything, even the Riemann zeta-function, since the third ingredient $m(z)$ in an $L$-datum can't be described in any direct way. – KConrad Aug 08 '18 at 23:01
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    For reference, the conference page is https://heilbronn.ac.uk/2017/08/08/perspectives-on-the-riemann-hypothesis/, where apparently videos will be uploaded at some point. – David Roberts Aug 09 '18 at 00:59
  • @KConrad and it looks as if $m$ can be an arbitrary function? Certainly it must be discontinuous. – David Roberts Aug 09 '18 at 01:00
  • This may be interesting (and other talks too). Waiting for videos. – Alex Gavrilov Aug 09 '18 at 10:00
  • I interpret this equivalence (for Dirichlet series with functional equation) between Euler product and non-vanishing for $\Re(s) > 1$ as a theorem about linear combinations of L-functions. Maybe interestingly, there is a Riemann hypothesis for those linear combinations, which is not about the zeros but the decay rate of $\sum_{n \ge 1} a_n n^{-s} 1_{\gcd(n,N)}$ as $N \to \infty$ – reuns Aug 28 '18 at 22:16
  • If anyone is still interested, the video is now available at https://vimeo.com/album/5303404/video/280769762 – Terry Tao Sep 18 '18 at 23:59