98 votes
Accepted

Is the Riemann zeta function surjective?

The Riemann zeta function is surjective. First, $\zeta(1/z)$ is holomorphic in the punctured disk $0<|z|<1$. Looking at $z=(1/2+it)^{-1}$ with $t\to\infty$ reveals that $\zeta(1/z)$ has an ...
GH from MO's user avatar
  • 96.5k
79 votes

Is the Riemann zeta function surjective?

$\zeta$ function has only one pole at $z=1$. It also has order $1$. If $\zeta$ omits $c\in C$ then $g:=1/(\zeta-c)$ is entire with one simple zero at $1$. As it is of order $1$, it must be $g(z)=(z-1)...
Alexandre Eremenko's user avatar
73 votes

"Long-standing conjectures in analysis ... often turn out to be false"

I don't know about analysis in general, but I think it's definitely fair to say "often" in functional analysis. My feeling is that we have a solid, thorough, elegant body of theory which usually leads ...
49 votes
Accepted

Why is so much work done on numerical verification of the Riemann Hypothesis?

People are interested in computing the zeros of $\zeta(s)$ and related functions not only as numerical support for RH. Going beyond RH, there are conjectures about the vertical distribution of the ...
user1728's user avatar
  • 818
48 votes

"Long-standing conjectures in analysis ... often turn out to be false"

If RH is "analysis", then surely Littlewood's 1914 theorem that $\pi(x)$ (the prime counting function) and $\mathrm{li}(x)$ (the logarithmic integral) alternate in size infinitely often... despite all ...
45 votes
Accepted

Could the Riemann zeta function be a solution for a known differential equation?

When posed properly, a long-standing open problem, but in the form you ask: Robert A. Van Gorder, MR 3276353 Does the Riemann zeta function satisfy a differential equation?, J. Number Theory 147 (...
Igor Rivin's user avatar
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43 votes
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Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$?

Following the suggestion I made in a comment, the integral can be rewritten as the contour integral $$ I_{3,2} = \frac{1}{2\pi i} \oint \frac{\operatorname{tanh}^3 z}{z^2} \log(-z) \, dz , $$ where ...
Igor Khavkine's user avatar
43 votes
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Why did Euler consider the zeta function?

This history is described in Euler and the Zeta Function by Raymond Ayoub (1974). In his early twenties, around 1730, Euler considered the celebrated problem to calculate the sum $$\zeta(2)=\sum_{n=1}^...
Carlo Beenakker's user avatar
41 votes

Heuristic argument for the Riemann Hypothesis

The Riemann hypothesis is true, if primes are random in certain ways.
Pace Nielsen's user avatar
39 votes
Accepted

Why these surprising proportionalities of integrals involving odd zeta values?

For $n\geq 1$ and $m\geq 0$, an application of integration by parts ($u=\log^n(1-x)$, $dv=\log^m(x)\,dx/x$) followed by the substitution $x\mapsto 1-x$ shows that $$ \frac{I_{n,m}}{I_{m+1,n-1}}=\frac{...
Julian Rosen's user avatar
  • 8,911
37 votes

Motivated account of the prime number theorem and related topics

To a certain extent, I think that analytic number theory really is magical, and there's a limit to how natural and motivated it can be. Of the accounts I have seen, the one in Donald Newman's book ...
Timothy Chow's user avatar
  • 77.9k
32 votes

Heuristic argument for the Riemann Hypothesis

There are many theoretical results that support the (generalized) Riemann hypothesis: zero density estimates for the zeros of certain $L$-functions infinitely many zeros on the critical line of ...
GH from MO's user avatar
  • 96.5k
30 votes

"Long-standing conjectures in analysis ... often turn out to be false"

The Riemann hypothesis is a conjecture in both analysis and number theory. Someone who tries to undermine it necessarily has to ignore the latter part or to declare it irrelevant. I am not suggesting ...
28 votes
Accepted

Riemann's attempts to prove RH

The short answer is no. If anyone were aware of such a record, it would surely have been Carl Siegel, who undertook a careful study of Riemann’s unpublished notes. However, Siegel wrote: Approaches ...
Timothy Chow's user avatar
  • 77.9k
28 votes

$\psi(x)-x$ on average

In Theorem 1 of Brent, Richard P.; Platt, David J.; Trudgian, Timothy S., The mean square of the error term in the prime number theorem, ZBL07569752. it is shown that for sufficiently large $x$ one ...
Terry Tao's user avatar
  • 107k
27 votes

Why is so much work done on numerical verification of the Riemann Hypothesis?

I would add a few more comments to the very pertinent ones above: 1: We are lucky to have two things that work in our favor - an excellent representation of $\zeta$ on the critical line by a simple ...
Conrad's user avatar
  • 1,795
26 votes

Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

As Peter Humphries points out, the precise claim is that "RH + Simple Zeroes" is stronger than "RH". Of course, this is formally trivial. So what's really meant is that "RH + ...
Will Sawin's user avatar
  • 133k
25 votes

Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

Regarding your second question, Apéry's amazing formula $$\zeta(3) = {5\over 2} \sum_{n\ge1} {(-1)^{n-1} \over n^3 {2n \choose n}}$$ has inspired the search for analogous formulas for other zeta ...
Timothy Chow's user avatar
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24 votes
Accepted

Representations of $\zeta(3)$ as continued fractions involving cubic polynomials

See NOTE below. This MO inquiry is over 3 yrs old now. By the date the question about the $\zeta(3)$ CF with $k=8/7$ was made (Feb, 2019), it can be answered in the negative nowadays, since it was '(...
Jorge Zuniga's user avatar
  • 2,165
24 votes

Why is so much work done on numerical verification of the Riemann Hypothesis?

Part of the point is that such numerical checks can be demonstrations of the efficiency of this or that new algorithm. However, it is also the case that a finite check (that all the zeroes of $\zeta(s)...
Nell's user avatar
  • 535
22 votes

A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula

The paper entitled Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru contains a derivation of the first formula in the OP (Benoȋt Cloitre's formula), and ...
Carlo Beenakker's user avatar
21 votes

Could the Riemann zeta function be a solution for a known differential equation?

The fact that $\zeta$ satisfy no algebraic differential equation is due to its famous relation with the Gamma function which was proved by Hölder not to satisfy such an equation. Detailed answer can ...
Duchamp Gérard H. E.'s user avatar
21 votes

Could the Riemann zeta function be a solution for a known differential equation?

The Riemann zeta function is "hypertranscendental" in the sense shown HERE It is not the solution $y(x)$ of a differential equation of the form $$ F(x,y,y',y'',\dots,y^{(n)})=0 $$ where $F$ is a ...
Gerald Edgar's user avatar
  • 40.1k
21 votes
Accepted

Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

The proof of irrationality of $\displaystyle\sum_{n=0}^{+\infty}\frac1{F_n}$ (where $F_n$ is the $n$-th Fibonacci number) by RIchard André-Jeannin is an adaptation of the original Apery's proof of the ...
joaopa's user avatar
  • 3,627
20 votes
Accepted

values of $\zeta$ function are linearly independent?

People have been exerting steady effort to prove/disprove irrationality of the Riemann zeta values in your list. Of course, $\zeta(3)$ is known to be irrational due to Roger Apery. Such investigations,...
T. Amdeberhan's user avatar
20 votes

Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$?

Rewrite the integrand and apply Taylor expansion to $\frac1{(1+e^{-2x})^3}$ so that $$\frac{\tanh^3x}{x^2}=\sum_{j\geq0}(-1)^j\binom{j+2}2 \frac{(1-e^{-2x})^3}{x^2}\binom{j+2}2e^{-2jx}.$$ Integrate ...
T. Amdeberhan's user avatar
20 votes
Accepted

$P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(4s)}-\sqrt{\frac{2}{\zeta(8s)}-...}}}}$

This has been answered in the comments by Lucia. The identity $$P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(4s)}-\sqrt{\frac{2}{\zeta(8s)}-\cdots}}}}$$ is false. By ...
20 votes

Algebraic independence of shifts of the Riemann zeta function

Hmm, it was more difficult than I expected to leverage universality to establish the claim. But one can proceed by probabilistic reasoning instead, basically exploiting the phase transition in the ...
Terry Tao's user avatar
  • 107k
20 votes
Accepted

Algebraic independence of shifts of the Riemann zeta function

$\zeta(s - z)$ has an Euler product $\prod_p \frac{1}{1 - p^{z-s}}$, and so a monomial $\prod_i \zeta(s - z_i)$ (with the $z_i$ not necessarily distinct) has an Euler product $$\prod_i \zeta(s - z_i) =...
Qiaochu Yuan's user avatar
19 votes
Accepted

On an asymptotic formula of Keating and Snaith involving the Riemann zeta function

There is a connection! (Though see the edit below.) Keating and Snaith make their conjecture by modeling the distribution of $\zeta(s)$ by the distribution of the characteristic polynomial of a random ...
Brad Rodgers's user avatar
  • 2,121

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