Collection of equivalent forms of Riemann Hypothesis This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include statements that imply RH or its negation.
Here is what I am suggesting we do:

Construct a more or less complete list of sufficiently diverse known reformulations of the Riemann Hypothesis and of statements that would resolve the Riemann Hypothesis.

Since it is in bad taste to directly attack RH, let me provide some rationale for suggesting this:
1) The resolution of RH is most likely to require a new point of view or a powerful new approach. It would serve us to collect existing attempts/perspectives in a single place in order to reveal new perspectives.
2) Perhaps the resolution of RH will need ideas from many areas of mathematics. One hopes that the solution of this problem will exemplify the unity of mathematics, and so it is of interest to see very diverse statements of RH in one place. Even in the event where no solution is near after this effort, the resulting compilation would itself help illustrate the depth of RH.
3) It would take very little effort for an expert in a given area to post a favorite known reformulation of RH whose statement is in the language of his area. Therefore, with very little effort, we could have access to many different points of view. This would be a case of many hands making light work. (OK, I guess not such light work!) 
Anyhow, in case this indeed turns out to be an appropriate forum for such a collection, you should try to include proper references for any reformulation you include. 
 A: I like Lagarias "elementary" reformulation of Robin's theorem: that RH is true iff
$\sigma(n)\leq H_n+e^{H_n}\log(H_n)$
holds for every $n\geq 1$, where $\sigma(n)$ is the sum of divisors function and $H_n$ is the nth harmonic number.
Its major appeal is that anyone with rudimentary exposure to number theory can play with it. Having spent the better part of my youth fiddling with this reformulation really brought out the enormous difficulty of proving RH. In a way I think this reformulation is evil, because it looks tractable, but is ultimately useless and perhaps even harder to work with than other more complex reformulations. On the other hand I hope a future proof of RH will involve this reformulation because then I might have a chance of understanding the proof!
A: See M. Balazard, Un siècle et demi de recherches sur l'hypothèse de Riemann.
A: I can't seem to find it in any of the answers here, so:
In three papers, Roesler presents an $n\times n$ matrix $\mathbf A_n$ whose determinant is $n! \sum_k \frac{\mu(k)}{k}$, (from which the statement that is equivalent to the hypothesis is $\det \mathbf A_n=O(n! n^{\varepsilon-\frac12})$ for all positive $\varepsilon$). He then proceeds to study the eigenvalues of this matrix to glean insights into the hypothesis from this viewpoint.
Relatedly, this paper by Barrett and Jarvis deals with a matrix originally studied by Redheffer, whose determinant is the Mertens (summatory Möbius) function $\sum_k\mu(k)$. They also then study the eigenvalues of this matrix.
A: On the arXiv this morning, The Landau function and the Riemann Hypothesis by Marc Deleglise and Jean-Louis Nicolas:

The Landau function $g(n)$ is the maximal order of an element of the
  symmetric group of degree $n$; it is also the largest product of
  powers of primes whose sum is $\le n$. The main result of this article
  is that the property 
For all $n > 0$ , $\log g(n) < li^{-1} (n))$
(where $li^{-1}(n)$ denotes the inverse function of the logarithmic
  integral) is equivalent to the Riemann hypothesis.

A: Two other articles, FWIW:  
http://arxiv.org/pdf/0808.0640.pdf (mainly a criteria of Riesz and Baez-Duarte)
http://arxiv.org/pdf/1003.3392.pdf (introducing the "Acid zeta function")
A: Let $C_{c}^{r}(\mathbb{R}^{\ast})$ denote the set of all functions $f \colon (0,\infty) \to \mathbb{C}$ that are $r$ times differentiable and have compact support. For any $y \in (0,\infty)$ and any $f \in C_{c}^{2}(\mathbb{R}^{\ast})$, let $m_{y}(f) := \sum_{n \in \mathbb{N}} y\phi(n) f(y^{\frac{1}{2}}n)$ where $\phi$ is the Euler totient function; in addition, let $m_{0}(y) := \int_{0}^{\infty} \left(\frac{6}{\pi^{2}}\right)u f(u)\, du$.    
In the paper Discrete measures and the Riemann hypothesis (Kodai Math. J. 17 (1994), no. 3, 596–608.), Prof. Alberto Verjovsky proved the folllowing:
The Riemann Hypothesis is true if and only if $m_{y}(f) = m_{0}(f) + o(y^{\frac{3}{4}-\epsilon})$ as $y \to 0$ for every $f \in C_{c}^{r}(\mathbb{R}^{\ast})$ with $r \in [2,\infty]$ and every $\epsilon > 0$.
You can find the aforementioned paper here: http://projecteuclid.org/euclid.kmj/1138040054
A: Polya-Hurwitz program.
This may become more interesting in light of the recent progress in the Polya-Jensen program by Griffin, Ono, Rolen, Zagier.
We will first provide definitions of some functions involved.
The Riemann Xi-function $\Xi(z)$ is related to the Riemann zeta-function $\zeta(s)$ via ([A], [B]): $\Xi(z)=\xi(\tfrac{1}{2}+iz)$,
$\xi(s)=\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$.
Riemann $\Xi(z)$ can be expressed as a Fourier transform of a positive, fast decaying, and even kernel $\Phi(t)$ ([A], [B]):
\begin{equation}
\Xi(z)=2\int_0^{\infty}\Phi(t)\cos(zt)\mathrm{d}t,\tag{1}
\end{equation}
where
\begin{equation}
\Phi(t)=\sum_{k\geqslant 1}\phi_k(t)=\Phi(-t),\tag{2}\\
\end{equation}
\begin{equation}
\phi_k(t)=\left(4\pi^2 k^4 e^{9t/2}-6\pi k^2e^{5t/2}\right)\exp\left(-\pi k^2 e^{2t}\right)\tag{3}.
\end{equation}
The Polya aspect of this approach is the following:
Truncate the Kernel $\Phi(t)$ of (2) and/or the integration range in (1) such that the resulting Fourier transform leads to a family of entire functions which only have real roots. 
One such candidate is given in [C]:
\begin{equation}
\Phi_{\color{red}n}(t)=(1/2)\sum_{1\leqslant k\leqslant {\color{red}n}}\left(\phi_k(t)+\phi_k(-t)\right)=\Phi_{\color{red}n}(-t)\tag{4}
\end{equation}
\begin{equation}
\Xi_{\color{red}n}(z)=2\int_0^{(1/2)\log {\color{red}n}}\Phi_{\color{red}n}(t)\cos(zt)\mathrm{d}t=\Xi_{\color{red}n}(-z),\tag{5}
\end{equation}
We refer to [D] and [E] for a near complete review on the zeros of entire functions as Fourier transforms.
The Hurwitz aspect of this approach is the following:
Corollary of Hurwitz's theorem in complex analysis (applied to our case) [F]:
If $\Xi(z)$ and $\{\Xi_n(z)\}$ are analytic functions on a domain $S_{1/2}(z)=\{z: 0<Im(z)<1/2\}$, $\{\Xi_n(z)\}$ converges to $\Xi(z)$ uniformly on compact subsets of $S_{1/2}(z)$, and all but finitely many $\Xi_n(z)$ have no zeros in $S_{1/2}(z)$, then either $\Xi(z)$ is identically zero or $\Xi(z)$ has no zeros in $S_{1/2}(z)$.
The functional equation for $\zeta(s)$ becomes $\Xi(-z)=\Xi(z)$. The candidate of $\Xi_n(z)$ in (5) automatically satisfies this functional equation.  
Another benefit of Polya-Hurwitz approach is that the entrance barrier is relatively low (comparing to other approaches that usually require the advance knowledge of analytical number theory).  
To get started, one only needs to know Fourier transform, basic complex analysis, some knowledge of entire functions, polynomials etc.  So anyone who has math training with the college undergraduate math major may start to work on the Polya-Hurwitz approach and learn other necessary new math as he/she goes.
The most difficult part of Polya-Hurwitz approach seems to be the following:(for example,) proving that all the zeros of $\Xi_n(z)$ in (5) are real in $S_{1/2}(z)$.  
One may need to have several iterations: guess one form of the Kernel like $\Phi_n(1,t)$ and complete the integration to get explicit expression for $\Xi_n(1,z)$.  If all the zeros of $\Xi_n(1,z)$ are found not to be all real in $S_{1/2}(z)$, then move on to $\Phi_n(2,t)$ and $\Xi_n(2,z)$...
References:
[A] Titchmarsh,"The Theory of the Riemann Zeta-Function",
(1986).
[B] Edwards, "Riemann's Zeta Function",  (1974).
[C] Shi, "On the zeros of Riemann Xi-function", (2017) arXiv:1706.08868.
[D] Dimitrov and Rusev, “ZEROS OF ENTIRE FOURIER TRANSFORMS” (2001), 108 page review paper.
[E] Hallum, “ZEROS OF ENTIRE FUNCTIONS REPRESENTED BY FOURIER TRANSFORMS” (2014), Master thesis.
[F] Conway, "Functions of One Complex Variable",(1978)
A: First formulation, the Möbius function:
From Wikipedia:
The statement that the equation:
$$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$$
is valid for every $s$ with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis.
This in turn can be rewritten as:
$$\displaystyle \frac{1}{\lim\limits_{k\to \infty } \, \left(\sum\limits_{a=1}^{k} \frac{1}{a^s}+\frac{1}{(s-1) k^{s-1}}\right)} =$$
$$\displaystyle \lim_{k \rightarrow \infty} \left( \underbrace{1 - \sum_{2 \leq a \leq k} \frac{1}{a^{s}} + \underset{ab \leq k}{\sum_{a \geq 2} \sum_{b \geq 2}} \frac{1}{(ab)^{s}} - \underset{abc \leq k}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2}} \frac{1}{(abc)^{s}} + \underset{abcd \leq k}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2} \sum_{d \geq 2}} \frac{1}{(abcd)^{s}} - \cdots}_{\text{number of alternating sums} > \frac{\log(k)}{\log(2)}} \right)$$
where:
$\Re(s)>0:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\zeta(s)=\lim\limits_{k\to \infty } \, \left(\sum\limits_{n=1}^{k} \frac{1}{n^s}+\frac{1}{(s-1) k^{s-1}}\right)$

Second formulation, the von Mangoldt function:
Reuns answer:
The Riemann hypothesis is equivalent to that the following sum converges:
$$\displaystyle \sum_{n=2}^{\infty} \left( \underbrace{-\frac{1}{\sqrt{n} \log^{3+\epsilon}(n)}+\underset{a = n}{\sum_{a \geq 2}} \frac{\log(a)}{\sqrt{n}\log^{3+\epsilon}(n)} - \underset{ab = n}{\sum_{a \geq 2} \sum_{b \geq 2}} \frac{\log(a)}{\sqrt{n}\log^{3+\epsilon}(n)} + \underset{abc = n}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2}} \frac{\log(a)}{\sqrt{n}\log^{3+\epsilon}(n)} - \underset{abcd = n}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2} \sum_{d \geq 2}} \frac{\log(a)}{\sqrt{n}\log^{3+\epsilon}(n)} + \cdots}_{\text{number of alternating sums} > \frac{\log(n)}{\log(2)}} \right)$$
Mathematica: https://pastebin.com/gxAE6ZgY

Third formulation, Liouville Lambda function:
The Riemann hypothesis is equivalent to:
$$\lim_{n\to \infty } \, \frac{\sum\limits_{k=1}^n \lambda (k)}{n^{\frac{1}{2}+\epsilon}}=0$$
where $\lambda(k)$ is the Liouville Lambda function.
This in turn can be rewritten as:
$$\lim\limits_{n \rightarrow \infty}\frac{1}{n^{\frac{1}{2}+\epsilon}}\left(\underbrace{\underset {a^2 \leq n} {\sum_ {a\geq 1}} 1 - \underset {a^2 b \leq n} {\sum_ {a\geq 1}\sum_{b\geq 2}} 1 + \underset {a^2 bc \leq n} {\sum_ {a\geq 1}\sum_ {b\geq 2}\sum_ {c\geq 2}} 1 - \underset {a^2 bcd \leq n} {\sum_ {a\geq 1}\sum_ {b\geq 2}\sum_ {c\geq 2}\sum_ {d\geq 2}} 1 + \cdots}_{\text{number of alternating sums} > \frac{\log(n)}{\log(2)}}\right)=0$$
A: Matiyasevich has reformulated RH as a computer science problem : "a particular explicitly presented register machine with 29 registers and 130 instructions never halts", see this reference .
A: The following is given without source here:  
RH is equivalent to the assertion that for all $n\ge3$ $$| \log \operatorname{lcm}(1,2,\dots, n) - n | < \sqrt{n}\log^2(n)$$
where $\operatorname{lcm}$ denotes the least common multiple.  
More details about this function (also called second Chebyshev function) can be found in this Wikipedia entry (thank you Wojowu!).
A: This one is not too bad though not big:
Equivalences to the Riemann Hypothesis,
ed. J. Brian Conrey and David W. Farmer
Yet there are many (above a hundred at least)  and it depends on the type you are looking for. Analytic elementary number theory ....
There is the two volume work "Equivalences of the Riemann hypothesis: vol I Arithmetic Equivalences, vol II Analytic Equivalences" Cambridge, 2017, by Kevin Broughan. Information is linked to the site
https://web.archive.org/web/20120731034246/http://aimath.org/pl/rhequivalences . Volume III is currently being drafted and should contain some recent equivalences, such as those relating to the number of divisors function, the de Bruijn-Newman constant and Jensen polynomials.
ADDED LATER : My favorite is very elementary:
Among the square free integers below $N$:
Let $D(N)$ denote the absolute value of the difference between the number of those divisible by an even number of primes and the number of those divisible by an odd number of primes .
R.H. says that $D(N)$ comes close to the square root of $N$.
More precisely:  for any $\epsilon > 0 $ there is $N_0$ such that  any $N > N_0$ verifies $ {D(N)} \leq N^{1/2+\epsilon}$.
A: Robin's criterion has been written in various places in MO: define Gronwall's function
$$   G(n) = \frac{\sigma(n)}{n \log \log n}. $$ In 1984, Robin showed that RH is equivalent to
$$  G(n) < e^\gamma, \; \; \forall n \geq 5041.$$
Robin's adviser was Jean-Louis Nicolas. There is a new equivalence due to Nicolas, G. Caveney, and J. Sondow. Define a positive integer $N$ to be $GA1$ if $N$ is composite and $G(N) \geq G(N/p)$ for all primes $p |N.$ Let $N$ be called $GA2$ if $G(N) \geq G(aN)$ for all positive integers $a,$ where in this case we allow $N$ to be prime or composite. Then RH is equivalent to the assertion that the only number that is both $GA1$ and $GA2$ is 4. See arXiv  and arXiv 
I learned of this because Sondow wrote to me asking for a pdf of Robin 1984. And I wrote back. Which people ought to do.
A: Lapidus and Maier show that “One can hear the shape of
a fractal string of dimension $D \neq \frac12$” if
and only if the Riemann hypothesis is true. 
A: Found in this question
DMR 1974: http://books.google.ca/books?id=4lT3M6F745sC&pg=PA335
$$\eta(j)=p \text{ if } j=p^k, \; p \text{ is prime}$$
$$\eta(j)=1 \text{ otherwise}$$
$$ \delta(x)=\prod_{n < x }\prod_{ j \le n} \eta(j) $$
RH is equivalent to the assertion that
$$\left( \sum_{k \le \delta(n)}\frac1k - \frac{n^2}{2}\right)^2 < 36n^3$$
for $n \ge 1$.
A: Using Corollary 1 in Schoenfeld's 1976 paper "Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II", we see with a bit of numeric work that the Riemann Hypothesis is equivalent to the following inequality:
$$|\pi(x)-\mathrm{li}(x)|<\sqrt{x}\log x,\qquad x\geq 2.$$
A: Li's criterion ?
A: A good resource is The Riemann Hypothesis by Borwein, Choi, Rooney, and Weirathmueller, CMS, 2008. It has equivalences plus much more.
A: Two statements I find quite interesting involve Farey sequences, and, very roughly, they can be interpreted as saying that elements of Farey sequence are "not far" from being evenly distributed in the unit interval.
To be precise, suppose $n$-th Farey sequence is $0=a_{0,n}<a_{1,n}<\dots<a_{m_n,n}=1$ and let $d_{k,n}=a_{k,n}-\frac{k}{m_n}$. Then RH is equivalent to any of the two statements below:


*

*$\sum_{k=0}^{m_n} d_{k,n}^2=O(n^{-1+\varepsilon})$ for any $\varepsilon>0$ as $n\rightarrow\infty$.

*$\sum_{k=0}^{m_n} |d_{k,n}|=O(n^{1/2+\varepsilon})$ for any $\varepsilon>0$ as $n\rightarrow\infty$.

A: There are many matrix formulations of the RH. 
This one is my favorite, due to Jean-Paul Cardinal: A sequence of symmetric matrices related to the Mertens function. 
(There is also an English translation.)
One considers the set 
$$S_n = \left\{ s \in \mathbb{N}\ \left| \ \  s = \left\lfloor \frac{n}{k}\right\rfloor , \ k \in \mathbb{N} \right.\right\}$$ 
and the $|S_n| \times |S_n|$ matrix $A^{(n)}_{ \ \ ij} = \text{Mertens}\left(\left\lfloor \frac{n}{s_i s_j} \right\rfloor\right)$. 
Given that $\text{Mertens}(n) \le \|A^{(n)}\| = \max_{\|u\|=1} \|A^{(n)} u\|$ , the Riemann hypothesis would be implied by 
$$ \|A^{(n)}\| = \mathcal{O}(n^{1/2+\epsilon}) \tag{$\star$} .$$
What is amazing is that the sequence $\|A^{(n)}\|$ is one of the smoothest you'll ever see being related to the Riemann hypothesis, and the fact that $\text{Mertens}(n)$ only depends on the $|S_n|{\scriptstyle-}1 < 2\sqrt{n}$ previous values of $\text{Mertens}\left(s_i\right)$ which is perfectly encoded in that matrix sequence.
A: By Value Distribution Theory Related to Number Theory, Riemann's Hypothesis is true if and only if
$$\frac{1}{\pi}\int_0^{\infty} \log\left|\frac{\zeta(\frac{1}{2}+it)}{\zeta(\frac{1}{2})}\right|\ \frac{dt}{t^2}=\frac{\pi}{8}+\frac{\gamma}{4}+\frac{\log 8\pi}{4}-2$$
And a more general theorem has be proved in this book:
Take $a\in R$ with $\frac{1}{2}\leq a<1$. Riemann's $\zeta$-function has no zeros in $\Re(s)>a$ if and only if
$$\frac{1}{\pi}\int_0^{\infty} \log\left|\frac{\zeta(a+it)}{\zeta(a)}\right|\ \frac{dt}{t^2}=\frac{\zeta'(a)}{2\zeta(a)}-\frac{1}{1-a}$$
A: I met a guy today who I convinced to sign up on MathOverflow.  His name is Kevin Broughan and he has a couple of volumes on this subject, divided into arithmetic and analytic equivalents.  You might check it out.
Gerhard "Oh, The People You Meet...." Paseman, 2018.08.02.
A: In the article Seized opportunities (Notices of the AMS, April 2010), Victor Moll gives the following, which he credits to V.V.Volchkov. Establishing the exact value
$$\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{1/2}^{\infty}\log|\zeta(\sigma+it)|~d\sigma ~dt=\frac{\pi(3-\gamma)}{32}$$
is equivalent to the Riemann Hypothesis. Moll cheekily adds that evaluating that integral might be hard. 
A: Not especially sophisticated, but there is the note by Tuck: When does the first derivative exceed the geometric
mean of a function and its second derivative?. This is studied some more in

*

*M V Berry and P Shukla, Tuck's incompressibility function: statistics for zeta zeros and eigenvalues,
2008 J. Phys. A: Math. Theor. 41 385202, doi:10.1088/1751-8113/41/38/385202, arxiv:0807.3474.

