Question

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.

Answer 1

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.

It's 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!

Answer 2

An article in the Notices last April by Victor Moll gave 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.

Answer 3

This one is not too bad though not big .

http://aimath.org/pl/rhequivalences

Yet there are many (above a hundred at least) and it depends on the type you are looking for. Analytic elementary number theory ....

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)} <= N^{1/2+\epsilon}$.

Answer 4

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.

Answer 5

Li's criterion ?

Answer 6

See M. Balazard, Un siècle et demi de recherches sur l'hypothèse de Riemann.

Answer 7

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

Tuck's incompressibility function: statistics for zeta zeros and eigenvalues
M V Berry and P Shukla 2008 J. Phys. A: Math. Theor. 41 385202

Answer 8

A good resource is The Riemann Hypothesis by Borwein, Choi, Rooney, and Weirathmueller, CMS, 2008. It has equivalences plus much more.

Answer 9

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 $n \ge 1$.