# The Lumber Room

"Consign them to dust and damp by way of preserving them"

## Equivalent forms of the Riemann hypothesis

1. Let $H_n$ be the nth harmonic number, i.e. $H_n = 1 + \frac12 + \frac13 + \dots + \frac1n$. Then, the Riemann hypothesis is true if and only if

$\displaystyle \sum_{d | n}{d} \le H_n + \exp(H_n)\log(H_n)$

The left-hand side, which is the sum of the divisors of $n$, is also denoted $\sigma(n)$.
See Jeffrey Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis [PDF, arXiv.

2. Define the Redheffer matrix $A = A(n)$ to be the $n \times n$ 0-1 matrix with entries $A_{ij} = 1 \text{ if } j=1 \text{ or } i \text{ divides } j$ (and 0 otherwise). Then the Riemann hypothesis is true if and only if $\det(A) = O(n^{1/2+\epsilon})$ for all $\epsilon$.

For more, see Equivalences to the Riemann hypothesis (by J. Brian Conrey and David W. Farmer), and Consequences of the Riemann hypothesis (MathOverflow)

For fun: claimed [dis/]proofs.

Quote found via rjlipton:

The Riemann Hypothesis is the most basic connection between addition and multiplication that there is, so I think of it in the simplest terms as something really basic that we don’t understand about the link between addition and multiplication.
Brian Conrey

Written by S

Tue, 2010-07-06 at 22:03:54

Posted in mathematics

### 3 Responses

1. #1 is really beautiful. Wow! Something like e^x*log(x) is inherently interesting, [valleygirl] like you know, the product of taking something ‘higher’ with taking it ‘lower’. [/valleygirl] I wonder if there’s some name to a function g(x) = f'(x)*Integral(f(x),0,x).

Can #2 be derived from #1 by any simple way?

KVM

Sun, 2011-01-16 at 06:10:57

• Wait why did this post from 7th July 2010 appear in Reader now?

KVM

Sun, 2011-01-16 at 06:11:50

• Because it was ‘private’ and I just set it ‘public’… and the former probably because there’s no context or completeness or anything.

I actually don’t know how to prove either of these, or whether it’s possible to derive one from the other… just a pair of curiosities to me, for now at least!

S

Sun, 2011-01-16 at 07:14:38