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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