Equivalent forms of the Riemann hypothesis
- Let
be the nth harmonic number, i.e.
. Then, the Riemann hypothesis is true if and only if
The left-hand side, which is the sum of the divisors of
, is also denoted
.
See Jeffrey Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis [PDF, arXiv. - Define the Redheffer matrix
to be the
0-1 matrix with entries
(and 0 otherwise). Then the Riemann hypothesis is true if and only if
for all
.
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
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#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