The Franel-Landau theorem and JI scales

This is a rather far-fetched connection, since it involves JI scales
as the number of notes to the octave goes to infinity. It's about
Farey sequences, but you can work it to make it apply to distibutions
which are logarithmically more or less even, rather than linearly.

If Fn is the nth Farey sequence, then this theorem involves the estimate

Sum |Fn(i)-i/M| = o(n^e)

Here Fn(i) is the ith element of the nth Farey sequence, M is the
number of elements in the nth Farey sequence, and e is any number
greater than 1/2. The little o notation G(x)=o(H(x)) means
G(x)/H(x)-->0 as x-->infinity. The Franel-Landau theorem is that the
truth of the above estimate is equivalent to the Riemann hypothesis,
which says all the nontrivial zeros of the zeta function are on the
critical line.

What it means is that as n goes to infinity, we have a fairly tight
bound on how irregularly relatively low-denominator fractions can be
distributed if the Riemann hypothesis is true. JI scales involving
low-denominator fractions can be smooth.

>This is a rather far-fetched connection, since it involves JI scales
>as the number of notes to the octave goes to infinity. It's about
>Farey sequences, but you can work it to make it apply to distibutions
>which are logarithmically more or less even, rather than linearly.
>
>If Fn is the nth Farey sequence, then this theorem involves the estimate
>
>Sum |Fn(i)-i/M| = o(n^e)
>
>Here Fn(i) is the ith element of the nth Farey sequence, M is the
>number of elements in the nth Farey sequence, and e is any number
>greater than 1/2. The little o notation G(x)=o(H(x)) means
>G(x)/H(x)-->0 as x-->infinity. The Franel-Landau theorem is that the
>truth of the above estimate is equivalent to the Riemann hypothesis,
>which says all the nontrivial zeros of the zeta function are on the
>critical line.
>
>What it means is that as n goes to infinity, we have a fairly tight
>bound on how irregularly relatively low-denominator fractions can be
>distributed if the Riemann hypothesis is true. JI scales involving
>low-denominator fractions can be smooth.

Fascinating...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Fascinating...

If you liked that, you might like that

Sum |Fn(i)-i/M|^2 = o(n^e) with e>-1 is also equivalent to the Riemann
hypothesis.