Japanese version
In 23 Oct.2011 Mathematical Physics Seminar the second I gave a talk about "Zeta Functions and Statistical Dynamics," And now as part II "2,The origin of conjunction of Phys. and NT." is posted. The talk is related Mr. Konishi's talk at the first seminar.
The contents are the following :
0,Introduction
1,Distribution of prime numbers (Riemann zeta function)
2,The origin of conjunction of Phys. and NT.
3,Various zeta functions
4,Statistical Dynamics and Zeta Functions
App. A story of (Thermo) Statistical Dynamics
Zeta Functions and Statistical Dynamics II
On en.wikipedia about Farey sequences.
ReplyDeleteFarey sequences are used in two equivalent formulations of the Riemann hypothesis. Suppose the terms of $F_n$ are $\{a_{k,n}:k=0,1,\dots,m_n\}$. Define $d_{k,n}=a_{k,n}-k/{m_n}$, in other words $d\{k,n}$
is the difference between the $k$th term of the $n$th Farey sequence, and the $k$th member of a set of the same number of points, distributed evenly on the unit interval. Franel and Landau proved that the two statements that
\[ \sum^{m_n}_{k=1} |d_{k,n}|=\mathcal{O}(n^r) \]
for any $r > 1/2$, and that
\[ \sum^{m_n}_{k=1} d^2_{k,n}=\mathcal{O}(n^r) \]
for any $r > -1$, are equivalent to the Riemann hypothesis.