Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 15 Oct 1999 08:39:37 +0300 (EET DST)
Hi all number theory afficionados!
I decided to look at the Riemann hypothesis from p-adic point of view
and noticed very remarkable coincidence (confession: I know
practically nothing about Riemann hypothesis and I had to check
the basic facts from web).
1. Rieman zeta as thermodynamical partitition function
Riemann Zeta has the expression
Zeta(s) = prod_p 1/(1-p^(-s))= prod_p Z_p(s),
Z_p(s) = 1/1-x_p, x_p ==p^(-s).
Product is over primes and each factor is partition function for
harmonic oscillator with frequency/temperature ratio
omega/T= log(p)/s. Infinite number of oscillators with frequencies
log(p) in thermal equilibrium if s is identified as temperature.
2. Phase preserving canonical identification maps real quantum TGD
to its p-adic counterpart
Phase preserving canonical identification is defined as follows.
i) Let
z = rho*exp(i*phi)
be a p-adic number and restrict phi to the set of angles
for which real and imaginary parts of exp(iphi) are *rational numbers*
(this means that phi corresponds to pythagorean triangle, orthogonal
triangle with integer valued sides).
ii) Define canonical identification mapping as a mapping
which maps rational phase factors numerically to themselves but
interpreted as p-adic numbers. Rationals are indeed "common" to both
p-adics and reals and p-adics and reals are different completions
of rationals.
iii) Map rho having pinary expansion
rho = sum_nx_np^n
to reals using canonical identification
rho = sum_nx_n p^n---> sum_n x_np^(-n).
This map is continuous and single valued when one selects
the pinary expansion of rho to have finite pinary digits when
this is possible (1=.99999.. tells that there are two possible
manners to select the pinary expansion for finite number of digits).
iv) Canonical identification thus maps only a subset of
complex plane to p-adics since only rational phases are mapped
to their p-adic counterparts. This is crucial for the canonical
identification. In fact, one must pose pinary cutoff in order to map
real structures to p-adic structure satisfying corresponding
defining equations. But this is not important for the argument.
The map is defined for *p=4 mod 3* only since only in this case
sqrt(-1) does not exist as "p-adically real" number.
3. Under what conditions on s real partition function Z_p(s) is
canonical image of corresponding p-adic partition function?
Consider partition function as function of argument x_p
Z_p(x_p) == 1/(1-x_p),
x_p== p^(-Re(s))* p^(iIm(s))
=p^(-Re(s)) exp(ilog(p)*Im(s))
== X_p U_p(Im(s)).
Question is: under what conditions on s real Z_p(x_p) can be
regarded as phase preserving canonical image of
p-adic partition function defined by mapping p-adic
counterpart of x_p to its real counterpart by phase
preserving canonical identification? Or briefly:
**When canonical identification commutes with the property of
being partition function?**
If square root allowing algebraic extension of p-adics is used
this requires that
i) *X_p is half integer power of p* and
ii) that *U_p is complex rational phase*.
5. Re(s)=1/2 is smallest value of Re(s) for which
canonical identification commutes with property of being
of partitition function
Consider the conditions for X_p, the modulos of x_p in 1/1-x_p.
a) Res(s)=n is special
X_p can be directly mapped by canonical identification
to its real counterpart if Re(s_1) is integer:
Re(s)=n. (p^n ---> p^(-n)) in canonical identification.)
This is not enough: one wants Res(s)= n/2.
b) For square root allowing extension Re(s)=1/2 is special
sqrt(p) exists in algebraic extension of p-adics
if one uses algebraic extension allowing square root of a "p-adically
real"
p-adic number, which is 4-dimensional (!) for p>2 and 8-dimensional(!)
for p=2. For instance, for p>2 and p mod 4=3 one has
Z= x+iy+sqrt(p)u+ isqrt(p)v.
What is essential is that sqrt(p) exists. This means
that one can map the exponent X_p to its p-adic counterpart for
Re(s_1)=n/2.
c) Even more: Re(s_1)=1/2 is the smallest value of Re(s) for which
p-adic counterpart of Z_p(x_p) exists!! This gives direct
connection with Riemann hypothesis since
***re(s_1)=1/2 is the line at which the nontrivial zeros of Riemann
function lie according to Riemann hypothesis!!***
6. Can one sharpen Riemann hypothesis?
An attractive guess is that the phases
U_p== exp(iIm(s)log(p))
exist as rational complex phases for all values of p mod 4 =3
when Im(s) corresponds to zero of Riemann Zeta.
This would mean that the log(p2):th power of any rational phases
U_p1 would be also rational
(U_p1(x_p1))^(log(p2/p1)) = U_p2(x_p2) rational
when UP_1(x_p1) rational.
It might be that *very simple number theoretic considerations exclude this
possibily*. I apologize my number theoretic dummyness.
*If not* one would have infinite number of conditions on each zero
of Riemann function and much sharper form of Riemann hypothesis:
***The zeros of Riemann Zeta lie on axis Re(z)= 1/2 and correspond
to values of Im(z) such that exp(ilog(p)Im(z)) is rational
complex number for all values of prime p mod 4=3!!***
Zeros would thus correspond to very very special Pythagorean triangles.
Probably this condition is however quite too strong.
Best,
MP
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