Updates
- December 18, 2013: Jeff Lagarias made me aware of a paper
J. Milnor,in Enseign. Math. 1983. What I had called Birkhoff renormalization
has been known for a long time as Kubert relation Tf=f
Milnor proved for example that the solution space of T(f)=f is two dimensional.
I had once tried in vain to prove once that the cot function is the
only nonconstant solution to the Kubert relation
(1/n) sum_k=0^n-1 f((x+k)/n) = f(x). Milnor has shown this, and
much more.
- December 24, 2013: Isn't it curious that the root curve
of the circular zeta function touches the axes sigma=1 around the point
where the first root of the Riemann zeta function is? The first root
of the Riemann zeta function is 0.5 + 14.1347 i.
(Tweet). Now, if one looks at
the local maxima of the interpolation of the circular root line,
they appear to be close to the place where the roots of the Riemann
zeta function are. Interesting. While it is reasonable to expect such
a thing if the eta function were entire, this is far from obvious.
But it suggests that it might be possible to probe
what is behind the abscissa of
convergence using the circle zeta functions (of course with hard
analysis and not only with the elementary calculus tools as done so far).
- December 24, 2013: for theorem 11 on page 20, one should
add that the K-deriative of g'' and g'''' etc are all bounded. We have to
iterate the Rolle estimate.