Comments on
the Riemann Hypothesis
- There exist series related to zeta(s) that
do not have their non-trivial zeros having real part 1/2; these functions do
not have an Euler product. Thus it is believed, though it need not be
so, that the Euler product will play a key role in a correct proof. How is the
Euler product used in your proof? Where is it used? See articles on the
Hurwitz zeta function and its zeros for more:
https://en.wikipedia.org/wiki/Hurwitz_zeta_function
- When you set conditions / create equations
that you want zeta(s) to satisfy to deduce facts about its zeros, see if other
functions satisfy all of these, in particular functions whose zeros are not
all on the critical line. If you can, that indicates a problem.
- Test your equations to see if they are
reasonable; if they are not that indicates an algebra error may have happened.
For example, if the right hand side is an absolute value and it is less than
something which can be negative, there is a mistake.
- Be careful interchanging limits, taking
limits, .... Respect the limit laws.
- Be careful not to use an analytic
continuation outside the realm where it holds.
- 100% is not the same as all; it is possible
for 100% of the zeros to be on the line and still have infinitely many off;
for example, in the limit 100% of integers are composite BUT there are still
infinitely many primes.
- Note in the explicit formulas the sum over
zeros is often only conditionally convergent, and thus you must be careful,
especially
if you try to interchange any operation with this sum.
Comments
on specific problems.
Main
page:
https://web.williams.edu/Mathematics/sjmiller/public_html/jntnewpolicy/