[seqfan] A057641, A094348 and the Riemann Hypothesis
peter.luschny
peter.luschny at googlemail.com
Thu Nov 18 21:05:08 CET 2010
While considering to propose my favourite sequence A057641
for the sequence of the day I came up with the question:
/
| (*) Is it true that A057641(A094348(n)) >= 0 for n > 0
| is equivalent to the Riemann Hypothesis?
\
What is known is that
(**) Robin/Lagarias: A057641(n) >= 0 and n > 0
is equivalent to the Riemann Hypothesis.
By the way, A057641(A094348(n)) =
0, 0, 0, 0, 0, 1, 6, 9, 2, 14, 6, 23, 33, 34, 78, 114, 105,
227, 281, 207, 492, 1536, 1667, 3036, 5155, 6502, 5206, 7682,
15861, 15396, 9051, 21295, 22160, 36300, 58657, 71186, 81276,
91901, 165315, 154544, 199731, 143828, ...
I have not spend much time on (*), but it may be not difficult
to prove (*) => (**). In any case, I think there is a connection
between A094348 and the Riemann Hypothesis worth to be formulated carefully.
To give you the idea without words I plotted T. D. Noe's table of
A057641 which you can see here [1]: All that grey stuff is just
noise from the Lagarias formula (the quantum chaos? :), all what
counts are the red points of A094348 and the fact that they are
below the green line (and if I am right they are the only points
below that line). Maybe you like exploring this.
Cheers, Peter
[1] http://www.luschny.de/math/primes/RH-A057641-A094348.png
(File will be removed after some time.)
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