[seqfan] Inequalities for Euler's phi(n) and the Riemann hypothesis
Vladimir Shevelev
shevelev at bgu.ac.il
Tue Dec 20 12:58:44 CET 2016
Dear SeqFans,
Define P=e^gamma*loglog(n), where gamma is
Euler's constant.
In 1909, Landau proved that for each eps>0,
there exist infinitely many n for which phi(n)< n/P',
where in P' e^gamma is replaced by e^(gamma-eps).
In 1983 J.-L. Nicolas beautifully strengthened Landau's
result showing that there exist infinitely many n for
which phi(n)<n/P. In connection with this I submitted
A279229 which lists numbers n for which phi(n) < n/P.
It begins 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ...
Denote, further, c(n) = (n/phi(n) - P)*sqrt(log(n)).
In 2012 Nicolas proved that the Riemann
hypothesis is equivalent to the inequality: for n>=2,
c(n)<=c(N), where N is the product of the first 66
primes, such that c(N)=4.0628356921...
Let n be in A279229, such that c(n)>0. I submitted
sequence A279291: {floor(c(n))} beginning with
1, 1, 0, 2, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2,...
By Nicolas' result, assuming the R. H., we see that
a(n)<=4.
On the other hand, I conjecture that a(n)<=4 is an
absolute result which is independent of the validity
of the R. H. If this conjecture is true, then the
statement that the R. H. is false is equivalent to the
existence of n for which c(n) is in interval (c(N),5).
I was surprised that neither A279229 nor A279291
were not in OEIS.
All remarks are welcome.
Best regards,
Vladimir
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