[seqfan] Re: Inequalities for Euler's phi(n) and the Riemann hypothesis

[Vladimir Shevelev]

Dear Jean-Paul,

Many thanks for this very interesting article!
In particular, Theorem 2.1 yields that in A279229
all terms are even, except for 3,5,9 and 15.

Best regards,
Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of jean-paul allouche [jean-paul.allouche at imj-prg.fr]
Sent: 20 December 2016 17:55
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Inequalities for Euler's phi(n) and the Riemann hypothesis

Dear Vladimir, dear all

May be there is something to do in the same spirit
with Robin's criterion for the Riemann hypothesis
(for this criterion see, e.g.,
http://www.numdam.org/item?id=JTNB_2007__19_2_357_0
)

best
jean-paul

Le 20/12/16 à 12:58, Vladimir Shevelev a écrit :
> 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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