[seqfan] Re: Constant C = ?

[Tomasz Ordowski]

Yes, Maximilian, you may be right,
thank you for taking part in the discussion.

Thomas

P.S. I already have a much better idea for the next post,
so I asked Amiram Eldar to calculate a new constant.


czw., 23 mar 2023 o 01:28 M. F. Hasler <oeis at hasler.fr> napisał(a):

> Tomasz,
> I think your conjecture is wrong.
> From Pierre Dussard (arxiv:1002.0442) we know that
> pi(x) = (x / L)(1 + D)
> with L = log x,  D = 1 / L + d / L² ,  2 < d < 2.4 for x large enough.
> With this you get
> L / x - 1 / pi = (1/x) (1 + (d - 1 - d / L) /(L + 1 + d/L))
> = 1 / x + (d - 1 + o(1)) / (x log x)
> The first term gives you log N + O(1) when summed up to N,
> but using d > 2 for x large enough, the sum over the second term is
> divergent.
> (Primitive of 1/x log x is log log x.)
> - Maximilian
>
>
> On Tue, Mar 21, 2023 at 10:13 AM Tomasz Ordowski <tomaszordowski at gmail.com
> >
> wrote:
>
> > Hello everyone!
> >
> > Let's start with the conjecture
> >
> > Sum_{k=2..n} (log(k)/k - 1/pi(k)) = log(n) - C + o(1).
> >
> > Note that log(x)/x - 1/pi(x) ~ 1/x, since log(x) - x/pi(x) ~ 1.
> > Cf. https://en.wikipedia.org/wiki/Legendre%27s_constant
> >
> > Let's define (without proof of convergence) the constant
> > C = lim_{n->oo} (log(n) - Sum_{k=2..n} (log(k)/k - 1/pi(k))).
> > What is the approximate value of this limit?
> > Is C a new math constant?
> >
> > Best,
> >
> > Thomas
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>