[seqfan] Re: Constant C = ?

[M. F. Hasler]

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
>
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