[seqfan] Re: "Euler-Mertens" constant

[Tomasz Ordowski]

Cf. A083343,
see my new comment:
https://oeis.org/history/view?seq=A083343&v=61

Note that
lim_{n->oo} (Sum_{k=1..n} 1/k - Sum_{p<=n} (q - p)/p) = gamma + C,
where q is the next prime after prime p.

Hence C = - gamma + Sum_{n=1..oo} a(n)/n,
where a(n) = 1 - (q - p) if n is prime p or a(n) = 1 otherwise.

Approximately, the constant C = 5/8 - gamma = 0.0477...

T. Ordowski

sob., 25 lut 2023 o 20:21 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Dear readers,
>
> I have something interesting:
>
> C = lim_{n->oo} (log p(n) - Sum_{k=1..n} g(k)/p(k)),
> where p(n) = prime(n) and g(k) = p(k+1) - p(k).
>
> Amiram Eldar calculated that C = 0.0477967...
>
> Since g(k)/p(k) = p(k+1)/p(k) - 1 and log p(n) - p(n)/n ~ 1, so
> C = lim_{n->oo} (n + p(n)/n + 1 - Sum_{k=1..n} p(k+1)/p(k)).
>
> Is this a new math constant?
>
> Best regards,
>
> Thomas Ordowski
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