Number-Divisors Almost = ln(m) + 2c-1

[Pfoertner, Hugo]

> -----Ursprüngliche Nachricht-----
> Von: Leroy Quet [mailto:qq-quet at mindspring.com]
> Gesendet am: 02 December, 2003 03:18
> An: seqfan at ext.jussieu.fr
> Betreff: Number-Divisors Almost = ln(m) + 2c-1
>
> In the sci.math thread (posted about a month back) "two 
> number-theoretical limits 
> (& bonus sum)":

See http://mathforum.org/discuss/sci.math/a/t/549557 
(replace the weird Google groups link)

> I figured (and Martin Cohen confirmed {referencing Hardy & Wright theorem 
> 320}) that d(m), the number of positive divisors of m, was such that
>
> limit{n-> infinity} (1/n) (sum{m=1 to n} d(m))   - ln(n)
>
> = 2*c - 1, where c is Euler's constant (.5772...). (=gamma) A001620

This also makes a new sequence:

n such that abs ( (sum_m (m=1..n) d(m)) / n - log(n) - 2*gamma + 1) is a
decreasing sequence.

 n  delta
 1  0.8455687
 2  0.652421519
 3  0.413623078
 5  0.236130788
 7  0.185372837
11  0.084037064
17  0.071178885
19  0.059024458
23  0.01442231
47  -0.004578902
89  -0.002618232

Neil: I erroneously wrote this as a comment for the other sequence
proposed by Leroy and extended by Hans Havermann submitted last night.
Please delete.

Hans: Could you again try to extend the sequence given above?

What about the ratio of primes / composites in this sequence?

Hugo Pfoertner