Sums-Of-Powers over Minimum Divisors

[Leroy Quet]

{Posted to sci.math}


Let s(r,m) =

---
\      r
 >    k
/
---
k|m
1<= k <= sqrt(m)

(which is, in linear-mode)

sum{k|m,1<= k<= sqrt(m)}   k^r.

So, we have s(r,m) is 
the sum of the r-powers taken over the lower half of the positive 
divisors of m.

For example, s(1,m) is:
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A
num=A066839
 

 If r is > 0 (r = any *positive* real), then:


limit{m -> oo} 
           m
          ---
  1       \
-------    >   s(2r,k)       =
m^(r+1)   /
          ---
          k=1


    1
----------    (?)
2 r (r+1)


Linear-mode:

limit{m->oo} 
      (1/m^(r+1)) sum{k=1 to m} s(2r,k)  = 

1/(2 r (r+1))     (?)



(I am err-prone today, so I hope I thwarted fate...) 

Example: If I am right, the sum of the first m terms of the EIS's A066839 
divided by m^(3/2) approaches 2/3.



thanks,
Leroy Quet