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## Here's a related theorem (primes between squares)

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• I wrote a paper covering some aspects of this problem a while ago. To this day, no one who looked at it or received it has said anything to me about it, but I
Message 1 of 3 , Jan 3, 2004
I wrote a paper covering some aspects of this problem a while ago.
To this day, no one who looked at it or received it has said anything
to me about it, but I certainly would appreciate that. The Less Than
Double Theorem states, in essence, that if you sum the number of
divisors of each of the first 2x integers, then double the result and
subtract x, you get about the sum of the number of divisors of the 2x
integers between x^2 and (x+1)^2.
The link is

http://www.imathination.net/D_Boland_V2.PDF

It establishes upper and lower bounds on the sum of the number of
divisors of the (2n+1) consecutive integers (n^2+1),...,(n+1)^2. The
theorem relates that summation to the same summation based over the
first 2n+1 integers {1,2,...,2n+1}. The curve of the actual values
approaches a constant between the two curves of the upper and lower
bounds. c=0.92... such that (actuals ~= lower bound + 2nc)

The constant is probably a natural result related to the Euler-
Mascheroni constant, it certainly needs to be fleshed out and
investigated further, but I hit a roadblock on converging the
constant. The technique in the theorem can be used to create other
divisor sum models with different spans or perhaps modular
restrictions as might be needed for application to other problems or
continued work on this problem. I haven't looked at this in awhile,
but I do recall thinking the bounds could likely be tightened quite a
bit more with some quadratic residue theory applied, we'll see.

Relating divisors to primes is the next big step, anyone got any
ideas or appropriate references?

-Dick
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