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Here's a related theorem (primes between squares)

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  • richard042@yahoo.com
    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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