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<DIV> Instead of the sequences of
least *<STRONG>increasing</STRONG>* integers that satisfy (*), another
set of very interesting sequences would be the least positive integers
<STRONG>not used previously</STRONG> that satisfy (*). These would
form a <STRONG>permutation of the natural numbers</STRONG>, one for
each Riemann Zeta zero. Anyone like to isolate these?</DIV>
<DIV> </DIV>
<DIV> Of course, there are
probably many other complex numbers for which one could select
integers to satisfy (*), but the Zeta zeros are simply more interesting.</DIV>
<DIV> </DIV>
<DIV> Thanks Much,</DIV>
<DIV>
Paul</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>On Fri, 23 May 2003 02:31:18 -0400 Paul D Hanna <<A
href="mailto:pauldhanna@juno.com">pauldhanna@juno.com</A>> writes:</DIV>
<BLOCKQUOTE dir=ltr
style="PADDING-LEFT: 10px; MARGIN-LEFT: 10px; BORDER-LEFT: #000000 2px solid">
<DIV></DIV>
<DIV> I would like to describe a set
of sequences that could be generated from the non-trivial zeros of
the Riemann Zeta function.</DIV>
<DIV> </DIV>
<DIV> While it is obvious that the
following sum does not converge:</DIV>
<DIV> sum(n>=1,
1/n^(1/2 + i*y) )</DIV>
<DIV>where (1/2 + i*y) is a Riemann Zeta Zero, perhaps certain subsets of the
integers would allow this to converge to zero.</DIV>
<DIV> </DIV>
<DIV> Suppose we define such an
integer sequence {a(n)} such that</DIV>
<DIV> </DIV>
<DIV><STRONG>(*) </STRONG>
<STRONG>sum(n>=1, 1/a(n)^(1/2 + i*y) ) =
0 </STRONG> </DIV>
<DIV> </DIV>
<DIV>by requiring that the modulus of the partial sums be always decreasing in
magnitude, so that the sum approaches zero as a limit. The
n-th term a(n) is to be the least positive integer that
causes the n-th partial sum of <STRONG>(*)</STRONG> to be less in
magnitude than the (n-1)-th partial sum.</DIV>
<DIV> </DIV>
<DIV> Further, it would be
interesting to derive a table of related row
sequences satisfying <STRONG>(*)</STRONG> for the
same Zeta zero, such that they collectively would form a permutation
of the natural numbers. There would then be a separate table
defined for each Riemann Zeta zero. </DIV>
<DIV> </DIV>
<DIV> Perhaps someone would like to
derive some of these tables? I can generate the first row sequences, but
the subsequent rows are beyond my calculating capability at present.</DIV>
<DIV> </DIV>
<DIV> Here are a few sequences that
satisfy <STRONG>(*)</STRONG> for six different Zeta zeros, and would
form the first row of the above tables.</DIV>
<DIV> </DIV>
<DIV> Thanks,</DIV>
<DIV>
Paul</DIV>
<DIV>------------------------------------------------------</DIV>
<DIV>[snip]</DIV></BLOCKQUOTE></BODY></HTML>