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<DIV>Dear Seqfans,</DIV>
<DIV> In extending sequence <A
href="http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A084816">A084816</A>,
I seem to have found a threshold to a series
of integers with imaginary powers where the sum equals
zero. </DIV>
<DIV> </DIV>
<DIV> Is this a new constant, or a limitation of my
program?</DIV>
<DIV> </DIV>
<DIV> Appended below is the DEFINITION,
EXAMPLES, CONJECTURE, and the behavior of the sum BEYOND THE
THRESHOLD. </DIV>
<DIV> </DIV>
<DIV> Any comments or counter-examples would be
very welcomed.</DIV>
<DIV>
<DIV> </DIV>
<DIV> Regards,</DIV>
<DIV> Paul</DIV></DIV>
<DIV><STRONG></STRONG> </DIV>
<DIV><STRONG>DEFINITION:</STRONG></DIV>
<DIV>Consider the sequence of positive integers {a(n)} that satisfy:</DIV>
<DIV>(*) sum(n=1,infinity, 1/a(n)^z ) = 0</DIV>
<DIV>where z is a given complex number. </DIV>
<DIV> </DIV>
<DIV>The sequence that satisfies (*) is generated by a greedy
algorithm that requires the moduli of the successive partial
sums to be monotonically decreasing in magnitude for the given
z.</DIV>
<DIV> </DIV>
<DIV><STRONG>EXAMPLES.</STRONG></DIV>
<DIV>Examples of such sequences are: <A
href="http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A084589">A084589</A>-<A
href="http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A084589">A084593</A> (z=non-trivial
zeros of Riemann zeta function), <A
href="http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A084813">A084809</A> (z=(1+I*sqrt(3))/2), and <A
href="http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A084816">A084816</A> (z=2*I).</DIV>
<DIV> </DIV>
<DIV>If we restrict z in (*) to be purely imaginary (z=y*I), then there
appears to be a new constant associated with this type of series.
</DIV>
<DIV> </DIV>
<DIV><STRONG>CONJECTURE</STRONG> (probably not
true)<STRONG>:</STRONG> </DIV>
<DIV>There exists a maximum real value to y beyond
which there does not exist a sequence of positive integers
{a(n)} that satisfy:</DIV>
<DIV> sum(n=1,infinity, 1/a(n)^(y*I) ) = 0;</DIV>
<DIV>
<DIV>if the sequence is to exist, then </DIV>
<DIV> y <= c where c
= <STRONG>2.09093376429203249731769984806</STRONG>...</DIV>
<DIV> </DIV>
<DIV>The sequence that satisfies (*) at this
threshold z=c*I begins:</DIV>
<DIV>
<STRONG>1,3,6,15,53,194,729,2753,10410,39381,148991,563688,2132651,8068666,...</STRONG></DIV>
<DIV> </DIV>
<DIV><STRONG>BEYOND THE THRESHOLD.</STRONG></DIV>
<DIV>Increasing the size of the y parameter by any sufficiently small
quantity results in the sequence </DIV>
<DIV> 1,3,6,16</DIV>
<DIV>at which point it appears that no further terms
after n=4 can satisfy (*).</DIV>
<DIV> </DIV>
<DIV>I find this too hard to believe!</DIV>
<DIV> </DIV>
<DIV>The partial sum of the series (*) where
y=2.0909337642920324973176998480<STRONG>7 (>c)</STRONG> at
the 4th partial sum is:</DIV>
<DIV> S(4) = 0.39754879551669854439 + 0.28801966236045306842 *
I.</DIV>
<DIV> </DIV>
<DIV>It appears that after this, either there does not exist an
<STRONG>n </STRONG>(!)<STRONG>,</STRONG> or <STRONG>n</STRONG> is very large
(most likely), such that</DIV>
<DIV> modulus( S(4) + 1/<STRONG>n</STRONG>^(y*I) ) <
modulus( S(4) ).</DIV>
<DIV> </DIV>
<DIV>Does there exist another positive integer > 16 such that the resulting
modulus of the 5th partial sum is less than the 4th partial sum (given
above)? </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>My rough PARI program has been running for several hours
(with high precision) and has not found such a 5th
term. Here is the PARI program:</DIV>
<DIV> </DIV>
<DIV>x=2.09093376429203249731769984807;</DIV>
<DIV>a=1;S=0.0;w=1.0;for(n=1,40,b=a; while(abs(S+exp(-x*I*log(b)))>w,b++);
</DIV>
<DIV>S=S+exp(-x*I*log(b));w=abs(S);a=b+1;print1(b,","))<BR></DIV>
<DIV>Have I encountered a limitation of my program? Or is there some
threshold constant here?</DIV></DIV></BODY></HTML>