# [seqfan] Constant of Series of Integers with Imaginary Powers?

Paul D Hanna pauldhanna at juno.com
Tue Jun 17 08:50:22 CEST 2003

```Dear Seqfans,
In extending sequence A084816, I seem to have found a threshold to a
series of integers with imaginary powers where the sum equals zero.

Is this a new constant, or a limitation of my program?

Appended below is the DEFINITION, EXAMPLES, CONJECTURE, and the
behavior of the sum BEYOND THE THRESHOLD.

Any comments or counter-examples would be very welcomed.

Regards,
Paul

DEFINITION:
Consider the sequence of positive integers {a(n)} that satisfy:
(*)  sum(n=1,infinity, 1/a(n)^z ) = 0
where z is a given complex number.

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.

EXAMPLES.
Examples of such sequences are: A084589-A084593 (z=non-trivial zeros of
Riemann zeta function), A084809 (z=(1+I*sqrt(3))/2), and A084816 (z=2*I).

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.

CONJECTURE (probably not true):
There exists a maximum real value to y beyond which there does not exist
a sequence of positive integers {a(n)} that satisfy:
sum(n=1,infinity, 1/a(n)^(y*I) ) = 0;
if the sequence is to exist, then
y <= c where c = 2.09093376429203249731769984806...

The sequence that satisfies (*) at this threshold z=c*I begins:
1,3,6,15,53,194,729,2753,10410,39381,148991,563688,2132651,8068666,...

BEYOND THE THRESHOLD.
Increasing the size of the y parameter by any sufficiently small quantity
results in the sequence
1,3,6,16
at which point it appears that no further terms after n=4 can satisfy
(*).

I find this too hard to believe!

The partial sum of the series (*) where y=2.09093376429203249731769984807
(>c) at the 4th partial sum is:
S(4) = 0.39754879551669854439 +  0.28801966236045306842 * I.

It appears that after this, either there does not exist an n (!), or n is
very large (most likely), such that
modulus( S(4) + 1/n^(y*I) ) < modulus( S(4) ).

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)?

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:

x=2.09093376429203249731769984807;
a=1;S=0.0;w=1.0;for(n=1,40,b=a; while(abs(S+exp(-x*I*log(b)))>w,b++);
S=S+exp(-x*I*log(b));w=abs(S);a=b+1;print1(b,","))

Have I encountered a limitation of my program?  Or is there some
threshold constant here?
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