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

[Olivier GERARD]

(Please do not reply to this message on seqfan with a full unedited copy of
the original, thanks; the administrator :-)

Dear Paul,

What you have found is very interesting (at least to me), and a
potential source of many constants and sequences but
what seems to happen is both a little easier and a little more complex than
what you describe.

Here (except mentionned) the exponent z is understood to be used
multiplied by I as in your mail. This case can be very different from
the general case.

The 'greedy' procedure you give has indeed a qualitative behavior which
can be very different for very close value of the base exponent.  This is 
not uncommon in "integer" packings.

A few remarks on your presentation
First it doesn't seem to converge always to zero even for a z < c.

Experiments and a few inequalities let me think that in the case
of z > 2, there are an infinite number of z for which the limit of the
absolute value of the process could be most reals between 0 and 1/2.

It converges (quickly !) to zero for an infinite number of
'minima' z > c whose first values are

2.859600867380127269
4.532360141827193936
6.043146855769591746
6.798540212740790714
...

The exact values are  Pi/Log[3], Pi/Log[2],  4/3 Pi/Log[2], 3/2  Pi/Log[2], ...
They simply correspond to the various solutions of finite equations  
    sum( (b_i)^(-I z) ) == 0.
For instance the third value corresponds to 1+2^(-I x)+4^(-I x) == 0.

Being superior to c, this part of your conjecture is false.

The process seems to stay put at many branching 'maxima' z
whose values corresponds to roots of equations in z
abs( sum( 1/a_i, i <= n) ) == abs( sum(1/a_i, i<= n+1))
(where the a_i are members of the fixed local greedy sequence
and of the form b^(- I z) )

One the first examples is the root of 
abs( 1+ 3^(- I x) + 6^(-I x) ) == abs( 1+ 3^(- I x))

i.e. 2.39283754700677674818481197643339619962527...

where the sum is   0.5071919630209197...

About your constant  c

There are many values (beside the one you give, here is a more precise version:
c = 2.0909337642920324973176998480675671084310008622992212956027...)
for which the process is instable by perturbation of z, that is, the number of terms inferior
to a given integer and sometimes the final limit is a discontinuous fonction of the exponent z.
Your c is a left accumulation point, but some points seem isolated .

What I can see is that c is close to be an even root of the equation between the two sums,
abs(1+ 3^(-I z) + 6^(-I z) + 16^(-I z)) = abs(1+ 3^(-I z) + 6^(- I z)+ 15^(-I z)+ 53^(-I z) + ...)
one being finite, the other infinite on the two sides of the threshold.
But having enough terms of the right side requires a lot of precision

================
A little generalization gives similar constants when you restrict or
enlarge the admissible factors to odd, even, half, third... integers.
For instance, when the process is run on integers and half integers,  there is a 
threshold at

c2 = 2.77042540878476795140709449134

and for the multiples of 1/3

c3 = 3.80253970046170080964010789562

Of course when z is not used as pure imaginary value, the organization of roots
and values differ, other phenomena occurs, and the most common behavior
of your process changes.

Olivier

PS: the ISC of our member Simon Plouffe
at http://pi.lacim.uqam.ca/  helped me a lot during the preparation of this post.


----- Original Message ----- 
From: Paul D Hanna 
To: seqfan at ext.jussieu.fr 
Sent: Tuesday, June 17, 2003 8:50 AM
Subject: [seqfan] Constant of Series of Integers with Imaginary Powers?


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?

[...]

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.

[...]

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. 

c = 2.09093376429203249731769984806...

[...]