[seqfan] A004018, the barely lacunary series of the khe function

[Pierre Abbat]

Since December 2021 I've been studying the khe function, which I defined by 
these functional equations:

խ(z)+խ(z+πi)=2խ(2z)
խ(z)*խ(z+πi)=խ(2z+πi)²

and the conditions that it approaches 1 as Re(z)→-∞ and it is analytic and 
nonconstant on the left half-plane. Alternatively, one could define 
Խ(z)=խ(ln(z)), which is analytic inside the unit circle. I wrote code to draw 
the loops of this function and watched them grow from tiny circles to 
fantastical shapes to a bunch of circles all passing through the origin (the 
khe function never actually reaches the origin) in two sets perpendicular to 
each other.

Because of ambiguity in the square root, computation of խ(z) was prone to 
splicing of loops. In February 2023, I got it to compute khe correctly, using 
A099957.

The khe function has a singularity at every rational angle on the imaginary 
axis (i.e. every rational number times 2π). Of these, 1/3 are real poles of 
degree 1, 1/3 are imaginary poles of degree 1, and 1/3 are essential 
singularities resembling exp(-1/x). If an analytic function has singularities 
which are dense on a curve that is either closed or unbounded and has no ends, 
then it cannot be analytically extended past this barrier. It is, of course, 
possible to define a function on the right half-plane that satisfies the 
functional equations, but it is not possible to define it on both sides and 
make the singularities match.

I've seen functions which can't be continued past some barrier, such as 
Σx^(n!), but they are obviously contrived to show that such functions exist. 
Such a series, which has increasingly long gaps between nonzero terms, is 
called a lacunary series, and a function defined by a lacunary series which 
can't be extended past a barrier is a lacunary function.

So I computed the coefficients of the power series of e^z that generates the khe 
function, by taking the Fourier transform of a loop. I already knew that 
a[0]=1 by definition and had to compute a[1]=4 to calculate the function. The 
first few terms are 1,4,4,0,4,8,0,0, which I looked up in OEIS and found 
A004018 to be the best match.

Then I considered the series 1+4e^z+4e^2z+0e^3z+4e^4z+..., whether it satisfies 
the functional equations. The sum equation is 1+4e^z+4e^2z+0e^3z+... + 
1-4e^z+4e^2z-0e^3z+... = 2(1+4e^2z+4e^4z+0e^6z+...), which simplifies to 
A004018[n]=A004018[2n], which is clearly true. The product equation I coded 
up, and it appears to be true as well.

I noticed that some other sequences, such as A104794, mention the arithmetic-
geometric mean, so maybe I'm not the first to study the khe function. However, 
when I posted graphs of khe on MAA Connect, no one recognized them. Maybe 
someone else has studied the real khe function, but not the complex khe 
function.

The nth term of A004018 (where n>0) is zero iff n has a prime congruent to 3 
raised to an odd power. The fraction of natural numbers which have 3 raised to 
an even power is 3/4, the fraction which have 7 raised to an even power is 
7/8, and so on. So the fraction of terms that are nonzero is 
3/4×7/8×11/12×19/20×.... As Σ1/p diverges to infinity slowly, so this product 
series diverges to zero slowly. So the proportion of nonzero terms in A004018 
is zero, but just barely, which makes the series of the khe function barely 
lacunary.

The program is at https://github.com/phma/agm . Unlike Quadlods, it has no 
user interface; I make it do things by uncommenting function calls.

Pierre
-- 
Por H o por B, los campos magnéticos se difieren dentro de un imán.