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For years umbral calculus have fascinated me. Bernoulli numbers (which represent powers of Bernoulli umbra) are involved in many classic power series expansions.

Yet, it still remains mistery what Bernoulli umbra represents and why it is so important. There are elementary explanations on what is imaginary unit or Dirac Delta distribution, which play similarly important roles, but what is the fundamental role of Bernoulli umbra? We do not even have a descent martrix or power series representation of it.

Awhile ago I found that the algebra of divergent integrals and series can be constructed in such a way to be isomorphic to umbral calculus. In this representation, Bernoulli umbra corresponded to divergent series: $B=\sum_{k=1}^\infty 1=-1/2+\int_0^\infty dx$ and $B+1=\sum_{k=0}^\infty 1=1/2+\int_0^\infty dx$...

The problem was that to have divergent integrals/series to be isomorphic to umbral calculus, one had to define multiplication of divergent integrals in a special, umbral way, which was far from being the simplest and the most natural one (the more natural and simple multiplication is more in line with Hardy fields or formal power series).

This makes me think that those divergent series still do not represent the natural objects corresponding to the Bernoulli umbra.

Let's now discuss fractals. Fractal dimension is typically defined in the following way: you scale the fractal with a factor $a$. If the former fractal (before scaling) fits the new one $b$ times, the fractal dimension is $\frac {\ln a}{\ln b}$. Using this definition, we can see that the integer lattice, the set of all integers, as well as uni-directional lattices (set of all natural numbers, etc) are dimension -1 fractals. This is because when we scale a lattice twice, it becomes less dense and the new one twice fits the old one: $\frac{\ln 2}{\ln 1/2}=-1$.

Moreover, the set of all integers represents the unit circle of dimension $-1$, that is corresponds to the range $[-1,1]$ in dimension 1. The set of positive integers thus corresponds to $(0,1]$, non-negative integers to $[0,1]$, etc.

It is logical to assume that there should be other analogies between dimension -1 and dimension 1.

Since Bernoulli umbra represented as divergent series is the numerocity of positive integers, that is the sum of the indicator function over all positive integers, and $B+1$ is the numerocity of non-negative integers, it is natural to conjecture that the numerocity of the interval $(0,1]$ or $[0,1)$ (which is the same) somehow can be represented by Bernoulli umbra as well (given certain multiplication rules). $B+1$ then would be numerocity of $[0,1]$, a closed unit interval.

I currently do not know how to represent numerocities of non-countable sets, but there are attempts to this direction. In the notation from the linked post, we would be able to write $B=\int_{[0,1)} \overline{\delta}(0)dx$ and $B+1=\int_{[0,1]} \overline{\delta}(0)dx$

So, if the numerocity of the unit interval is Bernoulli umbra, it is a much greater number than the numerocity of integers!

There is some indirect indication to that effect.

Now take the simplest infinite number, $\omega$, which corresponds to the germ at infinity of the function $f(x)=x$ or the germ at $0^+$ of the function $f(x)=1/x$, can be represented as divergent integral $\int_0^\infty dx$, divergent series $\frac12\sum_{-\infty}^\infty1$, and represents half of the numerocity of integers. What will happen if we calculate $(-1)^\omega$? Nothing special. The result is $1$. If we take the numerocity of all integers, $2\omega$, instead, the result will be the same.

But what about Bernoulli umbra? If we take $(-1)^{B+1/2}$, the resulting number is special. It is not a real number any more. Its absolute value is $1$, but its "finite part" is $\pi/2$. Increasing this power by reals makes the finite part do circles on the complex plane, but not with radius 1, but $\pi/2$.

So, my question is:

  • Can it be that Bernulli umbra is in fact (as its main role), the numerocity of unit interval?

  • What are some references linking Zeta function with fractal dimensions, including negative and fractional, as well as properties of unit circle in those dimensions?

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