When researching divergent integrals, I decided to [introduce a concept][1] of "modulus" or "determinant" of divergent integral (and series). Basically, it is the exponent of the real part of finite(scalar) part of the logarithm of the object in question.
$$
\det w=\exp(\Re \operatorname{reg }\ln w).
$$
In this post I will call it "hypermodulus". It has many properties of normal modulus or determinant.

**EDIT: It turned out that my work on divergent integrals was isomorphic to umbral calculus.**

It turned out that some divergent integrals with umbral multiplication and Bernoulli umbra have this hypermodulus expressed via exponential of Euler-Mascheroni constant:

$\det \int_0^\infty dx= \frac{e^{-\gamma}}4$

$\det \sum_{k=0}^\infty 1=e^{-\gamma}$

etc.

In umbral terms this looks like the hypermodulus of $B+1$ is $e^{-\gamma}$, where $B$ is Bernoulli umbra.

But recently I found that the concept is applicable not only to divergent integrals, but also to other objects, whose logarithms normally diverge.

Let's see.

* **Zero**. Since the integral $\int_0^1 \frac1x dx$ can be regularized to $\gamma$, we can ascribe $-\gamma$ as the regularized value of logarithm at zero. Thus, $\det 0=e^{-\gamma}$. Notice that hypermodulus here does not coincide with normal modulus of zero.

* **Zero divisors**. For instance, the hypermodulus of zero divisor $j+1$ is $\det (j+1)=2e^{-\gamma/2}$, while of $1/2-j/2$ is $e^{-\gamma/2}$.

* **Improper elements**. Since divergent integral $\int_1^\infty \frac1x dx$ regularizes to zero, thus, logarithm at infinity also regularizes to zero, and we can in some sense write $\det \infty=e^0=1$. Another improper element with hypermodulus is diected infinity on the split-complex plane $\infty+j\infty$, which has hypermodulus $e^{-\gamma/2}$. 

Notice that unlike for divergent integrals, hypermodulus applied to zero, infinity and zero divisors does not satisfy some expected properties, for instance $\det uw=\det u\cdot \det w$. For divergent integrals this holds, for zero divisors, does not.

Having said this, I wonder, what other mathematical objects may possess hypermodulus that differs from thair usual modulus and what meaning it may carry.

  [1]: https://mathoverflow.net/questions/392295/what-intuitive-meaning-determinant-of-a-divergency-divergent-integral-series