When researching divergent integrals, I decided to introduce a concept 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.
