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.

It turned out that some divergent integrals 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.

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 $\det (j+1)=\sqrt{2}e^{-\gamma/2}$

• Infinity. 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 $\det \infty=e^0=1$

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.