Power series using the exponential

Question

My question is about the local zeta function

https://en.wikipedia.org/wiki/Local_zeta-function

what is the meaning of $e^{R(T)}=exp(R(T))$ with $R(T)\in \mathbb Q[[T]]$?

Only that is my doubt

Thanks you all

Answer 1

This will be longer than any comment may be, but there won’t be much content in.

Surely $\exp\bigl(R(T)\bigr)$ must mean just what you want it to, namely $$ 1+\sum_{i\ge1}\frac{R^n}{n!}\,. $$ The only question is whether or in what sense this is well-defined, namely whether the implied sequence of power series is convergent. It surely can’t turn out to be a series over $\Bbb Q$, but it should be a real series.

I see that you said nothing about any domain convergence of $R$, and for that reason, I will only show you what formal series over $\Bbb R$ the notation should be describing.

Well, since the formal series for $\exp(x)$ still satisfies the relation $\exp(x+y)=\exp(x)\exp(y)$, we may make the substitution $x\mapsto a\in\Bbb R$ and find $\exp(a+y)=e^a\exp(y)$. Call $R(x)=a_0+r(x)$, where $r(x)=\sum_{i\ge1}a_ix^i$, so that we get $$ \exp\bigl(R(x)\bigr)=e^{a_0}\exp\bigl(r(x)\bigr), $$ where the expression $\exp\bigl(r(x)\bigr)$ may be considered purely formally, and is a power series over $\Bbb Q$, while of course $e^{a_0}$ will only exceptionally be rational, but is a positive real.

And that’s my answer, but I suppose you could take the exponential of a sum implied by the notation $\exp(R)$ and convert it to an infinite product, which would also be formally convergent, again a positive real times a formal series with rational coefficients.