Exactly as in the title. I am learning the $\mathbb{Z}_p^\times$ measure-theoretic construction of $p$-adic $L$-functions and was wondering if there was an `easy' way produce some example computations with these in the specific case of the interpolated $p$-adic Riemann zeta function.
Any code for this or suggestions for how one might algorithmically implement this would also be very welcomed!
Thank you in advance!
On
Interestingly, I just completed my dissertation on this exact topic and implemented the Kubota-Leopoldt $p$-adic zeta function into Sage. I'm not sure if this is the way you wanted to implement it, but I worked through Jacinto-Williams' fantastic notes on the topic - which I highly recommend (http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/Number-Theory---Full-Lecture-Notes-2017-18.pdf).
Note that this more measure-theoretic approach is quite different to Koblitz's which @Merosity cites in another answer.
My 'code' is not at all efficient (rather its purpose was more illustrative) but it did seem to agree with expected results provided the precision was kept sufficiently large.
I've included my construction for it in the pictures attached if you'd like to improve it!
Cheers :)
According to the p-Adics section of the reference manual there is nothing I could find to do with zeta functions nor integration.
At any rate, here's a suggestion on algorithmically getting an arbitrary level of accuracy on your integral. In order to be called a p-adic measure we require it to be a bounded p-adic distribution, and this is enough to imply that the limit of Riemann sums of a continuous function is independent of the choices of representatives. It can then be easily shown that we can approximate our integral by an integral of a locally constant function that approximates our function.
By continuity we know there exists a $M$ so that $|x-a|\le p^{-M}$ that gets us $|f(x)-f(a)|\le p^{-N}$, and so in particular the most convenient choice is for $a \in \{0,1,..., p^M-1\}$ define the locally constant function $g(x)=f(a)$ for $|x-a|\le p^{-M}$.
Specifically, let's suppose your measure is bounded $|\mu(U)|_p\le p^{-K}$ for all compact open $U \in \mathbb{Z}_p^\times$ then $$\left|\int f \mu -\int g \mu\right|\le p^{-K-M}$$ and we have the exact expression for a sum that approximates $\int f \mu$ which shouldn't be difficult to code,
$$\int g \mu = \sum_{\substack{0\le a <p^N \\ a+p^N \mathbb{Z}_p \subseteq \mathbb{Z}_p^\times}} g(a)\mu(a+p^N \mathbb{Z}_p) = \sum_{a_0=1}^{p-1} \sum_{a=0}^{p^{N-1}-1} f(a_0+ap) \mu(a_0+ap+p^N \mathbb{Z}_p) $$
I can elaborate on more details here, but I highly recommend you see chapter 2, sections 3 and 5 of Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions for more information and proofs on the theorems above.