# A p-adic analogue for a formula of Riemann?

This might be naive question but I was wondering whether a p-adic analogue of the following (shockingly) beautiful formula $$\zeta(s)\Gamma(s) = \int_0^\infty \frac{t^{s-1}}{e^t-1} dt$$ (vaild for $\mathrm{Re}(s)> 1$ and with all the usual notations) of Riemann is (well) known.

So with a p-adic analogue I'm of course aiming at an integral expression for the product of the classical p-adic zeta function $\zeta_p$ that interpolates $\zeta(s)$ in its values at the negative even integers and $\Gamma_p$ might be taken to be Morita's p-adic Gamma function. (Note that up to a power of $t$ the right hand side of the above formula can be said (in a catchy way) to be the Mellin transform of the generating function of the Bernoulli numbers (which essentially give the values of $\zeta(s)$ at the negative even integers...)).

What is "of course" lurking/hidden behind the above formula is the functional equation of the (completed) Riemann zeta function (for the time being I haven't learned/understood yet how this works in the p-adic setting, but I heard that in the work of Perrin-Riou, e.g., questions concerning functional equations of zeta functions in the p-adic world are dealt with (at least conjecturally)).

So, in a broader sense my question aims at understanding how the functional equation works in the p-adic setting and whether there are "nice functions" behind, that implement the mechanism (in the complex, global setting this is related, e.g., to theta functions and Bernoulli numbers, and the above formula is one incarnation of the eternal beauty of this area of the (mathematical) universe).

Thank you very much in advance for any help!

EDIT: I'd like to make one aspect of my question more precise after having a glance at the nice Bourbaki talk of Colmez suggested by Olivier below.

For example, the p-adic zeta function $\zeta_p$ of Kubota and Leopoldt can be constructed by constructing a "complicated" measure $\mu_{\zeta_p}$ on $\mathbb Z _p^\times$ by (making use of the Coleman map) and showing then that $$\int _{\mathbb Z_p^\times} \chi(g)^k d\mu_{\zeta_p} = (1-p^{k-1}) \zeta(1-k),$$ for $k$ an even and positive integer. (For $k$ odd we get 0). In the same spirit it is also possible to approximate p-adically special values of the completed zeta function (suitably normalized) which then satisfy a corresponding functional equation. My (very vague) question here is (maybe it's not what one should ask) whether one can write a (completed) p-adic zeta function in terms of an integral over a nice space with a simple measure but more interesting functions. In the description above of $\zeta_p$ the measure is very complicated but the function one integrates is quite easy. For example, is there some sort of p-adic analogue of theta functions that would do the job (suitable interpreted perhaps)?

(I know that to $\mu_{\zeta_p}$ there corresponds a certain (formal) power series via the Mahler transformation but I don't see a nice interpretation of this "function" either at the moment...)

(My apologies if this is way too vague for you...)

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And in surmising correctly that finding a $p$-adic analogue of this formula will provide an understanding of the functional equation of $p$-adic $L$-functions, you have just got a glimpse of what are now called the Coleman map, the Block-Kato exponential map, explicit reciprocity laws, and from then on the $p$-adic Langlands program. The article Théorie d'Iwasawa des représentations de de Rham d'un corps local Annals of Math 148 or the survey Fonctions $L$ $p$-adiques Séminaire Bourbaki 851 will tell you (much much) more.