For an discrete valuation field $K$ we can define the tame symbol:
$$(\,,\,)_K:K^\times\times K^\times\to \overline K^\times$$ $$(a,b)\mapsto(-1)^{v(a)v(b)}\overline{a^{v(b)}b^{-v(a)}}$$
Consider now a smooth projective curve $X$ over a finite field $k$ and let $K=K(X)$. We denote with $(\,,\,)_x$ the tame symbol for the complete discrete valuation field $K_x$.
For any element in the space of differential forms $K_xdt$ we have also the notion of residue. Suppose that
$$\omega:=dt\sum a_it_i\in k(x)((t))\,.$$ then we define $\operatorname{res}_x(\omega):=a_{-1}\in k(x)$.
What is the relation between $(\,,\,)_x$ and $\operatorname{res}_x$?
The both satify reciprocity laws: Weyl reciprocity law, and Tate reciprocity law. So I guess that we can express:
$$(a,b)_x=\operatorname{res}_x(f(a,b))\,,$$ where $f:K_x^\times\times K_x^\times\to K_x^\times dt$ is "a function". Is this true, can we calculate explicitly the tame symbol by using the residue? For example we know that for $a$ in the invertible part of the local ring of $K_x$:
$$(a,t)_x=\operatorname{res}_x\left(\frac{a}{t}dt\right)$$
Many thanks in advance
Usually the picture is the other way round.
You propose to express the tame symbol using a residue. This is probably not possible. Here is a suggestive argument: Over a finite field $\mathbf{F}_q$, the value of the tame symbol lies in the multiplicative group of a finite field extension of your base field, so in an abelian group of order $q^n - 1$ for some $n$. The order of this group is coprime to the characteristic $p$. But the residue takes values in the additive group, so an abelian group of exponent $p$. Thus, all maps from the additive group to the multiplicative group must be trivial. So, it is impossible that there would just be a group homomorphism mapping the value of some residue expression to the output of the tame symbol.
On the other hand, there are connections. You might wish to look up the Contou-Carrere symbol. This is like a "fancy" version of the tame symbol. It contains both the tame symbol and the residue as special cases. Also, there is a reciprocity law for the Contou-Carrere symbol on curves which simultaneously implies the residue theorem and the Weil reciprocity law. You can find this in
Anderson, Pablos Romo - Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an artinian local ring
Beilinson, Bloch, Esnault - Epsilon factors for Gauss-Manin determinants. In this paper the material is a bit hidden, https://arxiv.org/pdf/math/0111277.pdf Section 3.4
There is another connection. There is a proof of Gillet of Weil reciprocity using Algebraic $K$-theory. Replacing $K$-theory by Cyclic Homology, the same proof gives the residue theorem (see for example https://arxiv.org/pdf/2111.11580.pdf Appendix B). Both Algebraic $K$-theory as well as Cyclic Homology are examples of localizing invariants in the work of Blumberg-Gepner-Tabuada (https://arxiv.org/abs/1001.2282). Looking at the situation through the lens of their work, one could say that there is a single reciprocity law for all localizing invariants on curves (that is: applied to the categories of coherent sheaves with zero-dimensional support) and using $K$-theory as the invariant yields Weil reciprocity in $\pi_2$ and using Cyclic Homology as the invariant yields the residue theorem in $\pi_1$. The theorem that the sum of orders of poles and zeros of any non-zero rational function is zero sits in degree $\pi_1$ for $K$-theory.
Finally, all this can be interpreted in terms of geometric class field theory. There is a book about this by Serre (Algebraic Groups and Class Fields). The book is a bit outdated by now. Modern interpretations go in the direction of Geometric Langlands or Motives with Modulus. There is also the concept of a reciprocity sheaf (as in the work of Kahn, Rulling, etc)., which axiomatizes a sheaf having a reciprocity law similar to the ones you mention.