A multiplicative character weights the zeta series and creates an Euler product sensitive to progressions. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.
Statement
Dirichlet series \(\sum a_nn^{-s}\) turn multiplicativity into Euler products. The complex variable \(s=\sigma+it\) lets analytic continuation and zero-free regions control arithmetic sums.
The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).
Worked algebra
A worked instance is useful here because it exposes every index that the compressed statement hides.
Conceptual compression
What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.
Caveat
An Euler product converges absolutely only in a right half-plane. Formal rearrangement outside that region can destroy the argument.
The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.