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This seems like a very standard notation in analytic number theory, and I see it a lot. But I was confused with it and I would greatly appreciate any clarification.

When one writes sum of the shape $$ \sum_{q \leq Q} \ \sum_{\chi (mod \ q)} ' $$ where $\sum_{\chi (mod \ q)} '$ is the sum over the primitive characters, I am wondering how the principal characters being taken into account here.

My questions regarding this: Is the convention to take the principal character as the primitive character mod 1 (so it appears only when $q=1$ but for no other $q$?), or are the principal characters mod q are primitive character mod $q$ so they actually appear for all $q$? or is the sum simply an empty set when $Q <2$?

Thank you very much.

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By definition, a Dirichlet character is primitive if it is not induced by a Dirichlet character of smaller modulus. In particular, the trivial Dirichlet character modulo $1$ (i.e. the constant function $\mathbb{Z}\to\{1\}$) is primitive.

The above definition is convenient in the sense that every Dirichlet character (including the principal Dirichlet characters) is induced from a unique primitive Dirichlet character. This fact generalizes to automorphic forms, e.g. every Hecke eigenform on the upper half-plane comes from a unique primitive Hecke eigenform, but only when the level $1$ forms are regarded primitive.

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