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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.