T2. 10. Dirichlet L-Functions

Question 1

Define the Dirichlet L-function for character modq

Answer 1

L(s, χ) = SUM_1,∞ χ(n)/n^s

Question 2

Write the Dirichlet L-function in product form

Answer 2

L(s, χ) = PROD_p (1- χ(p)/p^s)^-1

Question 3

How does the Dirichlet L-function behave for χ=χ_0 and R(s) >0?

Answer 3

It is holomorphic and converges, however is not absolutely convergent, hence we cannot take the product rep as with χ= χ_0