(of a non-trivial character) at 1 is nonzero. The proof of this statement requires some calculus and analytic number theory . The particular case 1=''a'' = 1 (i.e., concerning the primes that are congruent to 1 modulo some n) can be proven by analyzing the splitting behavior of primes in cyclotomic extensions, without making use of calculus .

Although the proof of Dirichlet's Theorem makes use of calculus and analytic number theory, some proofs