In analytic number theory, a fundamental problem is to understand the distribution of the primes number . One important result is Dirichelt’s theorem, which states that for any two positive coprime integers and , the are infinitely many primes of the form , where is a non-negative integer.
Theorem. Let and be fixed an integer with and coprime (thus ), then there are infinitely many primes in the arithmetic progression .
Remark. If , then there is at most one prime meeting the condition $latex mod , since such would be divisible by the . Thus, the necessity of the gcd condition is obvious.
We first explain Dirichlet’s innovation, the use of group characters to isolate in a specified congruence class module .
Definitoin. Let be a finite abelian group, the dual group or group of character of a abelian group is
This is itself an abelian group under the operation on character defined for by
Definition. Let be a cyclic group of order with specified generator . Then is isomorphic to the group of -th roots of unity . That is, an -th root of unity gives the character such that
In particular, is a cyclic group of order .
Proposition. Let be a direct sum of finite abelian groups. Then there is a natural isomorphism of the dual groups
by , where is the character defined by
Remark. Combining this proposition and the structure theorem for finite abelian groups, we conclude that for a finite abelian group $G$.
Theorem 1. (Dual version of cancellation) For in a finite ableian group,
Proof. If , then the sum counts the characters in .
One the other hand, given in , by previous proposition let be in such that . The map on
is a bijection of to itself.
Proof of Dirichlet’s theorem
Definition. A Dirichlet character module is a group homomorphism
extended by to all of , that is by defining if is not invertible modulo . The trivial or principal character module is the character which takes only the value and .
Remark. This extension by zero allows us to compose with the reduction-mod- map and also consider as a function on . Even when extended by the function is still multiplicative in the sense that
where is the Euler’s totient function
Proof. The proof is easy, by changing variables.
Dirichlet’s dual trick is to sum over character module evaluated at fixed in . From Theorem 1 we see that
We also have for invertible module ,
Definition. (Dirichlet -function) Given a character module , the corresponding Dirichlet -function is
Since we have the multiplicative property , each -function has an Euler product expansion
The next step is to take logarithm of and equate it to the logarithm of the Euler product. This must be done with some care since complex logarithms can be insidious.
As seen from the Euler product, the -function does not vanish in the half-plane , so the half plane being simple connected, the logarithm is well defined (for instance, ).
However, when taking the logarithm of the Euler product, one can not immediately infer that one arrive at the sum of the logarithm ( is not necessarily equal to ; they might different by ). Summing the principal logarithm of the factor in the Euler product, and using the branch it holds that
for , we arrive at the following series
which converges for , uniformly on compact set, as it is dominated by . And since exponential function behaves better that the logarithm, i.e., it it continuous and always holds, this series is a logarithm of (but may not be the principal one); that is, one has
We still use to denote this branch of logarithm, then we have
The second sum on the right will turn to be subordinate to the first, so we aim our attention at the first sum where .
To pick out the primes with mod , we use the sum over trick to obtain