Dirichlet’s theorem a real variable approach

Although our proof is standard (essentially the same as presented by Serre [2], whose was the first account that I understood) we eschew complexvariable theory. Instead we use simple estimates from real-variable theory. Only one new difficulty is introduced: proving that L(1, χ) 6= 0 for nontrivial real-valued Dirichlet characters χ. We use an argument of Monsky [1] to prove this fact. Our arguments use complex valued functions of a real variable. A typical function will map an interval I ⊆ R to C. Such a function may be considered as a pair, consisting its real part and its imaginary part. Standard notions of real analysis transfer to complex-valued functions by considering the real part and imaginary part. For instance a complex-valued function is continuous if and only if both its real and imaginary parts are continuous. We shall use such notions without further comment. We use big-O notation from time to time. In all cases the reader may interpret this as follows: