Peter Gustav Lejeune Dirichlet

Dirichlet made deep contributions to mathematics, especially in number theory by his work on primes in arithmetic progressions, introducing Characters and L-functions, Class number formula, Units in number fields, Divisor problem, Diophantine approximations, reciprocity laws, special cases of Fermat’s last theorem. In analysis he introduced important ideas like Dirichlet Kernel (Fourier convergence), Dirichlet boundary conditions, principle in PDE.

Life and Work of Dirichlet

Proof of Fermat’s Last Theorem for $n=5$

Primes in a given arithmetic progression (1837).

Main ideas: Introduces the Dirichlet characters (He does it explicitly using primitive roots modulo $q$ and $q-1$ root of unity, not abstractly as characters of the group $(\mathbb Z/q)^{*}$ , which helps to detect primes in arithmetic progressions and access the sums $\displaystyle \sum_{p \equiv a \mod q} p^{-s}$ in terms of the logarithms of Dirichlet L-functions $L(s, \chi).$ That reduces the problem to showing non-vanishing of $L(1, \chi).$ for non-trivial characters. For non-real characters occur in pairs, it’s easy to rule out their vanishing, and we are left to show $L(1, \left( \frac{\cdot}{p}\right)) \neq 0$ . This is a fundamental problem in analytic theory; We have some ideas to prove the non-vanishing (but quantitatively the results are weak because of a possible “Siegel Zero” close to $s=1$ ). Dirichlet provides a couple of proofs for non-vanishing.