Original proof of Dirichlet’s theorem on primes in AP

I had nothing better to do during my spare time in the last 3 days, so I wasted my time reading this ancient math paper:

There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime

This paper was authored by German mathematician Johann Peter Gustav Lejeune Dirichlet and published in 1837. The paper’s title says it all: it contains the first proof of the celebrated Dirichlet’s theorem on primes in arithmetic progression:

If a and d are relatively prime integers, then there are infinitely many primes in the arithmetic progression a, a+d, a+2d, a+3d, a+4d, \dotsc

or equivalently,

If a and d are relatively prime integers, then there are infinitely many primes which are \equiv a (mod d).

This easy-to-state result has a deep proof (deep in the sense of Hardy, i.e., we have to appeal to real and complex analyses to understand the structure). It is probably the only staple of elementary number theory that does not have an entirely elementary proof.

173 years on, this result remains (in a certain sense) the best possible that we have in the field of prime-valued functions. Despite advances in other prime-related conjectures — the proofs of which remain out of reach in the present state of mathematics — it can be said that our complete understanding of prime numbers only goes so far as to their ‘linear’ structure. We do not even know for sure whether there are infinitely many primes in the form n^2 + 1, let alone the far-reaching generalizations of it (cf. Schinzel’s Hypothesis H).

Reading through the original proof gives me a mixed feeling. For one, reading original masterworks, like reading a facsimile of a literary masterpiece, invokes a sense of romance. The feeling comes with an appreciation of the state of mathematics in the early 19th century, allowing us the full appreciation of the brilliance of the proof. Dirichlet did not have access to representation theory and algebraic characters, although he was among the first mathematicians to use the L-series in a truly significant way.

On the other hand, the mathematical anachronism in the paper is mildly amusing. Dirichlet took half a page to explain absolute convergence. Some proofs in the paper can be telescoped into a fraction of their length in a modern textbook.

This is the third proof of this theorem that I have read (my favorite is the one in Cohn’s miniature text Advanced Number Theory, which runs to 5 pages vs Dirichlet’s 22). Although the tools differ, the ideas used are quite similar, relying on properties of the L-series and an accurate estimation of error terms.


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