Dirichlet’s Theorem on the infinitude of primes in arithmetic progressions from 1837 is often viewed as the first result in analytic theory. To prove this result, Dirichlet shows that it suffices to prove the non-vanishing of (non-trivial) Dirichlet -functions
at the special point . Dirichlet’s theorem is without a doubt a beautiful result, but the reduction to is often introduced in a clumsy way. For example, the proof given in Apostol’s Introduction to Analytic Number Theory exposes the significance of while trying to prove the implication
(By this point in the proof, the first result has been proven and the second is known to imply Dirichlet’s theorem.)
In this post I’d like to give a different (and possibly new) proof of the reduction of Dirichlet’s theorem to the non-vanishing of which borrows some ideas from sieve theory. After this, I’ll show how the stronger hypothesis for can be used to prove an asymptotic for the size of the sieved set
— REDUCTION OF DIRICHLET’S THEOREM —
The method of sieving with Dirichlet series is an effective method of estimating sieved sets that exhibit some sort of multiplicative structure. Roughly, to estimate the size of a set , let denote the indicator function of and consider the Dirichlet series
Note that has an Euler product if and only if the indicator function is multiplicative.
In the case of the set from the introduction, we consider
This series is analytic in the right half-plane by comparison to the Riemann zeta function. To understand its analytic properties, we need a way to detect the condition . There’s really just one way to do this, which uses orthogonality of characters:
Lemma 1: We have
in which the sum runs over all characters mod .
in which is analytic and non-zero in . Of course, this implies that the analytic properties of in may be obtained from those of the Euler products
When is the trivial character mod , reduces to (up to some Euler factors which divide ). Thus has a pole at unless one of the terms corresponding to vanishes at .
Some remarks are in order:
1. If Dirichlet’s theorem is false, then is a finite product which extends analytically into the half-plane . Thus we prove Dirichlet’s theorem if we can show that has a pole at .
2. Except for the ugly addition of , the Euler products of the functions are exactly the Euler products of the Dirichlet -functions . Wouldn’t it be nice if we could relate the analytic properties of and ?
It turns out that relating and to each other isn’t actually that hard. The idea is simple (except for one technical result), so I’ll sketch the proof before filling in the last detail.
Lemma 2: The infinite product converges to a non-zero value if and only the series converges.
Assuming the lemma, note that converges to a non-zero value if and only if
converges. Of course, doesn’t affect convergence here, so we can factor it out, apply the Lemma in the reverse direction, and relate convergence of to !
There’s only one issue; namely, that the form of Lemma 2 you probably remember from calculus actually only holds when convergence is replaced with absolute convergence. And, alas, this is not satisfied at . This isn’t a fatal flaw, though — we just need a stronger form of Lemma 2:
Lemma 2.5: If , then converges to a non-zero value if and only if the series converges.
I first found this result in the AMM article Conditional Convergence of Infinite Products, by William Trench, but Trench notes that it appears in Knopp’s Theory and Applications of Infinite Series. In any case, we can use it to prove the following result:
Proposition: Suppose that . Then converges and is non-zero if and only the same holds for .
Corollary: If for all non-trivial , then has a pole of order 1 at . (This proves Dirichlet’s theorem by Remark 1.)
— ESTIMATING A CERTAIN SIEVED SET —
We can actually extract a fair bit more from these analytic preliminaries. In general, we expect to be able to study the asymptotic growth of
by studying the growth of the generating function about its dominant singularity. Our task is complicated by the fact that has a pole of fractional order at , but these concerns melt away with the right theorem. In this case, we apply a theorem due to Raikov (1954):
Theorem: Let be a Dirichlet series with non-negative coefficients. Suppose that converges in and that extends analytically to except at . Suppose further that
in which and is analytic and non-zero in . Then
The function satisfies the hypotheses of Raikov’s theorem provided that the Dirichlet series do not vanish in the region . But this follows from the previous Proposition and the well-known fact that on the line . We conclude that
for some which depends on and .
More generally, we can consider sets of the form
where is any subset of invertible residues mod . Since the Dirichlet series associated to factors as
we see at once that has a pole of order at . Raikov’s Tauberian theorem then implies that has
— EXERCISES —
Exercise 1: Let denote the set of integers which may be written as a sum of two squares. Prove that has elements .
Exercise 2: Use Euler’s classification of odd perfect numbers to show that the set of odd perfect numbers has density 0. (It’s actually known that the number of odd perfect numbers at most is for all .)
Exercise 3: Let be a finite set of primes and let denote the set of integers composed only of powers of elements of . Find the asymptotic size of .
Exercise 4: Let be any set of positive integers and let denote the set of integers which can be written as a product of th powers, as varies through . Find the asymptotic size of .
Exercise 5: We call an integer square-full if for every prime which divides . Find the asymptotic size of the set of square-full numbers. Along the way, show that the number of representations of as a product of a square-full number and a 6th power equals the number of representations of as a product of a square and a cube.