The prime number theorem can be expressed as the assertion

is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. One “elementary” proof of this theorem proceeds through the Selberg symmetry formula

where the second von Mangoldt function is defined by the formula

(We are avoiding the use of the symbol here to denote Dirichlet convolution, as we will need this symbol to denote ordinary convolution shortly.) For the convenience of the reader, we give a proof of the Selberg symmetry formula below the fold. Actually, for the purposes of proving the prime number theorem, the weaker estimate

In this post I would like to record a somewhat “soft analysis” reformulation of the elementary proof of the prime number theorem in terms of Banach algebras, and specifically in Banach algebra structures on (completions of) the space of compactly supported continuous functions equipped with the convolution operation

This soft argument does not easily give any quantitative decay rate in the prime number theorem, but by the same token it avoids many of the quantitative calculations in the traditional proofs of this theorem. Ultimately, the key “soft analysis” fact used is the spectral radius formula

for any element of a unital commutative Banach algebra , where is the space of characters (i.e., continuous unital algebra homomorphisms from to ) of . This formula is due to Gelfand and may be found in any text on Banach algebras; we will assume it here as a black box.

The connection between prime numbers and Banach algebras is given by the following consequence of the Selberg symmetry formula.

Theorem 1 (Construction of a Banach algebra norm)For any , let denote the quantityThen is a seminorm on with the bound

for all . Furthermore, we have the Banach algebra bound

We prove this theorem below the fold. The prime number theorem then follows from Theorem 1 and the following two assertions. The first is an application of the spectral radius formula (6) and some basic Fourier analysis:

Theorem 2 (Non-trivial Banach algebras have non-trivial spectrum)Let be a seminorm on obeying (7), (8). Suppose that is not identically zero. Then there exists such thatfor all . In particular, by (7), one has

whenever is a non-negative function.

The second is a consequence of the Selberg symmetry formula and the fact that is real (as well as Mertens’ theorem, in the case), and is closely related to the non-vanishing of the Riemann zeta function on the line :

Theorem 3 (Breaking the parity barrier)Let and . Then there exists such that is non-negative, and

Assuming Theorems 1, 2, 3, we may now quickly establish the prime number theorem as follows. Theorem 2 and Theorem 3 imply that the seminorm constructed in Theorem 1 is trivial, and thus

as for any Schwartz function (the decay rate in may depend on ). Specialising to functions of the form for some smooth compactly supported on , we conclude that

as ; by the smooth Urysohn lemma this implies that

as for any fixed , and the prime number theorem then follows by a telescoping series argument.

The same argument also yields the prime number theorem in arithmetic progressions, or equivalently that

for any fixed Dirichlet character ; the one difference is that the use of Mertens’ theorem is replaced by the basic fact that the quantity is non-vanishing.

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