The prime number theorem can be expressed as the assertion
as , where
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
(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; for sake of completeness we prove it below the fold.
The connection between prime numbers and Banach algebras is given by the following consequence of the Selberg symmetry formula.
for all .
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 (in particular, the observation that contains a plentiful supply of local units:
for 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 :
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.
— 1. Proof of Selberg symmetry formula —
From the integral test we have the estimates
for some absolute constants whose exact value is unimportant for us, and for any . We conclude that
for some further absolute constants . Replacing by and inserting this into (9), one obtains
The error term can be computed to be . The main term simplifies by Möbius inversion to , and the claim follows.
— 2. Constructing the Banach algebra —
We now prove Theorem 1. It is convenient to transform the situation from the classical context of arithmetic functions on (such as or ) to the more Fourier-analytic context of Radon measures on the real line . Define the discrete Radon measure
and for any , let denote the left translate of the measure by , thus
for any continuous compactly supported . We note in passing that the prime number theorem (1) is equivalent to the assertion that the translates converge in the vague topology to Lebesgue measure as .
where is the convolution of the Radon measures , and is the measure multiplied by the identity function . From (4) one has
or equivalently that
which we rewrite as
Since for , we thus have
which implies that converges vaguely to , and the claim follows.
Now we begin the proof of Theorem 1. Observe that the quantity can be rewritten as
for some limit point of the translates in the vague topology. From (12) we have
for any , where the decay errors are allowed to depend on . Since converges vaguely to , we already have from (10) that
so it suffices to show that
Let be Lebesgue measure on the half-line . Then , so converges vaguely to . The measure is equal to times the function , so by Mertens’ theorem this function also converges vaguely to . We conclude that
converges vaguely to , and so it suffices to show that
We rewrite this as
(The error term in can be controlled by using (15) with replaced by , and modifying the preceding arguments to replace by .)
From Fubini’s theorem we have
The integrand vanishes unless . By (11), we have
and the claim (15) follows.
— 3. Non-trivial algebras with many local units have non-trivial spectrum —
We now prove Theorem 2. Let be the Banach algebra completion of under the seminorm (thus is the space of Cauchy sequences in , quotiented out by the sequences that go to zero in the seminorm ). Since is not identically zero, is a non-trivial commutative Banach algebra (but it is not necessarily unital).
It is convenient to adjoin a unit to to create a unital commutative Banach algebra with the extended norm
for and ; one easily verifies that is a unital commutative Banach algebra.
Suppose that all elements of have zero spectral radius (as defined in (6)). Let be a Schwartz function with compactly supported Fourier transform. Then we can find another Schwartz function with compactly supported Fourier transform such that (by ensuring that on the support of ; thus is a “local unit” on the Fourier support of ). Thus for all . But has spectral radius zero, thus is zero in . By density this implies that is trivial, a contradiction.
Thus there is an element of with positive spectral radius. Then by (6), there is a character that is does not vanish identically on . Suppose that for each there exists in the kernel of whose Fourier coefficient is non-vanishing. Since the kernel of is a space closed with respect to convolutions by functions, some Fourier analysis and a smooth partition of unity then shows that the kernel of contains any Schwartz function with compactly supported Fourier transform, and thus by density is trivial, a contradiction. Thus there must exist such that contains all test functions with Fourier coefficient vanishing at . From this we conclude that on is a constant multiple of the Fourier coefficient map ; being a non-trivial algebra homomorphism on , we thus have
for all . Since characters have norm at most (as can be seen for instance from (6)), we obtain the claim.
— 4. Breaking the parity barrier —
We now prove Theorem 3. We divide into two cases, depending on whether or . If , we let be a continuous function that equals on and is supported on for some large . From Mertens’ theorem we have
for sufficiently large depending on , and thus
The claim then follows by taking sufficiently large.
Now suppose . In the language of Section 2, we have
for some limit point of the . We can write the right-hand side as
for some phase . From (14), is a real measure between and , so by the triangle inequality we have
Now we set , where is as before. Then
Since is periodic with period and has mean value strictly less than (in fact, it has mean ), we thus have
if is sufficiently large depending on . The claim follows.
— 5. The prime number theorem in arithmetic progressions —
Let be a non-principal Dirichlet character of some period . We allow all implied constants in the notation to depend on . In this section we sketch the changes to the above arguments needed to establish
which gives the prime number in arithmetic progressions by the usual Fourier expansion into Dirichlet characters.
We have the twisted versions
from which we obtain the twisted Selberg symmetry formula
If we define the twisted measures
and hence converges weakly to zero as . Introducing the twisted norms
we may verify that obeys the conclusions of Theorem 1.
By repeating the previous arguments, it will suffice that the analogue of Theorem 3 for holds. When , we can argue as in Section 4, where the role of Mertens’ theorem is replaced by Dirichlet’s theorem
which is ultimately a consequence of the non-vanishing of .
For , the argument in Section 4 works with minimal changes if is real-valued. If is complex valued, it still takes only a finite number of values in the unit disk. Then the limit measures appearing in Section 4 are equal to Lebesgue measure times a density taking values in the convex hull of this finite set of values, which is a polygon in the unit disk. One can then modify the arguments in Section 4 to bound
for some phase . If we set as before, we again observe that the function is periodic and has mean strictly less than one, and so we can again establish the required bound if is large enough.
— 6. Proof of Gelfand formula —
We now prove (6).
If is a character, then it has an operator norm:
But we may eliminate this norm by using the “tensor power trick”: replacing with and then taking roots we conclude that
and then on sending we have
Replacing by again, taking roots, and sending we conclude that
(The limit exists because is submultiplicative.) This gives one direction of (6). To give the other direction, suppose for sake of contradiction that we could find an such that
for some real number .
There are two cases, depending on whether we can find a complex number with and non-invertible. First suppose that such a exists; then generates an ideal of , which by Zorn’s lemma is contained in a maximal ideal , whose quotient is then a field. By Neumann series, any element of sufficiently close to the identity is invertible and thus not in ; since is a field, we conclude that the complement of is open, and so is closed. This makes a Banach algebra as well as a field. If is not a multiple of the identity, then is invertible for every and so (by Neumann series) is an analytic function from to which goes to zero at infinity, contradicting Liouville’s theorem. Thus is one-dimensional (this is the Banach-Mazur theorem) and thus isomorphic to ; this gives a continuous unital algebra homomorphism with in the kernel, thus , contradicting the second inequality in (16).
Now suppose that is invertible for all . Then, as in the preceding argument, is an analytic function from to which decays to zero at infinity, so we have the Cauchy integral formula
for any natural number . From the triangle inequality we conclude in particular that
which contradicts the first inequality in (16).