A Banach algebra proof of the prime number theorem

25 Oct

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The prime number theorem can be expressed as the assertion

$latex displaystyle sum_{n leq x} Lambda(n) = x + o(x) (1)&fg=000000$

as $latex {x rightarrow infty}&fg=000000$, where

$latex displaystyle Lambda(n) := sum_{d|n} mu(d) log frac{n}{d}&fg=000000$

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

$latex displaystyle sum_{n leq x} Lambda_2(n) = 2 x log x + O(x) (2)&fg=000000$

where the second von Mangoldt function $latex {Lambda_2}&fg=000000$ is defined by the formula

$latex displaystyle Lambda_2(n) := sum_{d|n} mu(d) log^2 frac{n}{d} (3)&fg=000000$

or equivalently

$latex displaystyle Lambda_2(n) = Lambda(n) log n + sum_{d|n} Lambda(d) Lambda(frac{n}{d}). (4)&fg=000000$

(We are avoiding the use of the $latex {*}&fg=000000$ symbol here to denote Dirichlet convolution, as we will need this symbol to denote ordinary convolution shortly.) For the convenience of…

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