# A Banach algebra proof of the prime number theorem

25 Oct

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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