A Banach algebra proof of the prime number theorem

What's new

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…

View original post 3,616 more words

Advertisements
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s