An improved Type I estimate

30 Jul

What's new

As in all previous posts in this series, we adopt the following asymptotic notation: $latex {x}&fg=000000$ is a parameter going off to infinity, and all quantities may depend on $latex {x}&fg=000000$ unless explicitly declared to be “fixed”. The asymptotic notation $latex {O(), o(), \ll}&fg=000000$ is then defined relative to this parameter. A quantity $latex {q}&fg=000000$ is said to be of polynomial size if one has $latex {q = O(x^{O(1)})}&fg=000000$, and bounded if $latex {q=O(1)}&fg=000000$. We also write $latex {X \lessapprox Y}&fg=000000$ for $latex {X \ll x^{o(1)} Y}&fg=000000$, and $latex {X \sim Y}&fg=000000$ for $latex {X \ll Y \ll X}&fg=000000$.

The purpose of this (rather technical) post is both to roll over the polymath8 research thread from this previous post, and also to record the details of the latest improvement to the Type I estimates (based on exploiting additional averaging and using Deligne’s proof of the Weil conjectures) which lead…

View original post 11,959 more words


Leave a Reply

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

You are commenting using your 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

%d bloggers like this: