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# analytic number theory

Analytic number theory uses the machinery of analysis to tackle questions related to integers and transcendence. One of its most famous achievements is the proof of the prime number theorem.

One concept that is important in analytic number theory is asymptotic estimates. Tools that are used to obtain asymptotic estimates for sums include the Euler-Maclaurin summation formula, Abel’s lemma (summation by parts), the convolution method, and the Dirichlet hyperbola method. Asymptotic estimates are important for determining asymptotic densities of certain subsets of the natural numbers.

Another one of Dirichlet’s contributions to analytic number theory is the Dirichlet series. As an example, the Dirichlet series of a Dirichlet character is a Dirichlet L-series. A tool that is helpful for studying any Dirichlet series is the Euler product. The most famous Dirichlet series is the Riemann zeta function, which is the Dirichlet series of the completely multiplicative function $1$. This leads up to what is possibly the most important unsolved problem in analytic number theory: the Riemann hypothesis. This states that all nontrivial zeros of the Riemann zeta function have real part equal to $\frac{1}{2}$. Its connection to prime numbers is made clearer by the Euler product formula.

## Mathematics Subject Classification

11N37*no label found*11M06

*no label found*11N05

*no label found*11-01

*no label found*

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## Comments

## please edit me

The topic ``analytic number theory'' has been available for everyone to edit for over a month. I know that my knowledge of analytic number theory is very limited, and I am certain that there are people active in PM that know a lot more about the subject than I do. I would greatly appreciate it if more information could be added to this topic. Thanks.

Warren

## suggested addition to PM

At the end of world-editable entries (such as this one, analytic number theory), the following sentences appear:

Anyone with an account can edit this entry. Please help improve it!

The words "Anyone with an account" automatically link to new membership. (By the way, I like the witty response that it gives when a member clicks on it.) I suggest that the words "edit this entry" link to same place that clicking on "edit content" (which is at the very bottom of the page and thus is harder to find) does. Does anybody agree with me? Does anybody know how to go about making this change?

Warren