Mangoldt summatory function MangoldtSummatoryFunction 2003-02-11 10:38:15 2005-04-14 19:32:33 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A number theoretic function used in the study of prime numbers; specifically it was used in the proof of the prime number theorem. It is defined thus: $$ \psi(x) = \sum_{r \leq x} \Lambda(r) $$ where $\Lambda(x)$ is the Mangoldt function. The Mangoldt Summatory Function is valid for all positive real x. Note that we do not have to worry that the inequality above is ambiguous, because $\Lambda(x)$ is only non-zero for natural $x$. So no matter whether we take it to mean r is real, integer or natural, the result is the same because we just get a lot of zeros added to our answer. The prime number theorem, which states: $$ \pi(x) \sim \frac{x}{\ln(x)} $$ where $\pi(x)$ is the prime counting function, is equivalent to the statement that: $$ \psi(x) \sim x $$ We can also define a ``smoothing function'' for the summatory function, defined as: $$ \psi_1(x) = \int_0^x \psi(t) dt $$ and then the prime number theorem is also equivalent to: $$ \psi_1(x) \sim \frac{1}{2} x^2 $$ which turns out to be easier to work with than the original form.