# Blog posts

## Arithmetic Functions and Dirichlet Convolution

$\def\bb{\mathbb}$An arithmetic function is a sequence $f\colon\bb N\to\bb C$. Some of the important arithmetic functions we study in number theory are: … Read more
A very useful result used often in analytic number theory is the so-called partial summation formula. It is a technique used to change sums into integrals, so that we can then use the nice results of calculus to attack the integral more effectively. We know that \begin{equation*} \sum_{n=1}^N 1 = N, \qquad \sum_{n=1}^N n = \frac{N(N+1)}2, \end{equation*} and we have similar results for $\sum_{k\leqslant N} n^k$ where $k$ is a positive integer, but can we get a nice formula for sums like \begin{equation*} \sum_{n=1}^N \frac 1n \qquad \text{or} \qquad \sum_{n=1}^N \log n? \end{equation*} What about \begin{equation*} \sum_{n=1}^N \sqrt{n}? \end{equation*} This is what partial summation will help us achieve. … Read more