- Analytic Number Theory
- Chebyshev functions
- Complex Analysis
- Dirichlet product
- Dirichlet series
- Divisor function
- Euler's totient function
- Little Picard theorem
- Möbius function
- Multiplicative functions
- Prime number theorem
- Ramanujan's sum
- Riemann zeta function
- Summation by parts
- Uncategorized
- Von Mangoldt function
abscissa of convergence Blaschke factors Borel-Carathéodory lemma bounds Cauchy's theorem characteristic function of divisors Chebyshev function Dirichlet series divisor function entire function Euler's totient function holomorphic function Jensen's inequality Kronecker's lemma Menon's identity Möbius function Picard theorem prime number theorem Ramanujan's sum Riemann zeta function summation by parts units zero-free region
Category Archives: Divisor function
Bounds for divisor and Euler’s totient function
The divisor function $d(n)$ counts the number of divisors of an integer $n$. It is a multiplicative function and so can be written as\[d(n) = \prod_{p^a || n} (a + 1).\] We will now show that $d(n) \ll_{\varepsilon} n^{\varepsilon}$ for … Continue reading
Problems about Ramanujan’s sum
Below we discuss some problem about Ramanujan’s sum. Problem 1. Let us denote $e(\alpha) = e^{2 \pi i \alpha}$. Show that\[\frac{1}{q}\sum_{a = 1}^{q} e \left( \frac{an}{q} \right) =\begin{cases}1 & \text{if $q \, | \, n$}, \\0 & \text{otherwise}.\end{cases}\] Solution. Note … Continue reading