- 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
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- Von Mangoldt function
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Tag Archives: Ramanujan’s sum
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