- 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: Complex Analysis
Homotopy version of Cauchy’s theorem
Let $\gamma_0$ and $\gamma_1$ be piecewise smooth curves defined on the interval $[a, b]$ with the same end points, i.e., $\gamma_0(a) = \gamma_1(a)$ and $\gamma_0(b) = \gamma_1(b)$. If $U \subset \mathbb{C}$ is an open set, then $\gamma_0$ and $\gamma_1$ are … Continue reading
Jensen’s inequality and Borel-Carathéodory lemma
The Jensen’s inequality bounds the number of zeros of an analytic function in a small disc in terms of size of the function in a slightly larger disc. Although Jensen’s inequality is a simple consequence of Jensen’s formula but we … Continue reading
Posted in Complex Analysis
Tagged Blaschke factors, Borel-Carathéodory lemma, Jensen's inequality
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Little Picard theorem
In this article we prove the little Picard theorem assuming the existence of a nonconstant holomorphic function $\lambda : \mathbb{C} \backslash \{0, 1\} \to \mathbb{C}$ which satisifies $\text{Re} \lambda(z) \leq 0$. In the proof below we will repeatedly use the … Continue reading