Author Archives: numbertheorist
Parseval’s identity and Liouville’s theorem
Unlike polynomials power series do not grow uniformly. For instance, if $n \geq 1$, \[P(z) = a_n z^n + \cdots + a_1 z + a_0, \] and $a_n \neq 0$, then $P(z) \sim a_n z^n$. In particular, $|P(z)| \to \infty$ … Continue reading
Evaluation of integrals using Cauchy’s residue theorem
Let $R$ be a rational function of two variables in $\mathbb{C}$, i.e., $R \in \mathbb{C}(x, y)$. Our goal is to evaluate integrals of the form\[\int_{0}^{2 \pi} R(\cos \theta, \sin \theta) \, d\theta.\] Let $z = e^{i \theta}$. Then\[\cos \theta = … Continue reading
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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Landau’s theorem on Dirichlet series
Let $\alpha(s) = \sum_{n = 1}^{\infty} a_n n^{-s}$ be a Dirichlet series with abscissa of convergence as $\sigma_c$. Then it is natural to think that $\alpha(s)$ must have some kind of singularity on the line $\sigma = \sigma_c$ which causes … Continue reading
Posted in Analytic Number Theory, Dirichlet series
Tagged abscissa of convergence, Dirichlet series
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Integral representation of Dirichlet series and Kronecker’s lemma
Let $\alpha(s) = \sum_{n =1}^{\infty} a_n n^{-s}$ be a Dirichlet series and let $A(x) = \sum_{n \leq x} a_n$. In this article we will establish a relationship between $\alpha(s)$ and $A(x)$. Theorem. Let $\alpha(s)$ and $A(x)$ be as above. If … Continue reading
Posted in Analytic Number Theory, Dirichlet series
Tagged Dirichlet series, Kronecker's lemma
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Absence of zeros of $\zeta(s)$ on the line $\sigma = 1$ under prime number theorem
In this article we show that prime number theorem implies nonvanishing of $\zeta(s)$ on the line $\sigma = 1$ and the argument we present here follows closely the approach taken in Ingham’s book The Distribution of Prime Numbers. The key … Continue reading
Bounds for the Riemann zeta function in the critical strip
In this article we obtain upper bounds for $\zeta(s)$ in the strip $\delta \leq \sigma \leq 2$, where $\delta > 0$. We will first show that for a fixed $0 < \varepsilon < \delta \leq 1$ we have \[\zeta(s) \ll … Continue reading
Posted in Analytic Number Theory, Riemann zeta function
Tagged bounds, Riemann zeta function
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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
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
A Möbius function formulation of prime number theorem
The prime number theorem states that\[\pi(x) \sim \frac{x}{\log x}.\] It is equivalent to $\psi(x) \sim x$ or $\theta(x) \sim x$. Let $M(x) = \sum_{n \leq x} \mu(n)$. In this article we will show that prime number theorem is also equivalent … Continue reading
Lifting of units and Menon’s identity
Let $d$ be a divisor of $n$. It is natural to ask the following question: Does a unit $a$ modulo $d$ lifts to a unit modulo $n$, i.e., if $a$ is a unit modulo $d$, then does there exist a … Continue reading
Posted in Euler's totient function
Tagged Euler's totient function, Menon's identity, units
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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
Chebyshev functions
The Chebyshev’s $\psi$-function and Chebshev’s $\theta$-function are defined as\[\psi(x) = \sum_{p^k \leq x} \log p, \qquad \theta(x) = \sum_{p \leq x} \log p.\] We can rewrite $\psi(x)$ in terms of von Mangoldt function as\[\psi(x) = \sum_{n \leq x} \Lambda(n).\] The … Continue reading
Dirichlet series
A Dirichlet series is a series of the form\[\alpha(s) = \sum_{n = 1}^{\infty} a_n n^{-s}.\] It is a general theme in analytic number theory to study a sequence ( arithmetic function) by means of its Dirichlet series. By studying analytic … Continue reading
Abel’s summation by parts formula
The Abel’s summation by parts formula is one of the most important and ubiquitous results in analytic number theory which is frequently employed to estimate the partial sums of an arithmetic functions weighted by some smooth function. Theorem. (Abel’s summation … Continue reading
Dirichlet product and multiplicative functions
The Dirichlet product (or Dirichlet convolution) of two arithmetic functions $f$ and $g$ is defined as\[(f * g)(n) = \sum_{d | n} f(d)g(n/d).\] The Dirichlet product arises when multiplying two Dirichlet series, that is, if two Dirichlet series\[\sum_{n = 1}^{\infty} … Continue reading
Euler’s totient function
The Euler’s totient function, denoted $\varphi$ (or $\phi$), is defined at $n$ to be the number of positive integers not exceeding $n$ that are relatively prime to $n$. We can rewrite $\varphi(n)$ in the summation notation as\[\varphi(n) = \sum_{\scriptstyle k … Continue reading
Möbius function
The Möbius function is one of the most important functions in number theory. It is defined as$$ \mu(n) = \begin{cases} 1 & \text{if } n = 1, \\(-1)^k & \text{if $n = p_1, \dots p_k$, where $p_i$ are distinct primes}, … Continue reading
Posted in Analytic Number Theory, Möbius function
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