Articles tagged [real-analysis ]
Infinity, axiom of choice, and continuity
We compare two notions of infinity and relate it to the axiom of choice and the continuity of functions on the real line.
21 Jan 2025
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Maier's chain of large gaps between consecutive primes
We present Maier's matrix method that generates a chain of consecutive primes with large gaps.
12 Nov 2024
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On the number of lattices in a right triangle
We count the non-negative solutions ( m , n ) (m,n) ( m , n ) α m + β n ≤ x \alpha m+\beta n\le x α m + β n ≤ x
04 Jun 2024
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The distribution of cyclic numbers
We show that the number of cyclic numbers ≤ x \le x ≤ x e − γ x / log log log x e^{-\gamma}x/\log\log\log x e − γ x / log log log x
02 May 2024
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On n n n n n n
It is known that every group of prime order is cyclic. In this article, we present an arithmetical criterion on n n n n n n
17 Apr 2024
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Brun's sieve and its applications
We derive Brun's sieve and show that every large even integer is a sum of two integers, each having at most 9 prime factors.
25 Feb 2024
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On the factorization of polynomials
We derive Gauss's lemma and Eisenstein's criterion. Then we use them to prove that every cyclotomic polynomial is irreducible over Q [ x ] \mathbb Q[x] Q [ x ]
30 Jan 2024
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On the coefficients of cyclotomic polynomials
We prove that when n n n n n n Φ n ( x ) \Phi_n(x) Φ n ( x ) ± x k \pm x^k ± x k
22 Jan 2024
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On Legendre's differential equation
We prove that the eigenvalues of the Legendre differential operator are exactly ℓ ( ℓ + 1 ) \ell(\ell+1) ℓ ( ℓ + 1 ) ℓ ∈ Z \ell\in\mathbb Z ℓ ∈ Z
16 Jan 2024
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On the spectrum of Fredholm integral equations
We give an elementary argument showing that the spectrum of Fredholm integral equations is discrete.
23 Dec 2023
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On the growth order of derivatives
We prove theorems that produce asymptotic bounds for derivatives of functions and discuss some applications and generalizations.
16 Nov 2023
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Cycloids and the tautochrone problem
We study cycloid and show that it is the unique solution to the tautochrone problem.
03 Nov 2023
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On differentiation under integral signs
We derive a generalization of Leibniz integral rule to higher dimensions
19 Oct 2023
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Riemann mapping theorem and Dirichlet's principle
We give an account of Riemann's original proof of his famous mapping theorem and discuss its connections to Dirichlet's principle of the calculus of variations
10 Oct 2023
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When theta functions meet Dirichlet characters
We study theta functions twisted by Dirichlet characters and present their applications to Dirichlet L L L
08 Sep 2023
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Introduction to the transformations of elliptic functions
We give an elementary derivation of Jacobi's imaginary transformations and Landen's transformations and discuss their consequences in the theory of elliptic functions.
25 Jul 2023
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Asymptotic theory of integer partitions
We present two proofs of the Hardy-Ramanujan asymptotic formula of p ( n ) p(n) p ( n )
16 Jul 2023
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Applications of the axiom of choice
We discuss several important applications of the axiom of choice in analysis and algebra.
12 Jun 2023
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Well-ordered sets and the axiom of choice
We launch a systematic investigation into well-ordered sets.
27 May 2023
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Absolute values and Ostrowski's theorem
We investigate all possible versions of absolute values over Q \mathbb Q Q
11 May 2023
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On the discriminant of polynomials
We explore connections between polynomial discriminants and polynomial coefficients and apply them to study the elliptic discriminant.
09 May 2023
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Introduction to Weierstrass elliptic functions
We develop Weierstrass's theory of elliptic functions
09 Apr 2023
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Jacobian elliptic functions and sum of squares
We use Jacobian elliptic functions to study the representations of integers as sum of squares
09 Apr 2023
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The foundation of Jacobian theta functions
We derive definitions of theta functions from Jacobian elliptic functions.
07 Apr 2023
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Infinite products for Jacobian elliptic functions
We deduce infinite product representations for Jacobian elliptic functions.
05 Apr 2023
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A modern introduction to Jacobian elliptic functions
We introduce the classical Jacobian elliptic functions from a modern perspective.
02 Apr 2023
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Euler's pentagonal number theorem and Dedekind eta function
We give a direct, elementary proof of the functional equation for η ( τ ) \eta(\tau) η ( τ )
19 Mar 2023
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The distribution of prime divisors and Erdös-Kac theorem
We study the statistical behavior of ω ( n ) \omega(n) ω ( n ) n n n
05 Nov 2022
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Explicit formula for Gauss circle problem
We express the error term of the Gauss circle problem in terms of Bessel functions
26 Sep 2022
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Number of integers that are a sum of two squares
We investigate a question parallel to Gauss circle problem
23 Sep 2022
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Improvement on Gauss Circle Problem
We improve the error term in the asymptotic expansion of S ( x ) S(x) S ( x )
16 Sep 2022
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Representing Integers as Sum of Two Squares
Theory from Z [ i ] \mathbb Z[i] Z [ i ] Z \mathbb Z Z
30 Aug 2022
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Euclid's Algorithm and Unique Factorization of Gaussian Integers
We prove unique factorization theorem for Z [ i ] \mathbb Z[i] Z [ i ]
26 Aug 2022
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Analytic Proof of the Prime Number Theorem
Taking down ψ ( x ) \psi(x) ψ ( x )
25 Mar 2021
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The Prime Number Theorem and Its Equivalences
Demystifying the counting function π ( x ) \pi(x) π ( x )
24 Mar 2021
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Borel-Caratheodory Lemma and Its Application
Tactical estimation of logarithmic derivatives
16 Feb 2021
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Riemann-von Mangoldt formula for ζ ( s ) \zeta(s) ζ ( s )
A powerful tool for us to study nontrivial zeros of ζ ( s ) \zeta(s) ζ ( s )
19 Jan 2021
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Alternative Approach to Redefine ζ ( s ) \zeta(s) ζ ( s )
Functional equation but the hardcore approach
05 Jan 2021
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Bernoulli Numbers and Their Associated Polynomials
One of the most brilliant tools in asymptotic analysis!
26 Dec 2020
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Mertens' Formula
A brilliant association between primes and logarithms
24 Dec 2020
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A Complete Investigation on Fourier Transform
A comprehensive derivation of different forms of Fourier transform
23 Dec 2020
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Jensen's Formula and Its Implications
Does maximum modulus tells distribution of roots?
14 Dec 2020
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Stirling's Formula
An efficient way to calculate factorial?
10 Dec 2020
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Analytic Continuation of the Riemann Zeta Function
Finally, we can catch a glimpse of the million-dollar math puzzle.
28 Nov 2020
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Factorial, Gamma Function, and More
One of the most commonly used way to implement factorials for complex numbers
13 Nov 2020
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Riemann-Stieltjes Integration and Asymptotics
Summation, but calculus approach
02 Nov 2020
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Möbius Inversion and Beyond
Arithmetic functions are fun!
14 Oct 2020
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Use Pycharm to Emulate Vim
What will you do if you don't know what vim is?
23 Jan 2018
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How to Create a Flashing Desktop
For those people who wanna make tricks
21 Jan 2018
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Welcome to my new website
Hello world
20 Jan 2018
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