66
votes
Accepted
What is the simplest proof that the density of primes goes to zero?
I'm summarising the discussion in GH from MO's answer as a separate answer for clarity.
The fact that the primes have (natural) density zero can be deduced from a (seemingly) more general statement:
...
52
votes
Accepted
Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)
The complex-analytic proof of the prime number theorem can help inform the elementary one.
The von Mangoldt function $\Lambda$ is related to the Riemann zeta function $\zeta$ by the formula
$$ \sum_n ...
46
votes
What is the difference between elementary and non-elementary proofs of the Prime Number Theorem?
The PNT is indeed equivalent to $\lim_{x\to\infty} \frac{\psi(x) -x}{x}=0$ which Von Mangoldt's formula and some trivial estimates reduce to proving $$ \lim_{x\to\infty} \sum_{ \zeta(\rho)=0} \frac{x^{...
44
votes
Accepted
What is the simplest proof that the density of coprime pairs does not go to zero?
I would say that the standard proof is elementary enough, but here is an argument that avoids the Möbius function and the Riemann zeta function.
Let $A_d$ be the set of pairs $(a,b)$ where $a,b\le x$ ...
37
votes
What is the limit of $a (n + 1) / a (n)$?
The value is close to $e$ but not. It's actually the positive real root of $p(t) := t^3 - 2t^2 + t - 8$. This is solvable via ACSV (see book by Pemantle and Wilson 2013). To summarize, the ...
36
votes
Alternative proofs sought after for a certain identity
$\DeclareMathOperator\prob{prob}$Alapan Das' clever argument may be rephrased on the probabilistic language.
Write $[m]=\{1,2,\dotsc,m\}$. Choose a random non-empty subset $A\subset [n]$ (all $2^n-1$ ...
34
votes
Which theorems have Pythagoras' Theorem as a special case?
The Law of cosines is the first that comes to my mind:
$$c^{2}=a^{2}+b^{2}-2ab\cos \gamma$$
(source: Wikipedia)
If $\gamma$ is a right angle, its cosine is 0 and all that remains is Pythagoras' ...
Community wiki
33
votes
Chebyshev polynomials of the first kind and primality testing
Wow. This deserves a separate answer.
As I mentioned in a comment, motivated by the question, in a previous comment, by Igor Rivin whether an efficient primality test can be made if the statement in ...
32
votes
Accepted
Unrigorous British mathematics prior to G.H. Hardy
Rigor and Clarity: Foundations of Mathematics in France and England, 1800-1840 explains in some detail how British mathematicians in the early 19th century viewed the role of rigor in the formulation ...
30
votes
What is the simplest proof that the density of primes goes to zero?
Here is a very much self-contained version of the argument discussed in the posts by GH from MO and Terry Tao.
The claim immediately follows from $H_k:=1+1/2+\ldots+1/k\to \infty$ and the following
...
29
votes
Accepted
Diophantine equation $3^a+1=3^b+5^c$
I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
$$
ap^x + bq^y = c+ dp^z q^...
29
votes
Which theorems have Pythagoras' Theorem as a special case?
Parseval identities in the theory of Fourier series and integrals.
Community wiki
29
votes
Accepted
What is the difference between elementary and non-elementary proofs of the Prime Number Theorem?
To complement Will Sawin's answer, in the specific context of the prime number theorem, there are historically well-established notions of "elementary" and "non-elementary" proofs, ...
28
votes
Accepted
What is the value of this double sum in closed form?
Consider the integral $$I=\int_0^1\int_0^1\frac{zdzdt}{(1-zt)(1-z(1-t))}=\sum_{k,j\geqslant 0} \int_0^1\int_0^1 z^{k+j+1}t^k(1-t)^jdzdt=\\
\sum_{k,j\geqslant 0} \frac1{k+j+2}\cdot \frac{k!j!}{(k+j+1)!}...
28
votes
Accepted
A Putnam problem with a twist
$\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\tup}[1]{\left( #1 \right)}
\newcommand{\ive}[1]{\left[ #1 \right]}
\...
27
votes
Accepted
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Here is an elementary proof. We rewrite the series as
$$\frac{1}{4}\int_0^1\frac{1-x^4}{1-x^6}\,dx=\frac{1}{8}\int_0^1\frac{dx}{1-x+x^2}+\frac{1}{8}\int_0^1\frac{dx}{1+x+x^2}.$$
It is straightforward ...
27
votes
Unrigorous British mathematics prior to G.H. Hardy
The excellent answers by Carlo Beenakker and Padraig Ó Catháin have inspired me to do some reading, and I have come to the understanding that the contrast between English and Continental mathematics ...
26
votes
Geodesics on the sphere
Imagine that the sphere is the surface of the earth, and that an earthquake happens at the south pole, point $p$. A seismic wave goes out from the point $p$, so that the wave front is a circle of ...
26
votes
Unrigorous British mathematics prior to G.H. Hardy
These examples relate to algebra rather than analysis, but might in any case be useful.
The papers of JJ Sylvester are full of deep insight and entertaining prose, but are also full of unsupported ...
24
votes
Accepted
A conjectural infinite series for $\frac{\pi^2}{5\sqrt{5}}$
More generally, if $1\le k\le N-1$ is an integer, where $N$ is a positive interger,
$$S_{N,k} := \sum_{n=0}^\infty\biggl( \frac{1}{(N n+N-k)^2} + \frac{1}{(N n+k)^2}
\biggr) = \frac{\pi^2}{N^2\sin^...
22
votes
Accepted
Roots and relation between polynomials and their derivatives
Suppose that we have
$$P(x)=x^n-ax^{n-1}+bx^{n-2}+\cdots$$
where we can take $P$ to be monic since it doesn't affect $V_n(P)$. From Vieta's formula we have
$$a=\sum_{i=1}^n x_i \quad , \quad b=\sum_{...
22
votes
Accepted
The constant $e$ represented by an infinite series
Your sum actually equals $\frac{\pi\sqrt{3}}{2}$, so it's more like $\pi$ all over again, not $e$. To see this, note first that by definition $\mathrm{sgn}_2$ is multiplicative, hence
$$
A=\sum_{n=1}^{...
22
votes
Which theorems have Pythagoras' Theorem as a special case?
The parallelogram law says that if $\mathcal{V}$ is an inner product space and $\mathbf{v},\mathbf{w} \in \mathcal{V}$, then
$$
2\|\mathbf{v}\|^2 + 2\|\mathbf{w}\|^2 = \|\mathbf{v} + \mathbf{w}\|^2 + \...
Community wiki
21
votes
Accepted
Nonstandard proofs of the fundamental theorem of arithmetic
To summarize the comments, this is also known as Zermelo's proof. A version can be found on wikipedia. I will give the proof here to avoid link rot.
The proof is by contradiction. If FTA did not ...
Community wiki
21
votes
Which theorems have Pythagoras' Theorem as a special case?
The Pythagorean theorem is a limit of the general formula for spherical or hyperbolic space: $\cos(a\sqrt{\kappa})\cos(b\sqrt{\kappa})=\cos(c\sqrt{\kappa})$, where $\kappa$ is the curvature.
Community wiki
21
votes
Simpler proofs using the axiom of choice
In Division by Three, Peter G. Doyle and John Horton Conway show, without invoking the Axiom of Choice, that for any sets $A$ and $B$, if there is a bijection between $3 \times A$ and $3 \times B$, ...
Community wiki
20
votes
What is the simplest proof that the density of primes goes to zero?
Using the Chinese Remainder Theorem, one can reduce the statement to $\prod_p(1-1/p)=0$, which in turn is equivalent to $\sum_p 1/p=\infty$. For the last statement a short (but clever) proof was given ...
20
votes
What is the simplest proof that the density of primes goes to zero?
The proof by GH from MO is, of course, correct. As noted in the comment of Fedor Petrov,
there is no need to pass to the series $\sum_p 1/p$ and it is easier to analyze the product $\prod_p (1-1/p)$ ...
20
votes
What is the difference between elementary and non-elementary proofs of the Prime Number Theorem?
The fundamental problem with using formulas of the kind you described for $\psi(x)$ is that the series $\sum_\rho x^\rho/\rho$ is hard to estimate on its own.
The crux of the problem here is that ...
19
votes
Accepted
roots of higher derivatives of exponential
The (physicists') Hermite polynomials are
$$ H_n(x) = (-1)^n e^{x^2} D^n e^{-x^2}$$
And their roots are real. For that you don't need to know they are Hermite polynomials: just Rolle's theorem. ...
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