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1500 questions
54
votes
6 answers
Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real part = $\frac12$ ?
The function $\Gamma(s)$ does not have zeros, but $\Gamma(s)\pm \Gamma(1-s)$ does.
Ignoring the real solutions for now and assuming $s \in \mathbb{C}$ then:
$\Gamma(s)-\Gamma(1-s)$ yields zeros at:
$\frac12 \pm 2.70269111740240387016556585336 i$
…
Agno
- 4,179
54
votes
9 answers
Nice proofs of the Poincaré–Birkhoff–Witt theorem
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$, with an ordered basis $x_1 < x_2 < ... < x_n$.
We define the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ to be the free noncommutative algebra $k\langle…
user332
- 3,878
54
votes
6 answers
Which of Quillen's Papers Should I read?
I just heard that Daniel Quillen passed on. I am not familiar with his work
and want to celebrate his life by reading some of his papers. Which one(s?)
should I read?
I am an algebraic geometer who is comfortable with cohomological methods in his…
jlk
- 3,254
54
votes
2 answers
Walsh Fourier transform of the Möbius function
This question is related to this previous question where I asked about ordinary Fourier coefficients.
Special case: is Möbius nearly orthogonal to Morse
August Ferdinand Möbius (November 17, 1790 – September 26, 1868), Harold Calvin Marston Morse…
Gil Kalai
- 24,218
54
votes
8 answers
Proper way to deal with papers you've already refereed.
This question is anonymous for obvious reasons.
I referee what feels like a decent number of papers (though I don't know how many is normal!), and I try to take it seriously. Sometimes, based on something I explicitly said or something implicit in…
anonymous
- 181
54
votes
2 answers
How many relations of length $n$ can exists in a group without enforcing shorter relations?
Let $G$ be a group with two generators. Suppose that all non-trivial words of length less or equal $n$ in the generators and their inverses define non-trivial elements in $G$.
Question: How many of the $4\cdot 3^{n}$ words of length $n+1$ in the…
Andreas Thom
- 25,252
54
votes
3 answers
cube + cube + cube = cube
The following identity is a bit isolated in the arithmetic of natural integers
$$3^3+4^3+5^3=6^3.$$
Let $K_6$ be a cube whose side has length $6$. We view it as the union of $216$ elementary unit cubes. We wish to cut it into $N$ connected…
Denis Serre
- 51,599
54
votes
8 answers
Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
I have spent some time using gp-pari. There is, of course, a formal power series solution to
$ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure whether it is a function of anything.
On the other…
Will Jagy
- 25,349
54
votes
7 answers
Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community?
String theory (and related areas of purely theoretical quantum gravity, like loop quantum gravity) has a unique position within the academic physics community. Many academic physicists don't really consider string theory to be physics at all (due to…
tparker
- 1,243
54
votes
4 answers
When do binomial coefficients sum to a power of 2?
Define the function $$S(N, n) = \sum_{k=0}^n \binom{N}{k}.$$
For what values of $N$ and $n$ does this function equal a power of 2?
There are three classes of solutions:
$n = 0$ or $n = N$,
$N$ is odd and $n = (N-1)/2$, or
$n = 1$ and $N$ is one…
John D. Cook
- 5,147
54
votes
4 answers
How many square roots can a non-identity element in a group have?
Let $G$ be a finite group. Let $r_2\colon G \to \mathbb{N}$ be the square-root counting function, assigning to each $g\in G$ the number of $x\in G$ with $x^2=g$. Perhaps surprisingly, $r_2$ does not necessarily attain its maximum at the identity for…
alpmu
- 785
- 5
- 10
54
votes
8 answers
Questions about analogy between Spec Z and 3-manifolds
I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed links. How would that be with Spec Z? Then,…
Thomas Riepe
- 10,731
54
votes
1 answer
In the two-person Killing the Hydra game, what is the winning strategy?
My question is which player has a winning strategy in the
two-player version of the Killing the Hydra game?
In their amazing paper,
Kirby, Laurie; Paris, Jeff, Accessible independence results for Peano arithmetic, Bull. Lond. Math. Soc. 14, 285-293…
Joel David Hamkins
- 224,022
54
votes
2 answers
Automatically solving olympiad geometry problems
Warning: I am only an amateur in the foundations of mathematics.
My understanding of this Wikipedia page about Tarski's axiomatization of plane geometry (and especially the discussion about decidability) is that "plane geometry is decidable".
The…
Kevin Buzzard
- 40,559
54
votes
6 answers
What is the smallest unsolved Diophantine equation?
If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq N$ when $d$ varies. What size would be the…
Zidane
- 917