All Questions
Tagged with reference-request nt.number-theory
1,408 questions
3
votes
1
answer
216
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Reference request: Serre's Groupes discrets
I'm reading some articles and at some point they both reference:
J-P. Serre: Groupes discrets (in collaboration with H. Bass),
Collège de France, 1969
However I have trouble finding this reference. ...
- 2,224
2
votes
2
answers
308
views
Reference for zero sum estimates of Dirichlet L functions
Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$.
I am reading a paper by Ihara and Murty where they use following estimate :
$\...
- 41
5
votes
1
answer
386
views
Is there a simple expression for $\sum_{k =1}^{(p-3)/2} \frac{1\cdot 3\cdots (2k-1)}{2\cdot 4 \cdots 2k\cdot(2k+1)} \bmod p$?
Let $p \equiv 1 \pmod 4$ be a prime and $E_n$ denote the $n$-th Euler number. While investigating $E_{p-1} \pmod{p^2}$ I have encountered this summation (modulo $p$)
\begin{align*}
\sum_{k =1}^{\frac{...
- 707
7
votes
1
answer
707
views
How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
- 707
8
votes
2
answers
1k
views
Question about functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $x$ is prime whenever $f(x)$ is prime
Let $f: \mathbb{\mathbb{Z}^+} \to \mathbb{Z^+}$ be a function and suppose
$(\star)$ For all integers $x \geq 3$, if $f(x)$ is prime, then $x$ is prime.
A trivial example of such a function is the ...
- 707
3
votes
1
answer
331
views
Fully explicit Linnik's Theorem
Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < ...
- 1,663
2
votes
1
answer
111
views
Equations for $H_1(M)$ and $T$-Tate module of Anderson t-motive $M$ are equivalent: a reference?
What is a reference for the following construction?
Let $M$ be an Anderson t-motive of rank $r$ dimension $n$, i.e. a module over the Anderson ring $\mathbb{C}_\infty[T]\{\tau\}$ satisfying some ...
0
votes
0
answers
121
views
Quadratic residue problem involving prime divisors of a polynomial
Let $n$ be a square-free natural number, and let $f\in\mathbb{Z}[x]$ be monic and irreducible of degree $\geq2$. I am trying to determine whether there always exists a prime $p$, $p\nmid n$, ...
- 1
2
votes
3
answers
742
views
Asking for a proof for a sum of products of binomials: an "interesting" identity?
The following identity must have received alternative proofs, including a combinatorial argument by David Callan as found at Bijections for the Identity $4^n = \sum_{k = 0}^n \binom{2k}k\binom{2(n - k)...
- 43.2k
6
votes
2
answers
546
views
On circles and ellipses drawn on an infinite planar square lattice
Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
- 5,979
7
votes
2
answers
570
views
Finite generation of motivic cohomology of number fields
Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups
$$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$
...
- 1,790
18
votes
1
answer
1k
views
Distinct integer roots for a degree 7+ polynomial and its derivative
Question: Is there a polynomial $f \in \mathbb{Z}[x]$ with $\deg(f) \geq 7$ such that
all roots of $f$ are distinct integers; and
all roots of $f'$ are distinct integers?
Background:
I asked a ...
- 7,831
0
votes
0
answers
73
views
Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?
What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$,
every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...
- 20.2k
1
vote
2
answers
350
views
Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
5
votes
3
answers
300
views
Closed formula for $(-1)$-Baxter sequences
The number of the so-called Baxter permutations of length $n$ is computed by
$$a_n=\frac1{\binom{n+1}1\binom{n+1}2}\sum_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}.$$
There has also been a ...
- 43.2k
12
votes
1
answer
407
views
Looking for a "clever" argument for a $q$-series identity
Consider the below $q$-series identity. One of the things I like about this expansion is how nicely the difference on the left hand side factors to the right hand side of the equation.
$$\prod_{k\geq1}...
- 43.2k
1
vote
1
answer
147
views
Name for an easy combinatorial game
What is the name of the following combinatorial game:
Two players, moving in turn.
Positions: $0,1,2,\ldots$.
Moves: $n\longmapsto n-1$ or $n\longmapsto \lfloor n/2\rfloor$
if $n>0$.
No move for $0$...
- 17.9k
6
votes
1
answer
546
views
On Cramér's theorem about roots of Zeta function
Cramér proved the following theorem (see the announcement in [1] and [2]):
Consider the following function:
$$V(z)=\sum_k e^{\rho_kz}$$
Where $\rho_k$ runs through non trivial zeta zeros with $Im(\...
- 790
6
votes
2
answers
339
views
Sum of divisors and LCM in determinants
$\newcommand{\lcm}{\operatorname{lcm}}$Let $\gcd(i,j)$ and $\lcm(i,j)$ be the greatest common divisor and least common multiple of the pair of positive integers $i$ and $j$. Denote the sum of divisors ...
- 43.2k
9
votes
2
answers
547
views
Primes between $x$ and $x+x^\theta$
Iwaniec [1] proved that
$$
\pi(x+x^\theta)-\pi(x) < \frac{(2+\varepsilon)x^\theta}{\eta(\theta)\log x},\ x>x_0(\varepsilon,\theta).
$$
with
$$
\eta(\theta)=\frac{15\theta-2}{9}.
$$
(Actually, he ...
- 9,114
3
votes
1
answer
188
views
Density of numbers with a prime factor satisfying a congruence
Let $S$ be the set of integers with at least one prime factor in the arithmetic progression $km+d$, $(m, d)=1$. I am looking for results on the density of $S$. I found this post which talked about the ...
- 31
9
votes
3
answers
659
views
Vinogradov-Korobov for Dirichlet L-functions?
Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
- 20.2k
40
votes
5
answers
8k
views
Is $\zeta(3)/\pi^3$ rational?
Apery proved in his paper from 1979 that $\zeta(3)$ is irrational, and we know that for all integers $n$,
$\zeta(2n)=\alpha \pi^{2n}$
for some $\alpha\in \mathbb{Q}$. Given these facts, it seems ...
- 7,013
2
votes
0
answers
109
views
Action of Galois group on the lattice of a Drinfeld module - a reference?
What is a reference for the following construction?
Let $K$ be a finite extension of $\Bbb F_q(\theta)$ and $M$ a Drinfeld module of rank $r$ defined over $K$. Its lattice $L(M)$ is a free $\Bbb F_q[\...
3
votes
0
answers
142
views
Reference for formula expressing products of two Fibonacci numbers in Zeckendorf-basis
It is well-known folklore that every natural integer has a unique Zeckendorf expansion as a
sum over a finite set of Fibonacci numbers containing no pair of consecutive Fibonacci numbers.
It is easy ...
- 17.9k
10
votes
2
answers
849
views
Schur's proof of Hilbert's inequality: streamlining?
TL;DR: Is there a way to make Schur's (elegant) proof of Hilbert's inequality feel like
less of a trick/miracle?
Longer version: Let me go quickly over Schur's proof to show what I mean. Actually, let ...
- 20.2k
7
votes
0
answers
266
views
"Reference Request" for a lecture note by C. Skinner: Galois Representations, Iwasawa Theory, and Special Values of $L$-functions
This was originally posted on math.stackexchange as https://math.stackexchange.com/questions/4589793, where I was suggested to move it here.
I'm searching a lecture note by C. Skinner named "...
- 71
31
votes
7
answers
6k
views
English reference for a result of Kronecker?
Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference:
Let $f$ be a monic polynomial with integer ...
- 873
9
votes
2
answers
629
views
How to read the paper of Arthur on trace formula on general reductive groups
My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily.
I would like ...
- 93
8
votes
1
answer
1k
views
Quick reference for general Weyl's inequality in number theory
I would like a reference for the result here. Having that $t$ there makes me happy. I would prefer not to have to, in my paper, run through and (not trivially but not too greatly) alter the proof of ...
- 1,354
38
votes
5
answers
10k
views
Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
- 2,350
31
votes
5
answers
8k
views
Fermat's proof for $x^3-y^2=2$
Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original ...
- 3,866
4
votes
1
answer
95
views
Limiting values of particular functions
Let's define the functions
$$A_n(q)=\sum_{k=0}^n(-1)^k\cdot\frac{(1+q)q^k}{1+q^{2k+1}}\cdot\frac{2k+1}{n+k+1}\binom{2n}{n-k}.$$
I'm interested in the following:
QUESTION. Let $n\geq1$ be integers. ...
- 43.2k
2
votes
0
answers
197
views
Mumford's computation of the determinant of cohomology of a relative curve
In Integral Grothendieck-Riemann-Roch theorem, Pappas mentions that Mumford computed the determinant of cohomology of $f:X\to S$ a relative curve integrally, and thus proved an integral version of GRR ...
- 2,044
10
votes
5
answers
771
views
Reference request: Diophantine equations
I am looking for a textbook, or preferably lectures, on the subject of Diophantine equations. I am familiar with the basic principles of modular arithmetic, conics and the Hasse Principle, and the ...
- 2,811
8
votes
1
answer
355
views
The distribution of certain Galois groups
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}_p(y)$ and $G_p$ be the Galois group of the ...
- 7,193
14
votes
1
answer
424
views
Unpublished result of Rosser in Sieve Methods book
Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert.
...
- 24.8k
2
votes
0
answers
79
views
Which sets of natural numbers are "lambda-analytic"?
Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define
$$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$
for all real numbers $x \in ...
- 13.3k
9
votes
1
answer
400
views
The difference between consecutive primes in arithmetic progressions
Let $\pi(x)=\sum_{p\leq x}$ denote the prime counting function. A well known result of Baker, Harman, and Pintz on prime gaps states that for $x\geq y\geq x^{0.525}$ we have that
$$\pi(x+y)-\pi(x)\gg \...
- 11.4k
3
votes
1
answer
492
views
Where can I find a rigorous proof of this statement in the literature? : $\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0$
In Wolfram MathWorld site at Moebius Function topic
there is identity number 10, which states that $\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0$.
Could you help me find a rigorous proof of this statement ...
- 39
2
votes
0
answers
110
views
Asking for a generating function for an arithmetic sequence
For fixed integer $n\geq1$, let $c_m(n)$ be the number of divisors $d$ of $m$ such that $n<d\leq 2n$. Here is an experimental generating function for which I ask:
QUESTION. Is this true?
$$\sum_{m\...
- 43.2k
5
votes
1
answer
210
views
Results using a certain kind of identity
Recently, I've been reading about asymptotics for smooth numbers as well as smooth numbers in arithmetic progressions. One of the ideas I find especially pleasing among some of these results is the ...
- 1,990
32
votes
3
answers
12k
views
What is the Katz-Sarnak philosophy?
It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...
- 8,071
3
votes
1
answer
280
views
Computing mth power residue symbols
Let's say I have a two odd primes, $p, q$ and $K$ is the field $\mathbb{Q}(\zeta_{pq})$. Let's say $\alpha \in \mathcal{O}$ is an arbitrary element in the ring of integers of $K$, $\frak{b} \subset \...
- 565
2
votes
1
answer
112
views
Counting numerical semigroups by largest element of minimal generating set
For a given integer $n$, I am interested in the number of different numerical semigroups one can make with a generating set consisting only of integers in $[n]$.
I have done some small examples. For $...
- 807
4
votes
1
answer
539
views
A (mild?) question on the number of monomials
Let $[n]_q=\frac{1-q^n}{1-q}$ with $[0]_q=0$. Recall the $q$-factorials $[n]_q!=[1]_q[2]_q\cdots[n]_q$ (with $[0]_q!=1$) and the $q$-binomials
$$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\,[n-k]_q!}.$$
Now, ...
- 43.2k
17
votes
13
answers
6k
views
Probability in number theory
I am hearing that there are some great applications of probability theory (or more general measure theory) to number theory. Could anyone recommend some good book(s) on that (or other types of ...
10
votes
2
answers
1k
views
Reference request: Oldest number theory books with (unsolved) exercises?
Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the ...
19
votes
1
answer
1k
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Deligne's letter to Bhargava from March 2004
I am quite interested in moduli spaces for Rings and Ideals, a letter from Deligne to Bhargava is cited in Melanie Wood's thesis Moduli spaces for Rings and Ideals (pdf), studying the minimal free ...
- 461
3
votes
1
answer
206
views
The growth of certain continued fractions
I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ ...
- 1,990