All Questions
Tagged with reference-request nt.number-theory
1,409 questions
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Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.2k
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1
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292
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Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
- 3,074
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0
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75
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Existence of smooth integers in every residue class with large modulus
Let us say that a positive integer $x$ is $y$-power smooth, if the largest prime power divisor of $x$ is at most $y$. In what follows, let $C$ be any real number larger than $1$ and, for an integer $x$...
- 1,663
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$F$-structure implies regular singularities + unipotent local monodromy?
Let $(\mathcal{E},\nabla)$ be a vector bundle with an integrable connection on a smooth quasi-projective $K$ scheme $X$, with $K$ a $p$-adic number field of characteristic $0$. Let $F$ denote a semi-...
- 2,907
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A generalisation of theorem of Landau on sum of two squares?
Let
$r(B)$ be the number of integers $1 \leq n \leq B$ such that $n = x^2 + y^2$ for some $x, y \in \mathbb{Z}.$
Then it is a known theorem of Landau that
$$
r(B) \sim C \frac{B}{\sqrt{\log B}}
$$
...
- 3,625
29
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2
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Closed formula for a certain infinite series
I came across this problem while doing some simplifications.
So, I like to ask
QUESTION. Is there a closed formula for the evaluation of this series?
$$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{...
- 43.2k
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Questions about ray class groups
Let $K$ be an imaginary quadratic number field (so there are no real embeddings) with ring of integers $\mathcal{O}_K$ . Let $w$ be the number of units in $K$ and $h$ be the class number of $K$. Let $\...
- 1,990
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3
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536
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A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 43.2k
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2
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390
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Solving a recurrence relation for the prime counting function?
I have found some number sequence $c_n = 1+b_n$ for $n \ge 0$, where $b_n = $ A307977(n).
I am trying to solve the following recurrence relation for the prime counting function:
$$\forall n \ge 3: \pi(...
- 3,074
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Approximating $\zeta^{(r)}(s)$ by a sum
Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
- 20.2k
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What is the Perrin-Riou logarithm (or regulator)?
Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this workshop at the CRM in ...
- 3,309
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1
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What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite ...
- 2,959
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2
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Using Kolmogorov complexity to measure difficulty of problems?
We call the natural number $n$ a partition number $\iff$
$$
\exists d | n: \gcd\left(d,\frac{n}{d}\right)=1 \text{ and } \Omega(d) = \Omega\left(\frac{n}{d}\right)\;,
$$
where $\Omega$ counts the ...
- 3,074
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Kummer's congruence at $p=3$
Let $B_{2k}$ be the Bernoulli numbers of even index and $\varphi(n)$ be Euler's totient function. We recall one instance of Kummer's congruences: for each integer $m\geq1$ and a prime number $p\geq5$, ...
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Reference Request: Local decomposition of GGP period integrals of cuspidal forms on unitary groups
Setup: Let $E/F$ be a CM-extension of global number fields. Let $(V,\phi)$ be an Hermitian space of dimension $n$ over $E$. Let $(V^{\flat}, \phi^{\flat})$ be a subspace of $V$ of dimension $n-1$ on ...
- 639
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131
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Analytic properties of $L$-functions attached to a compatible system of $\ell$-adic Galois representations
Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We ...
- 663
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174
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Upper bound on sum of Lambda(n) over short interval
I am looking for a bound of type
$$\sum_{x<n\leq x+y} \Lambda(n) \leq \frac{\log(x+y)}{\log y} \cdot 2y$$
(or better). Of course such a bound has to exist: the idea of the proof of Brun-Titchmarsh (...
- 20.2k
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131
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Taking integer values of a sequence of Beurling primes
Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
- 51
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Video abstracts for mathematical papers
I recently published a video abstract of a manuscript of mine (number theory), finding that more people are interested in its content than when I uploaded the preprint on arXiv.
Now, my main question ...
- 1,451
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Congruences for power-sum of divisors
If $\sigma_k(n)=\sum_{d\vert n} d^k$, denote
$$F_1(q)=\sum_{n\geq1}\sigma_1(n)\,q^n \qquad \text{and} \qquad
F_3(q)=\sum_{n\geq1}n\cdot\sigma_2(n)\,q^n.$$
QUESTION. Assume the prime $p$ is either $2,...
- 43.2k
148
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4
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What are "perfectoid spaces"?
This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?
Edit: A bit ...
- 10.8k
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2
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Only odd primes?
For $k \ge 2$, let
$$u = \{\lfloor{(k - \sqrt{k})n}\rfloor : n \ge 1\}$$
$$v = \{\lfloor{(k + \sqrt{k})n}\rfloor : n \ge 1\}.$$
My computer suggests that $u$ and $v$ are disjoint if and only if $k$ is ...
- 479
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1
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171
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Integers $8k+3>0$ not represented by $2x^2+4y^2+4yz+9z^2$ over the integers
Oeis A306970 lists positive integers of the form $8k+3$ which are not reprented by
$$f(x,y,z):=2x^2+4y^2+4yz+9z^2$$
over the integers as $3,43,163,907$. It says this list may not be complete and ...
- 458
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346
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A combinatorial proof: where art thou?
Start by introducing the finite sums
$$A_n:=\sum_{m=1}^nq^m\prod_{j=1}^{m-1}(1-q^j) \qquad \text{and} \qquad
B_n:=\sum_{m=1}^nq^m\prod_{j=m+1}^n(1-q^j).$$
An algebraic proof is facile: Clearly, $A_1=...
- 43.2k
5
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100
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Buchi's conditional proof of the non-existence of finite algorithm to decide solubility of system of diagonal quadratic form equations in integers
I am doing some literature review regarding Buchi's problem. In particular, I am reading the relevant section in this survey paper by Mazur (Questions of Decidability and Undecidability in Number ...
- 26.9k
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2
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349
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Geometric interpretation of Iwasawa algebras: $\mathbb{Z}_p[[T]]$ as a disk?
I am a student learning Iwasawa theory. I am so sorry if this post is too trivial for this site. I posted it on math.stackexchange yesterday but obtained no responce.
A quite basic object is the ...
- 639
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245
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Integration against Eisenstein series can be regarded as a cup product
This summer, I was very fortunate and honored to attend the conference "Iwasawa 2023" at the University of Cambridge as a young Ph.D. student on Iwasawa theory. There, one of the speakers, ...
- 639
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votes
2
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806
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Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the ...
- 25.4k
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1
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197
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Hyperbolic fixed points of SL(2,Z)
The real hyperbolic fixed points of $\mathrm{SL}_2(\mathbb{Z})$ are the points $x\in\mathbb R\smallsetminus\mathbb{Q}$ with
$$
\frac{ax+b}{cx+d}=x
$$
for some $\left(\begin{array}{2} a&b\\ c&d\...
user473423
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votes
4
answers
707
views
Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?
Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
- 20.2k
7
votes
0
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222
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Projected polar chessboard measure convergence in total variation?
$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set
$$F_n:=\...
- 128k
7
votes
2
answers
906
views
Positivity of the coefficients of Taylor series associated to the Riemann hypothesis
The question below relates to the paper "Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier. I'm asking it here because I am sure the ...
- 7,067
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1
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757
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The $9$th tetration of $-\sqrt2$
Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and
$$^{n+1}a=a^{^na}$$
for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...
- 128k
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0
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109
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PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
- 383
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1
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Sum of square roots of natural numbers
Recently, I've encountered the following question:
Assume that $n_{1}, \ldots, n_{k}$ are (not necessary distinct) natural numbers. If
$$ (\sum_{i = 1}^{k}\sqrt{n_{i}}) \in \mathbb{N},$$ can we ...
7
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2
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816
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Well known applications of Roth's theorem
Roth's theorem in Diophantine approximation (1955) is a well known milestone. It has been generalised in the case of number fields for simultaneous approximations considering several places.
It is an ...
- 321
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0
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374
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Example of abelian variety over finite field which doesn't lift
What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given.
Note that ...
- 21.4k
9
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3
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725
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Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
5
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1
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340
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About an asymptotic behavior in number theory
Where can I read about the asymptotic behavior (with $N$ tending to infinity) of the sum of the fractional parts obtained from dividing $N$ by all prime numbers up to $N$ divided by the number of ...
11
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1
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Has this number-theoretic constant been studied?
Unless I made a mistake, the expected value of the largest exponent in the prime factorization of random positive integer (defined in the appropriate way) is $$\eta := \sum_{n=1}^\infty \Big(1-\zeta(n)...
- 1,356
2
votes
1
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177
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Discrete maximization of geometric mean - reference request
This is a follow-up to my previous MO question:
A discrete optimization problem related to the AM-GM inequality
Let $n,k$ be integers such that $1\le k\le n$. Define the quantity
$$
P(n,k):=\max\ a_1\...
0
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0
answers
115
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Conjecture that there are finitely many integer powers $a^x$ and $b^y$ such that $b^y - a^x = n$: who first came up with it?
I came up with an interesting mathematical conjecture: for every natural number $n$ there is only a finite number of integer powers $a^x$ and $b^y$ such that $b^y - a^x = n$.
I would like to find out ...
- 161
3
votes
2
answers
459
views
Short sequence beats long sequence
I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, ...
- 43.2k
1
vote
2
answers
127
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Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known ...
- 3,625
1
vote
1
answer
344
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Products involving exponents of tribonacci numbers
The Fibonacci numbers $F_n$ can be given by
$$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$
Among many many properties of this sequence, consider the following two results:
(1) the coefficients of the ...
- 43.2k
17
votes
1
answer
1k
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Catalan's constant fast convergent series
NOTE. UPDATE 2 introduces proven series for Catalan's constant that is possibly the fastest currently known.
Working with some conjectured continued fractions that were published here, I have found ...
- 2,836
7
votes
2
answers
615
views
Reference request for recurrence relation of division polynomials
The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...
- 265
1
vote
0
answers
59
views
A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
1
vote
0
answers
109
views
What is known about the average growth rate of the denominators of $n$ Egyptian fractions summing to one?
Motivation
In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the ...
- 4,829
5
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1
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788
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Is there a reference for these types of cubic identities?
I'm looking at the following generalizations for sums of two cubes.
$u^3+v^3=(u+v)(u^2-uv+v^2).$
$u^3+(u+r)^3+v^3+(v+r)^3=(u+v+r)(2u^2-2uv+2v^2+ru+rv+2r^2).$
$u^3+(u+q)^3+(u+r)^3+(u+q+r)^3+v^3+(v+q)^3+...
- 1,109