Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
4,689
questions with no upvoted or accepted answers
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Is the unipotent section map of hyperbolic curve over local field injective?
Let $X/\mathbb{Z}_p$ be a smooth hyperbolic curve and $\pi^{un}_1(X_{\overline{\mathbb{Q}}_p},b)$ denotes the pro-unipotent completion (over $\mathbb{Q}_p$) of the etale fundamental group of $X$ base ...
- 123
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185
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A formula in Ramanujan's lost notebook and its connection with Chudnovsky series for $1/\pi$
While studying Berndt's Ramanujan's Lost Notebook Vol. 2, page 369 (chapter on Springerlink), I found that Ramanujan gave values of a certain expression $$\frac{1}{\sqrt{Q_n}}\left(\sqrt {n} P_n-\frac{...
- 2,019
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153
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Average size of class groups of cyclotomic fields: three perspectives
Let $K$ be a number field. Let $h(K)$ denote the class number (i.e., the size of the ideal class group) of $K$, $R(K)$ be the regulator of $K$, and $\Delta_K$ the discriminant of $K$.
Let $\mathcal{F}$...
- 25.5k
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203
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On $\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}$ and $\sum_{n=1}^\infty\frac1{p_1\cdots p_n}$
For $n=1,2,3,\ldots$ let $p_n$ denote the $n$-th prime number.
Question. Are the two numbers
$$c_1=\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}\ \ \ \ \text{and}\ \ \ \ c_2=\sum_{n=1}^\infty\frac1{p_1\...
- 14.5k
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222
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To which value does this infinite sum of power series coefficients converge?
Context:
In this and this paper, J. Arias de Reyna shows that the RH follows when:
$$1.2663935... \le \sum_{n=1}^\infty A_n^2 \le 1.2723669...$$
where $A_n$ is the coefficient in the following power ...
- 4,169
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99
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Why is it at all reasonable that $0\leq\sum_{S\subseteq[Q]^*}\frac{(-1)^{|S|}}{\mathrm{lcm}(S)}{|S|\choose (|S|+\sum_{i\in S}\mu(i))/2}\leq1$?
Let $[Q]^*$ denote the set of all squarefree natural numbers $\leq Q$, and let $\mu(n)$ be the Mobius function. I recently discovered the identity
$$0\leq\sum_{S\subseteq[Q]^*}\frac{(-1)^{|S|}}{\...
- 2,817
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0
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210
views
What is the complexity class of this problem without Cramer's conjecture?
The problem 'Given $0<a<b$ is there a prime in the interval $[a,b]$?' is in $\mathsf{NP}$. If we assume Cramer's conjecture the problem is in $\mathsf{P}$ since if $b-a>(\log a)^{2+\epsilon}$ ...
- 13.7k
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363
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Dense sets in $\Bbb{R}^2$ with rational distance
We call a subset $S\subset \Bbb{R}^2$ rationally distanced if all $s_1,s_2 \in S$ have rational Euclidean distance.
The Erdos-Ulam conjecture asks if there is a dense subset of $\Bbb{R}^2$ which is ...
- 3,413
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174
views
Products of short elements in a field
Consider a field $F$ of characteristic zero. Let $L=F[\alpha]$ be an extension of degree $d.$ We call an element
$$
x=x_0 + x_1 \alpha +\ldots+ x_{d-1}\alpha^{d-1}\in L
$$
short if $x_{d-1}=0.$
Under ...
- 4,296
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211
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Reference for a functional equation of a Hecke L-function
Let $F$ be a degree $n$ totally real number field. Let $W = (W_1,\dots ,W_n)$ be a $\mathbb Q$-basis for $F$, and let $\Lambda = \mathbb{Z}W_1 + \dots + \mathbb{Z}W_n$. It is a fractional ideal in the ...
- 3,699
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186
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Waring's problem with pairwise coprime summands
According to https://www.imo-official.org/problems/IMO2019SL.pdf (p.100, Problem N7, Comment 2) for any $\alpha > 0$ there are infinitely many positive integers $n$ such that there are infinitely ...
- 41
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210
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Traces vs. Determinants in Artin's $L$-functions
Loosely put, my question is:
What happens if we swap determinant by the trace in an Artin $L$-function?
This question is not very precise and can be a little misleading, so I explain the specific ...
- 79
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129
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Consecutive integers which are products of Fibonacci numbers
Let $F_1, F_2, \dots$ be the sequence of Fibonacci numbers, and let $\mathcal{M}$ be the set of positive integers expressible as a product of Fibonacci numbers (OEIS A065108).
One can prove that
$$F_{...
- 141
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482
views
Collatz conjecture and a diophantine equation
Let $M \ge 2$. Inspired by the Collatz iteration / algorithm ($M=2$), I tried the following function:
$$C_M(n) = n/M, \text{ if } n \equiv 0 \mod M, \text{ otherwise } (M+1)n+\{(M-n) \mod M \}$$
We ...
- 2,462
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151
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Can we attach (formal) abelian varieties to $p$-adic modular forms?
The Jacobian of the modular curve $X_1(N)$ over $\mathbb Q$ $J_1(N)$ can be decomposed up to isogeny, as a product of abelian subvarieties $A_f$ corresponding to Galois conjugacy classes of Hecke ...
- 461
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129
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$\delta$-equidistributed polynomials over finite fields
I'm trying to show that a polynomial over finite (prime) field is "close enough" to being equidistributed over its range. A polynomial $p(\cdot)$ from $\mathbb{F}^n$ to $\mathbb{F}$ is $\...
- 181
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180
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Sato-Tate over function fields
Suppose we have an elliptic surface $\pi: \mathscr E \to C$ over a curve over a finite field $\mathbb F_q$. We consider only the places on $C$ over which we have good reduction.
Over any point $x\in C$...
- 7,646
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199
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Are there zeros of $\zeta(s)-\zeta(2s)$ on $\Re(s)=1$?
I am trying to answer this question on Math.SE which is asking if $P(s)$ has any zeros with $\Re(s)=1$, and if it were true that $\zeta(s)-\zeta(2s)$ had no zeros on $\Re(s)=1$ then I would be able to ...
- 2,817
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198
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Frey's elliptic curve and perfect numbers?
Let $E_n:y^2=x(x-\sigma(n)/2)(x+\sigma(n)/n)$ be a Frey-elliptic curve, where $\sigma$ denotes the sum of divisors of the natural number $n$.
If $n$ is a perfect number ($\sigma(n)=2n$) then the $j$-...
- 213
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275
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How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series?
How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series ?
We know that if $f(x)=\sum_{i=0}^{\infty} a_ix^i$ be a power series over $p$-adic field, then the Newton ...
- 870
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102
views
What do the eigenvalues of a random element of $\mathbb Z_\ell[\Gamma]$ look like?
Let $\Gamma = \varprojlim \Gamma_n$ be a profinite group with $\Gamma_n$ finite quotients. For concreteness, let us fix $\Gamma_n = \operatorname{PGL}_2(\mathbb Z/\ell^n)$ so $\Gamma = \operatorname{...
- 7,646
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197
views
Which sums-of-two-squares are totients?
Consider the two subsets of the natural numbers $A$, $B$ given by
$$A = \{n \in \mathbb{N} \mathrel: \text{$\exists x,y \in \mathbb{Z}$ s.t. $n = x^2 + y^2$}\}$$
and
$$B = \{n \in \mathbb{N} : \exists ...
- 25.5k
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votes
1
answer
316
views
Boolean ring of unitary divisors / Structure of unitary divisors?
I hope this question is appropriate for MO:
Let $n$ be a natural number, $U_n := \{ d | d \text{ divides } n, \gcd(d,n/d)=1\}$ be the set of unitary divisors.
We can make $U_n$ to a boolean ring:
$$a \...
user6671
4
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485
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Euler Systems and Coleman’s Conjecture
I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
- 99
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182
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Integer polynomial inducing a permutation of order $p$ on $\mathbb{Z}/p\mathbb{Z}$ for infinitely many $p$
Let $Q\in \mathbb{Z}[x]$ be a non-linear polynomial. Can there exist infinitely many primes $p$ such that $Q(\mathrm{mod}\:p)$ induces a permutation $\mathbb{Z}/p\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$ ...
user158636
4
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276
views
Galois representation with infinite image but finite image everywhere locally
Fix a prime $l$. Let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$ be a semisimple continuous representation. Assume $\phi$ has finite image when restricted to $\mathrm{...
user158636
4
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297
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On $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$
My question is related to https://oeis.org/A269839.
It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. ...
- 681
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538
views
Modern example of a reciprocity law and intuition behind it
I'm very new to the Langlands program and I was going through the Gauss reciprocity law, Hilbert's 9th problem, Artin's reciprocity law which allowed him to identify the Artin's L-functions with the ...
- 3,766
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96
views
Diophantine problem in n log n
Let $x_n = n \log n$ for $n \geq 1$. What is a good upper bound for the number of solutions $(n_1, n_2, n_3, n_4) \in \{1, \dots, N\}^4$ of
$$
|x_{n_1} - x_{n_2} + x_{n_3} - x_{n_4} | < \gamma,
$$
...
- 2,199
4
votes
0
answers
65
views
$3$-variable Jacobi style identity linked to generalised Frobenius partitions
I was fiddling around with a family of probabilistic models and came across two "identities", which appear to be linked to generalized Frobenius partitions (more on this below). I would be ...
- 562
4
votes
0
answers
250
views
What is the multiplicative order of this number
Let $q, r \in \mathbb{P}$ and $r$ is the next prime to $q$.
What is the multiplicative order of $r$ modulo $\displaystyle\bigg( \prod_{\substack{p \leq q \\\text{p prime}}} p \bigg)$ ?
In other word ...
- 521
4
votes
0
answers
252
views
Why is Haven's discovery important?
Today my attention was caught by one of those little stories that appear when you open a certain browser: an inmate achieved a number theoretic breakthrough
It is about continued fractions and I would ...
- 12.7k
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198
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Galois action of Weil restriction
Let $K/\mathbb{Q}$ be a quadratic field. Let $E$ be an elliptic curve defined over $K$ but not over $\mathbb{Q}$, and let $\bar{E}$ be the Galois conjugate of $E$. Then by the descent theory (for ...
- 451
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votes
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Finding integer solutions to $n=a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc)$?
Is there anything known how to find integer solutions to
$$n=a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc)$$
where $n$ is a natural number $\neq 3,6 \mod 9$ and $a,b,c \in \mathbb{Z}$?
Notice ...
user6671
4
votes
0
answers
909
views
What fraction of fractions does Cantor's famous sequence enumerate?
Cantor's famous sequence
$\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{1},\frac{1}{4}, \frac{2}{3},\frac{3}{2},\frac{4}{1}, \frac{1}{5},\frac{5}{1},\frac{1}{6}, ...$
provides a ...
user112109
4
votes
0
answers
143
views
Compute a cardinality using Chinese remainder theorem
I posted the question here but I got no response.
I am looking for computing this cardinality:
$$N(q)=\#\Bigg\{n \in \mathbb{N} \ | \ \gcd\bigg(n^2+1, \prod_{\substack{p \leqslant q \\ p\text{ prime}}...
- 521
4
votes
0
answers
234
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Symmetric spaces as the moduli spaces of Arakelov vector bundles
Over a function field of a curve $K = k(C)$, there is the Weil uniformization
$$\mathrm{Bun}_{GL_n}(C) = GL_n(K) \backslash GL_n(\mathbb{A}_K) / GL_n(\mathcal{O}_K).$$
This equality is (for example)...
- 950
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votes
0
answers
305
views
Beilinson regulator: a road map
I'm approaching to the Beilinson Conjecture and after studying some properties of the Deligne-Beilinson cohomology, I want to understand the regulator maps. But I don't know anything about K-theory ...
4
votes
0
answers
143
views
Moments of the prime counting function given the moments of the second Chebyshev function
I have read this article (Montgomery and Soundararajan: Primes in short intervals. http://arxiv.org/abs/math/0409258 ). In the second page of the article, it is stated that the mean and variance of $\...
- 141
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votes
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361
views
Kottwitz global gerbes
I've been trying to understand global gerbes constructed by Kottwitz in $B(G)$ for all local and global fields (arXiv:1401.5728). Scholze explained in $p$-adic geometry (arXiv:1712.03708) why I would ...
- 4,246
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97
views
When is $\lfloor C^n \rfloor \mod b$ efficiently computable?
For real irrational $C > 1 $ and natural $n,b$, define
$a(C,n,b)=\lfloor C^n \rfloor \mod b$
Q1 For which $C,b$ is $a(C,n,b)$ computable in time polynomial
in $\log{n}$?
Searching in OEIS ...
- 24.2k
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429
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The irrational numbers α such that n odd and m=⌊nα⌋ odd implies ⌊mα⌋ odd
This post is the analogous of that one (about $\sqrt{2}$) but with a much stronger expectation here.
We observed, and then this comment of Lucia proved, that for $\phi$ the golden ratio, if $n$ ...
4
votes
0
answers
142
views
Can this number be interpreted as a fractal dimension?
Under Goldbach's conjecture, let's denote for a large enough integer $n$ by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\}$ and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n))$.
Let's ...
- 6,932
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votes
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275
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Commutator algorithm
Let $M \in \mathrm{SL}(2, \mathbb{Z}).$ Is there an efficient algorithm to write $M$ as a commutator (group commutator, not algebra commutator) [or fail if this is impossible]?
Addendum: answering ...
- 95.6k
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139
views
Is it true that $|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|=(1-e^{-1})p+O(\sqrt{p})\ ?$
For each prime $p$, let us define
$$w_p:=|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|,$$
where $a\pmod p$ denotes the residue class $a+p\mathbb Z$.
Based on my computation, I conjecture that
$$w_p=...
- 14.5k
4
votes
0
answers
163
views
Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers
The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by
$$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$
and
$$L_0=2,\ L_1=1,\ \text{...
- 14.5k
4
votes
0
answers
193
views
A conjecture on the cardinality of minimal mediated sequences
For a sequence of integer numbers $A=\{0,q_1,\ldots,q_m,p\}$ (arranged from small to large), if every $q_i$ is an average of two distinct numbers in $A$, then we say $A$ is a mediated sequence.
...
- 133
4
votes
0
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123
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On the asymptotic character of a certain sum involving the Mobius function
Let $\mu$ represet the Mobius function. If i recall correctly, i heard a few years ago that $\sum_{n=1}^{\infty} \mu(n)e^{-n/x} =\Omega_{\pm}(x^{1/2})$. Can someone help in proving this, or provide a ...
- 41
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votes
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227
views
Does $\text{Bun}_G$ have the homotopy type of a classifying space in positive characteristic?
In these lecture notes by Jacob Lurie, he identifies the homotopy type of $\text{Bun}_G$ with that of a certain classifying space $B\mathcal{P}_{sm}$ when the group scheme $G$ is over $\mathbb{C}$. ...
- 1,964
4
votes
0
answers
185
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Overview of Combinatorial Technique of "Selberg’s Symmetry Formula"
In the paper entitled "A Computational History of Prime Numbers and Riemann Zeros" (on page 4, click here) it is written about "Selberg’s symmetry formula" that-
Until 1950 it was widely believed (...
