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

Filter by
Sorted by
Tagged with
6 votes
0 answers
662 views

Show this number always is composite number

Conjecture: Let $m$ be a positive integer. Then $$f(m)=(2m)^{2m+1}+m^{2m+1}\cdot (2m+1)^m+(2m+1)^{2m}$$ is not a prime number. One can prove it when $m$ is odd number, it is clear that $f(m)$ ...
math110's user avatar
  • 4,230
5 votes
0 answers
353 views

Is there a relationship between the zeta function of a Laplacian and the Selberg Zeta Function?

Let us define the zeta function of an elliptic differential operator $H$ with eigenvalues $\lambda_n$ like so: \begin{aligned} \zeta_H(s) := tr( H^{-s} ) \\ := \sum^\infty_{n=1} ...
Nico A's user avatar
  • 447
4 votes
0 answers
205 views

Explicit elements of the first cohomology of modular curves

Let $M$ be a modular curve and $\pi:E\to M$ the universal elliptic curve. For a prime $\ell$, let $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$. I am wondering whether there are any explicit ...
User0829's user avatar
  • 1,378
4 votes
1 answer
445 views

Moments of merit

The merit of a prime gap equals $(p_{n+1}-p_n)/\ln p_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very ...
David Feldman's user avatar
7 votes
1 answer
355 views

index of smooth varieties

What are the simplest examples of smooth, projective varieties defined over the fraction field of an Henselian DVR of characteristic $0$ which have index $>1$? EDIT: Also assume that the residue ...
user43198's user avatar
  • 1,949
4 votes
2 answers
887 views

Frobenius actions on de Rham cohomology, clarify questions on a paper of Kedlaya

I am reading the paper "$p$-adic cohomology: from theory to practice" by K. S. Kedlaya. I have several naive questions about section 2: Frobenius action on de Rham cohomology. As a physicist, I lack ...
Wenzhe's user avatar
  • 2,961
3 votes
1 answer
384 views

Moduli problem of stable nodal curves over the integers

Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}$ represents the functor $$\overline{\mathcal{M}}_{g,n}(...
Dmitry Vaintrob's user avatar
2 votes
0 answers
166 views

Möbius function in every short interval

I am looking for references on conjectures or heuristics concerning cancellations of the Möbius function in very short intervals, namely how small is it believed one can take $H$ so that $$\sum_{X \...
Rodrigo's user avatar
  • 1,235
9 votes
1 answer
458 views

Summing moments and Riemann zeta values

Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments $$\alpha_n=\frac{\int_0^{\pi/2}x^4d\mu_n(x)}{\int_0^{\pi/2}d\mu_n(x)}.$$ Then, I have encountered a curious evaluation $$\sum_{n=1}^{...
T. Amdeberhan's user avatar
8 votes
2 answers
938 views

How to compute Dedekind eta function efficiently?

According to wiki: https://en.wikipedia.org/wiki/Dedekind_eta_function, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic ...
Licheng Wang's user avatar
3 votes
1 answer
418 views

The degree of the cube root of the $j$-invariant

I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be ...
Shimrod's user avatar
  • 2,335
17 votes
1 answer
748 views

Congruences Ramanujan-style

Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by $$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$ The numbers $p_t(n)$ can be regarded as enumerating ...
T. Amdeberhan's user avatar
3 votes
1 answer
167 views

Diophantine equations and 'quasi-paucity'

Let $X,Y \geq 1$. I am interested in the number of solutions of the following diophantine equations: $$S_1\colon \, \, x_1y_1^3 = x_2 y_2^3 $$ Let $N_1(X,Y) $ denote the number of solutions to $S_1$ ...
leithian's user avatar
  • 163
1 vote
1 answer
137 views

How to compute the Müller modular polynomials?

According to R.A.Kazmi's dissertation "Isogenies and Cryptography" (Page 22), given an isogeny degree $l$, the Müller modular polynomials are defined as $$G_l(x,y)=\sum_{r=0}^{l+1}\sum_{k=0}^{v}a_{...
Licheng Wang's user avatar
15 votes
3 answers
1k views

Perfect numbers and perfect powers

This was asked earlier at MSE. The observation that 28 = 27 + 1 shows that it is possible to have consecutive perfect numbers and perfect powers. However, this must be extremely rare. Is it ...
user2052's user avatar
  • 1,401
0 votes
1 answer
140 views

Searching for matrices with some property

I don't know if this question is considered research-related. If not, I will move it to Math SE. I am searching for matrices with the property $$|A|_F^2 = \deg( \chi_A(t) ) = 2 \deg( m_A(t)), tr(A) ...
user avatar
1 vote
1 answer
308 views

A product of polynomials

Let $f(n)=1+x^n+x^{2n}+...+x^{n^2}.$ Let $p(x)$ be $1+x+x^2+x^5+x^7+...$ where the exponents are the pentagonal numbers. Let $a(n)$ be the sequence of integers such that the coefficients of the ...
David S. Newman's user avatar
4 votes
0 answers
190 views

Can one define certain 'interesting' topology for $\overline{\mathbb{Q}}$ on which the absolute Galois group $G_\mathbb{Q}$ acts continuously?

The story can be generalized to the case of projective variety, however, we consider the most basic case. Fix an embeding (i.e.'complex place') $\iota:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$, ...
Bonbon's user avatar
  • 806
3 votes
0 answers
187 views

Growth Rate of the Square-Free Part

In the course of considering this Diophantine equation, I convinced myself that the following question is interesting: If $n$ is large, must it be the case that the square-free part of $2^n-1$ is ...
Richard Voepel's user avatar
2 votes
0 answers
114 views

Sieving the values of an arithmetic sequence which is infinitely many times $1$

I have a sequence of positive integers $a_n$ which assumes infinitely many times the value $1$. I want to estimate the cardinality of the following set: $$\#\{n\leq x : a_n>1 \text{ and } (a_n, \...
The Number Theorist's user avatar
5 votes
0 answers
113 views

$m$-thick sets with small $n$-fold sumsets in finite cyclic groups

Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties: $(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...
Taras Banakh's user avatar
  • 40.8k
9 votes
2 answers
362 views

Consecutive polynomial non-residues modulo a prime

Given a polynomial $f(x)\in \mathbb{Z}[x]$ of degree at least 2 and a positive integer $t$, does there always exist infinitely many primes $p$ such that the range of $f(x)$ modulo $p$ does not contain ...
Marco's user avatar
  • 537
4 votes
0 answers
189 views

Restricted Iwasawa theory

Let $p$ be a prime number, let $K$ be a number field, and let $L$ be a Galois extension of $K$ such that the Galois group $\Gamma$ of $L$ over $K$ is (continuously) isomorphic to the (additive) group $...
Pablo's user avatar
  • 11.2k
5 votes
1 answer
212 views

Positive polynomial equations mod integers

Let $f(x)$ and $g(x)$ be polynomials with integer coefficients such that $f(x)>0$ and $g(x)>0$ for all real values of $x$. Suppose that for every integer $n$, if $f(x)=0$ has a solution mod $n$, ...
Marco's user avatar
  • 537
18 votes
1 answer
674 views

Arithmetic motivations for modularity in higher rank

The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching ...
Kimball's user avatar
  • 5,709
2 votes
2 answers
1k views

What is the natural density of hyper prime numbers?

What do we mean by hyper prime numbers? Well, roughly speaking they are natural numbers which are prime with respect to hyperoperators in arithmetic such as exponentiation, tetration, pentation, et ...
Morteza Azad's user avatar
6 votes
0 answers
681 views

What are the fastest ways to calculate class number of number fields?

Given a number field $K$, which approaches help us to calculate the class number $h(K)$ of $K$? I am aware that the question is broad but any argument would be helpful. Some basic approaches I know:...
Ninja's user avatar
  • 161
3 votes
1 answer
230 views

Questions about a product of trinomials

Let $f(n)=1+x^n+x^{2n}$ Let $p(x)$ be $1+x+x^2+x^5+x^7+...$ where the exponents are the pentagonal numbers. Let $a(n)$ be the sequence of integers such that the coefficients of the series $f(a(1)) f(...
David S. Newman's user avatar
3 votes
1 answer
292 views

Degeneracy maps and cusps

Let $N$ be a positive integer and $p$ a prime not dividing $N$. Let $X_0(N)$ be the modular curve (over $\mathbb{Q}$) associated to the congruence subgroup $\Gamma_0(N)$, which is the subgroup of $\...
user116950's user avatar
5 votes
2 answers
429 views

Is there any work on the Gauss circle problem over function fields? [closed]

I would be thankful if someone had references to provide...
Dr. Pi's user avatar
  • 2,939
19 votes
1 answer
883 views

What computer program for automorphic forms

This question has its origins in this entertaining discussion on MO. There are many programs (CAS) and libraries that are able to handle algebraic expressions. These are both a verification tool for (...
Desiderius Severus's user avatar
3 votes
0 answers
164 views

$\text{PGL}_n(\mathbf{Q}_p)$ and the Congruence Subgroup Property

Suppose $\Gamma$ is a torsion-free lattice of $\text{PGL}_n(\mathbf{Q}_p)$ for $n\geq 3$. Then I know that by the Margulis arithmeticity theorem, $\Gamma$ must be arithmetic. My question is does $\...
user avatar
-1 votes
1 answer
115 views

probability of m to be a primality radius of n

Disclaimer : this is a crosspost from MSE, as the question got one upvote but no comment or answer whatsoever. Assuming Goldbach's conjecture, let's denote by $r_{ 0}(n):=\inf\{r\geq 0,(n−r,n+r)\in\...
Sylvain JULIEN's user avatar
8 votes
1 answer
502 views

Computational complexity of finding the class number

Let $f(x)$ be an integral irreducible monic polynomial and $\alpha$ be its root. What is the computational complexity of finding representatives of ideal classes of the integral ring $\mathbb{Q}[\...
user avatar
4 votes
1 answer
231 views

Meromorphic continuation of local zeta integrals

Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi_0$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{f} =\langle ...
m.s's user avatar
  • 163
8 votes
2 answers
336 views

Let $f \in \mathbb{Z}[x]$. Does $\bar{f}$ have as many roots in $\mathbb{F}_p$ as $f$ has in $\mathbb{C}$ for infinitely many primes $p$?

Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}_p[x].$ Let $\rho_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}_p$, and let $\rho$ be the number ...
Andrew James Kelley's user avatar
5 votes
0 answers
124 views

How good are these probabilistic algorithms for the NP-hard problem gcd of sparse polynomials?

The paper NEW NP-HARD AND NP-COMPLETE POLYNOMIAL AND INTEGER DIVISIBILITY PROBLEMS David A. PLAISTED” defines sparse polynomial as set $\{(a_i,i)\}$ and $f=\sum a_i x^i$. On p.5: Theorem 3.3. The ...
joro's user avatar
  • 24.2k
2 votes
1 answer
143 views

Difference of powers

Prove or disprove: Any positive integer N can be expressed as N=a^x-b^y with a,b,x,y >1. (I don't know the answer or whether this conjecture has been addressed.)
user123120's user avatar
7 votes
1 answer
269 views

Why does the class equation have rational coefficients?

Let $j$ be the Klein modular invariant and let $\Gamma=SL_2(\mathbb Z)$ be the modular group. Consider the set of primitive quadratic forms $ax^2+bxy+cy^2$ of discriminant $d<0$. The root of a ...
Shimrod's user avatar
  • 2,335
1 vote
0 answers
108 views

Complex multiplication of Jacobians of quadratic twists of a fixed hyperelliptic curve

Let $C: z^2 = f(x,y)$ be a hyperelliptic curve defined over $\mathbb{Q}$, with $f$ a binary form with integer coefficients and non-zero discriminant. Let $A_C$ denote the Jacobian variety of $C$, and ...
Stanley Yao Xiao's user avatar
5 votes
0 answers
432 views

Why does Faltings' Siegel lemma imply Siegel lemma?

Recall the Siegel lemma: Let $A = (a_{ij})$ be an $N \times M$ matrix with rational integer coefficients. Put $a = \max_{i,j} |a_{ij}|$. Then, if $N < M$, the equation $Ax = 0$ has a solution $x\in\...
joaopa's user avatar
  • 3,655
7 votes
1 answer
638 views

How is the Eichler-Shimura congruence related to L-functions?

My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T_p$ in terms of the geometric Frobenius map. Specifically, $T_p = Frob + Ver$ for Frobenius map $Frob$ and it's ...
Nico A's user avatar
  • 447
5 votes
1 answer
519 views

Few questions regarding Heath-Brown's identity

Heath-Brown's identity states: Let $K \geq 1, z \geq 1.$ Then for any $n < 2 z^K$ we have $$ \Lambda(n) = - \sum_{1 \leq k \leq K} (-1)^k {{K}\choose{k}} \sum_{ \substack{ m_1 \cdots m_k n_1 \cdots ...
Johnny T.'s user avatar
  • 3,547
12 votes
0 answers
309 views

Writing integers as a product of as few elements of $\{\frac21, \frac32, \frac43, \frac54, \ldots\}$ as possible

Is the number of elements we need to construct $x$ equal to $\log_2(x) + O(1)$? This question is inspired by question 2 of the 2018 European Girls' Mathematical Olympiad. I previously posted it on ...
user133281's user avatar
8 votes
1 answer
764 views

Tate modules of commutative group schemes over finite field

Let $G$ be a finite type commutative group scheme over a finite field $k=\Bbb F_q$ with $\Gamma=Gal(\bar k/k)$ and $l$ be a prime such that $(l,q)=1$, we can also define the Tate module $T_lG =\...
sawdada's user avatar
  • 6,148
12 votes
2 answers
989 views

Polynomials dense with primes

Let $p(n)$ be a polynomial with integer coefficients. Define $\Delta( p(n) )$, the prime density of $p(n)$, to be the limit of the ratio with respect to $n$ of the number of primes $p(k)$ generated ...
Joseph O'Rourke's user avatar
5 votes
1 answer
457 views

Growth order of numbers whose prime factors are all congruent to +1 or -1 modulo 8

Here are the numbers whose prime factors all congruent to $\pm 1\pmod 8$: http://oeis.org/A058529 My questions are: (1) What is the order of growth of these numbers? That is, what is the order of ...
Haoran Chen's user avatar
8 votes
0 answers
295 views

Finding a cyclic cubic extension of a field

Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...
thierry stulemeijer's user avatar
15 votes
1 answer
530 views

Counting primitive lattice points

In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result): Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then $$\# ...
Daniel Loughran's user avatar
10 votes
1 answer
483 views

Estimating volumes in the trace formula

Let $G$ be a reductive group. In many instances of the trace formula, elliptic terms corresponding to the $G(k)$-conjugacy class of $\gamma \in G(k)$ are weighted by the following volumes $$v_\gamma = ...
user123519's user avatar

1
125 126
127
128 129
319