Prime numbers, diophantine equations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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2
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1answer
208 views

Weil Conjectures Analog for Multivariate Zeta Functions

We know that the Riemann zeta function can be generalized to multivariate zeta functions. Is there a multivariate analog of the Weil conjectures?
3
votes
0answers
116 views

How to show that $h(-D)\geq \displaystyle\sum_{a\in A_1\\, 1\leq a\leq{\frac{\sqrt D}{2}}} 1$?

Here $A_1=\{u;p|u\Longrightarrow \chi(p)=1\}$ with $\chi$ a real quadratic character and $h(-D)$ the class number of the imaginary quadratic field of the fundamental discriminant. This problem occurs ...
1
vote
1answer
131 views

Orbital integral for matrix coefficients

I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula $$O_\gamma(f) = \int_G ...
3
votes
1answer
114 views

Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$

I am working my way through the literature regarding the following conjecture: There is a positive integer $k$ such that for all positive integers $d$, $$P(\Phi_d(2))^k \gt \Phi_d(2).$$ I am ...
2
votes
0answers
111 views

Values of Bernoulli polynomials at roots of unity

I am wondering if there are any nice results on the values of Bernoulli polynomials at roots of unity, besides those at 1 or -1.
4
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0answers
93 views

Are there only finitely many near-perfect numbers with more than 4 distinct prime divisors?

Given a positive integer $n$, let $\sigma(n)$ denote the sum of divisors of $n$. We say that a positive integer $n$ is a near-perfect number if $\sigma(n) - 2n$ is greater than $0$ and a proper ...
2
votes
1answer
114 views

Maass form properties and their fourier coefficients

Some Maass form can be written ($K_{iR}$ is the K-Bessel function): $$f(x+iy)=\sum_{n \ne 0}^{\infty} a_n \sqrt{y} \;K_{iR}(2\pi |n| y) \; e^{2 i\pi nx}$$ with the $a_n$ multiplicative, but inversly ...
6
votes
0answers
135 views

Mahler's theorem in the primes

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. Denote by $A_F$ the area of the region $$\displaystyle \{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq 1\}.$$ It ...
11
votes
3answers
562 views

Philosophy behind cohomological representations

For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...
25
votes
3answers
723 views

Does the category of (algebraically closed) fields of characteristic $p$ change when $p$ changes?

Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all homomorphisms as morphisms. The question is: when is there an equivalence of categories between ...
11
votes
5answers
1k views

How much do I need to learn algebraic geometry to understand arithmetics over number fields

I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to ...
1
vote
1answer
173 views

On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.) Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...
0
votes
0answers
178 views

Zero balancing fractions, number theory

Motivation: The main property was found in a paper analyzing networks of the human brain but was likely rooted in the realm of social networks. In a social network the influence of a person/vertex can ...
6
votes
2answers
281 views

Integer solutions of (x+1)(xy+1)=z^3

Consider the equation $$(x+1)(xy+1)=z^3,$$ where $x,y$ and $z$ are positive integers with $x$ and $y$ both at least $2$ (and so $z$ is necessarily at least $3$). For every $z\geq 3$, there exists ...
3
votes
0answers
95 views

Minimization module $p$

Let $p$ be a prime number, let $m$ be a fixed number (for example $2^{20}$), and $i=k^{-1}\cdot (j-t) \pmod{p}$ where $j \leq m$ and $m<k$. In the general case we have $t=0$, and $k,j$ are ...
2
votes
1answer
152 views

Applications of Level Lowering

What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
7
votes
2answers
381 views

Famous results about the value of a given limit assuming it exists

Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
3
votes
0answers
58 views

Log-concavity of difference of theta functions

My knowledge on theta functions is limited, but I suspect that this is a quite challenging question. The 3rd Jacobian Theta function is given by \begin{equation} ...
2
votes
1answer
137 views

How many monoids with $n$ arrows exist?

How many monoids with strictly $n$ arrows exist? Is this known? I ask this only out of curiosity. Looking at $n=1,2,3,4$, this number doesn't appear to be very large relative to $n$.
1
vote
0answers
99 views

The Galois side of the Norm map

Let $K$ be an abelian extension of $\mathbb{Q}$. We know that $[x, K]|_{\mathrm{Gal}{\mathbb{Q}^{ab}}}=[\mathrm{N}^{K}_{\mathbb{Q}} x, \mathbb{Q}]$ where $[x, F]$ is the Artin reciprocity map. Given a ...
5
votes
1answer
192 views

A question regarding lines on a cubic surface

Let $X$ be a smooth cubic surface in $\mathbb{P}^3$. It is a classical theorem of Cayley and Salmon that $X$ contains exactly 27 lines over an algebraically closed field. In 2002, Heath-Brown proved ...
20
votes
1answer
1k views

How many primes can there be in a short interval?

Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$. What is $$ \limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} ...
1
vote
0answers
74 views

Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$

I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when: (1.) $q=p$ and/or (2.) $E$ has multiplicative reduction at $q$. Here, $E$ is an ...
3
votes
0answers
113 views

Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
2
votes
1answer
149 views

theta functions and Brownian motion

I did some plots of the theta function $\theta(z) = \sum q^{n^2}$ near the real axis, so $q = e^{2\pi i \, n z}$ and $z = 0.001 + i \mathbb{R}$. At first it looks like some random sine curve and then ...
16
votes
2answers
609 views

What is the smallest x such that [x^n] has the same parity as n?

A previous MO question asked for information about a number $x$ such that $[x^n]$ has the same parity as $n$ for all positive $n$. Answers to this post included two values of x which meet the ...
1
vote
0answers
94 views

Expressing Numbers with a Minimal Sum in Powers of 2 [closed]

The first 64 bits of pi are: 11.00100100001111110110101010001000100001011010001100001000110100 Computer multiplication can be sped up by looking for patterns and ...
4
votes
1answer
246 views

Do there exist infinitely many $n$ such that $n^3+an+b$ is squarefree?

Question: Assume that $a,b\in Z$, and $4a^3+27b^2\neq 0$. Prove that there exist infinitely many positive integers $n$ such $n^3+an+b$ is square-free. I have following There exist infinitely ...
18
votes
1answer
721 views

Sphere packings : what next after the recent breakthrough of Viazovska (et al.)?

Given the march 2016 breakthrough concerning sphere packings by Viazovska for the case of dimension 8, and by Cohn, Kumar, Miller, Radchenko and Viazovska for the case of dimension 24, it follows that ...
15
votes
1answer
639 views

An equality involving roots of unity which holds most of the times, but not always

Let $m$ and $n$ be distinct odd positive integers. The equality $$ \prod_{k=0}^{mn-1} \left( e^{\frac{2\pi i k}{m}} + e^{\frac{2\pi i k}{n}} \right) \ = \ 2^{\gcd(m,n)} ...
4
votes
2answers
103 views

Sturmian subword whose reverse is not a subword

Let ${\cal L}_n$ be the set of all subwords of length $n$ of a biinfinite Sturmian sequence, induced by a rotation coding with irrational angle $\theta$. Take a word $w \in {\cal L}_{2^n}$ and write ...
7
votes
0answers
349 views

Field of definition of a point in $[p]^{-1}E(K)$

Let $E$ be an ordinary elliptic curve defined over a non-perfect field $K$ of characteristic $p$. If $P \in E(K)$ satisfies $P \not\in [p]E(K)$, is it true that its $p^m$-division points of $P$ are ...
2
votes
0answers
59 views

Carlitz factorials and Euler-like series

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write ...
5
votes
2answers
166 views

How to construct particular De Bruijn sequences

For $n \ge 2$, there is at least one binary DeBruijn sequence beginning with $n$ zeros followed by $n$ ones. Is there a straightforward way to construct such a sequence for each $n \ge 2$? Examples: ...
2
votes
0answers
43 views

Functional equations about Conway's box function

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). The ...
6
votes
3answers
373 views

Tricky two-dimensional recurrence relation

I would like to obtain a closed form for the recurrence relation $$a_{0,0} = 1,~~~~a_{0,m+1} = 0\\a_{n+1,0} = 2 + \frac 1 2 \cdot(a_{n,0} + a_{n,1})\\a_{n+1,m+1} = \frac 1 2 \cdot (a_{n,m} + ...
16
votes
2answers
729 views

Special topics to include in course in algebraic number theory

I'll be teaching an introductory course in algebraic number theory this fall (stopping before class field theory). I'm looking for a good list of "special topics" I can include to illustrate the ...
2
votes
0answers
189 views

Quantum Mechanics derivation of Wallis' Formula?

Recently there was a proof of the Wallis Product using quantum mechanics on the arXiv. However, there are many proofs of the result, Wikipedia has 4. Fine Print the first proof has on Wikipedia, the ...
-1
votes
1answer
62 views

Equation with norms of cyclic extensions of coprime degrees

Let $\mathbb{K}$ be a quadratic extension of $\mathbb{Q}$ and $\mathbb{L}$ be a cyclic extension of $\mathbb{Q}$ of odd degree. Given a rational $r\neq 0$, does there always exist $k\in \mathbb{K}^*$ ...
1
vote
1answer
143 views

Local nontriviality of genus-one curves over extensions of degree dividing $6^n$

Suppose $p\geq 5$ is a prime, and $C$ a genus-one curve, defined over $\mathbf{Q}$. Is there always an extension $K/\mathbf{Q}_{p}$ whose degree divides a power of $6$, so that $C(K)$ is not empty? (I ...
11
votes
2answers
332 views

Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix?

Let $A,B$ be two rational rotations: $$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\ -\frac{4}{5} & \frac{3}{5} & 0 \\ 0 & 0 & 1 \end{array}\right] ...
-1
votes
1answer
128 views

Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$th ...
0
votes
2answers
160 views

Reference : Partition of integer

In algebraic number theory we come across following formula: $n= e_1f_1+\cdots+e_rf_r$ where all $e_i$ and $f_i$ are positive integers. I am sure writing a positive integer n as above must be ...
4
votes
1answer
239 views

The values of $n$ which satisfy an inequality about prime numbers

For which values of $n$ does the following inequality hold for? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the ...
7
votes
0answers
198 views

Does Stepanov's method extend to complete intersections?

Stepanov (circa 1970) created the polynomial method to limit the rational points of an algebraic curve over $\mathbb{F}_q$, leading to one of several alternative proofs of Weil's Riemann hypothesis ...
6
votes
2answers
226 views

What are the special parahoric subgroups in unitary groups?

Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special ...
0
votes
1answer
162 views

How to calculate $N_{L/k}$(roots of unity)?

Suppose that $L/k$ is a Galois extension of number fields and that $G$ is the corresponding Galois group. Further, for $\frak p$ a prime ideal of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : ...
5
votes
2answers
142 views

Minimum length of a convex lattice polygon containing k lattice points?

Let $f(k)$ denote the minimum length of a convex lattice polygon containing exactly $k$ lattice points (including lattice points on the boundary). It is not too hard to show that $k = \frac{1}{4\pi} ...
3
votes
0answers
148 views

Computing algebraic properties of trace fields, as given by SnapPy

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting: the minimal polynomial of the field over $\mathbb{Q}$, and a decimal ...
2
votes
1answer
173 views

Reduction of Abelian Varieties with Complex Multiplication have Complex Multiplication

Let $A$ be an abelian variety of dimension $g$ over $C$ with complex multiplication by a CM field $K$ where $[K:Q] =2g$. By this I mean that End($A$) $\cong \mathcal{O}_K$. Then, $A$ has a model over ...