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

3
votes
0answers
108 views

holomorphic continuation of motivic $L$-functions

The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...
2
votes
2answers
121 views

Is it possible to find the maximum value of S?

Let $a_1$, $a_2$, …, $a_n$ and $b_1$, $b_2$, …, $b_n$ be $2n$ strictly positive integers not greater than $M$, with $M$ is a given positive integer, such that $$a_1+ a_2+ \dotsb+ a_n=b_1+ b_2+ \dotsb+ ...
26
votes
2answers
1k views

Lagrange four squares theorem

Lagrange's four square theorem states that every non-negative integer is a sum of squares of four non-negative integers. Suppose $X$ is a subset of non-negative integers with the same property, that ...
2
votes
1answer
131 views

The largest number $y$ such that $(x!)^{x+y}|(x^2)!$

Since the multiplication of $n$ consecutive integers is divided by $n!$, then $(n!)^n|(n^2)!$ with $n$ is a positive integer. Are there any formula of the function $y=f(x)$ that shows the largest ...
1
vote
1answer
83 views

$\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$

The question is: N is an even positive integer, then $\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$. I thought the terms on the left are the solutions set of some polynomial ...
-4
votes
0answers
51 views

Does anyone recognize this sum? [on hold]

$\sum_{s=0}^n{x^sy^{n-s}}$ Does anyone know what this evaluates in the case $x, y \in N$, where can I read more about it?
2
votes
1answer
134 views

Are there infinitely many integers $m$ such that $a+m$ divides $a^m+1$?

Let $a$ be a positive integer. Does there exist a positive integer $m$ other than 1 such that: $$a+m \ |\ a^m+1 \ $$ If not, what are the conditions of $a$ for $m$ to exist? If there exist ...
13
votes
2answers
435 views

Does the equation $(xy+1)(xy+x+2)=n^2$ have a positive integer solution?

Does there exist a positive integral solution $(x, y, n)$ to $(xy+1)(xy+x+2)=n^2$? If there doesn't, how does one prove that?
2
votes
0answers
75 views

Rational points on a weighted projective surface

The equation $$\displaystyle y^2 = f(x_1, x_2, x_3)$$ with $f$ a non-singular quartic form in three variables defines a del Pezzo surface of degree 2. I am interested in the similar construction $$\...
4
votes
0answers
90 views

Congruence for the product of quadratic residues + the product of quadratic non-residues

My question has been here on MSE for a long time, but it has not received a full answer. I bring it here: Find a prime $p$ such that $p \equiv 1 \bmod 4$ and such that the product in the range $[...
0
votes
1answer
418 views

Two reasons why the Collatz conjecture could fail

Let $\mathbb{N}$ denote the set of positive integers. The Collatz function $f:\mathbb{N}\to\mathbb{N}$ is given by $f(n) = n/2$ for $n$ even and $f(n) = 3n+1$ for $n$ odd. Given $k\in\mathbb{N}$ we ...
1
vote
0answers
58 views

Local factors determine Weil representations - proof of the Artin representation case

This post can be seen as a continuation of this post I created on MathOverflow. I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
4
votes
0answers
62 views

Local properties of Galois representations attached to torsion classes

$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}}$ Let $F$ be a number field, and let $\Gamma$ be a congruence subgroup of $\...
2
votes
1answer
89 views

A question about distribution of fractional part of $2^k\alpha$

Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$. The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...
4
votes
0answers
42 views

Approximation of an irrational point from a given direction

Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...
3
votes
1answer
104 views

Density of numbers $n$ which are co-prime with their $\phi$-value

Let $n$ be a positive integer. The Euler $\phi$-function is defined by $$\displaystyle \phi(n) = \# \{1 \leq a \leq n-1 : \gcd(a,n) = 1\}.$$ It is in fact a multiplicative function, and one has the ...
2
votes
1answer
117 views

Decide if a system of arithmetic sequences is an $m$-cover of $\mathbb{N}$

Let $A = \{ a_i + b_i \mathbb{N} \}_{i=1}^{k}$, where $a_1, \ldots, a_k \in \mathbb{N} \cup \{0\}$ and $b_1, \ldots, b_k \in \mathbb{N}$ be a system of arithmetic sequences. For a positive integer $m$...
0
votes
0answers
78 views

gcd of polynomial values

Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
3
votes
1answer
251 views

Vandermonde determinant: modulo

There is a lot of fascination with the Vandermonde determinant and for many good reasons and purposes. My current quest is more of number-theoretic. QUESTION. Let $p\equiv 3$ (mod $4$) be a prime. ...
9
votes
2answers
195 views

Is there a clear-cut analogue of the strong form of Serre's Conjecture for residually reducible Galois Representations?

Let $p$ be a prime and $\mathbb{F}$ a finite field of characteristic $p$. The theorem of Khare and Wintenberger roughly states that an irreducible, odd Galois representation $\bar{\rho}:G_{\mathbb{Q}}\...
0
votes
0answers
47 views

Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)

(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here) ...
6
votes
2answers
1k views

Does this multiplicative function have a name? If so, what is known about it?

It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. ...
5
votes
0answers
189 views
+50

Irrationality of the values of the prime zeta function

Preamble: I asked this question on Math.SE with a bounty of 100, but received no replies. The bounty has now ended, and it was suggested I post this here instead. Since Apéry we know that $\zeta(3)$, ...
5
votes
0answers
51 views

Structure of invariant lattices and reductions of group representations with $\text{dim}>2$

Let $G$ be a finite group, $K$ be a $p$-adic field with an uniformizer $\pi$ and residue field $k \cong \Bbb F_q$, and $V$ be an irreducible representation of $G$ over $K$. Consider $X_{V}^G$ the set ...
2
votes
1answer
114 views

The number of numbers no greater than n that are divisible by all their suffixes

My question: what a formula for finding the number of numbers no greater than n that are divisible by all their suffixes. e.g: 5, 25, 125, 0125, 70125 are divisors of 70125. refinement: $\overline{0....
1
vote
0answers
28 views

Mean value estimates for general number fields

Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for ...
2
votes
1answer
206 views

Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?

Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...
0
votes
0answers
159 views

On the connection between Faulhaber's formula and identity $n^{2m+1}=\sum_{k=0}^{n-1}\sum_{j=0}^m A_{m,j}k^j(n-k)^j$

This question is part of series of the questions, as follows: Coefficients in the sum $\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j=n^{2m+1}$, Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\...
4
votes
1answer
178 views

Is total degree version and $x,y$ degree version of Coppersmith's theorem correct?

The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...
8
votes
2answers
275 views

Find an integer coprime to sequence sum

Given two positive integers $a<b$, can we always find an integer $c\in [a, b]$ that is coprime to $\sum_{a\le i\le b} i$?
6
votes
1answer
222 views

The outer automorphism of the dihedral group $D_4$ and quartic polynomials

Let $L$ be a dihedral quartic field. That is, we have $[L : \mathbb{Q}] = 4$ and the Galois closure $M$ of $L$ is a degree 8 extension of $\mathbb{Q}$ with Galois group isomorphic to the dihedral ...
7
votes
1answer
252 views

Relation between Fourier coefficients and Satake parameters

Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part: $$L(s) =...
3
votes
1answer
164 views

Are there Galois representations associated with any regular algebraic cuspidal automorphic representation?

Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $\mathrm{GL}_2(\mathbb{A}_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ ...
9
votes
2answers
381 views

Lower density of numbers not summable by consecutive integers

Let us call a positive integer $n\in\mathbb{N}$ consecutively summable if there are positive integers $m, k < n$ such that $$n=\sum_{i=0}^k (m+i).$$For $A\subseteq \mathbb{N}$ we set the lower ...
4
votes
0answers
151 views

Two congruence conjectures modulo prime p

How to prove the following two congruences? Question1: Let $p\equiv 1 \pmod 3$ be a prime, then $$\sum_{k=0\atop k\neq(p-1)/3}^{(p-1)/2}\frac{\binom{2k}k}{3k+1}\equiv 0 \pmod p.$$ ...
3
votes
0answers
79 views

Remainder term in an integral linked to the Riemann zeta function

Sorry if this is not research level, but the following problem occurs in my own research: it is trivial to show that for $k\ge2$ integral we have $\zeta(k)=(1/(k-1)!)\int_0^\infty t^{k-1}/(e^t-1)\,dt$ ...
7
votes
3answers
596 views

Counting configurations on a 2xn board under restrictions [closed]

Find the number of ways of selecting k cells from a $(2\times n)$-board such that no two selected cells share a side (non-adjacent). For $n=3$ and $k=2$, the answer is $8$; for $n=5$ and $k=3$, the ...
1
vote
0answers
65 views

Dimension sum “rules” in Lie algebras [on hold]

tr;dr intro: I came up with this question when I couldn't remember how many terms are in an $E_7$-ish (representing $\bigotimes$ adjoint) clebsch. Tried it on $G_2$, $7 \bigotimes 14=7+...$ argh, is ...
2
votes
0answers
137 views

Is a reductive group scheme always parahoric?

Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
7
votes
1answer
154 views

For which primes does this iterated function act transitively? (Sort of a finite analogue of Collatz conjecture.)

Background: I was trying to prove something having to do with cyclic group actions on matroids and was able to show that what I want holds if a particular elementary-looking number-theoretic property ...
3
votes
0answers
59 views

Distribution of 'square classes' of binary quadratic forms

Let $f$ be a binary quadratic form with integer coefficients and non-zero discriminant $D$. Suppose for simplicity that $D$ is a fundamental discriminant (which in particular implies that $f$ is ...
2
votes
0answers
41 views

Is there a result that connects the discriminant of an order to the discriminants of the generators of an integral basis for the order?

Let $\left\{ 1 , \omega_1, \dots , \omega_{n-1} \right\}$ be an integral basis for an order $\mathcal{O}$ of a number field of degree $n$ over $\mathbb{Q}$, let $\Delta $ be the discriminant of $\...
3
votes
1answer
143 views

$\psi(2,1/6),\psi(4,1/6)$ in terms of zeta and pi only and another closed form for zeta

Let $\psi(n,x)$ denote the polygamma function. In this answer Lucia gave linear relations for $\psi(m,1/3),\psi(m,1/6),\zeta(m+1)$. The computer managed to find closed form for $\psi(2,1/6)$ and $\...
5
votes
0answers
95 views

Explicit algebraic constructions of Parshin covers

Let $X$ be an algebraic curve defined over a number field $K$ and has genus $g \geq 2$. Let $P$ be a $K$-point of $X$. We say that $X_P$ is a Parshin cover of $X$ if $X_P$ is defined over a finite ...
1
vote
1answer
579 views

Is every prime greater than 5, less than the sum of the two previous primes?

Can one prove by elementary means (such as Paul Erdös' proof of Bertrand's Postulate) that every prime greater than 5 is less than the sum of the two primes immediately preceding it?
6
votes
1answer
252 views

$\pi$ in terms of polygamma

The computer found this, but couldn't prove it. Let $\psi(n,x)$ denote the polygamma function. With precision 500 decimal digits we have: $$ \pi^2 = \frac{1}{4}(15 \psi(1, \frac13) - 3 \psi(1, \...
0
votes
0answers
113 views

Typical size of $\infty$ norm of integer points in subspaces associated to a structured linear diophantine equation

Take natural numbers $A_1,B_1,A_2,B_2$ random pairwise coprime in $[n,2n]$ for $n$ large enough and consider the space of solutions to $A_1a+B_1b=0$ and $A_1A_2a+A_1B_2b+B_1A_2c+B_1B_2d=0$ spanned by $...
2
votes
0answers
81 views

Conjectural bound on gaps between values assumed by quadratic forms

Let $D$ be a discriminant, i.e., $D \equiv 1 \pmod{4}$ or $D \equiv 0 \pmod{4}$. Let $\mathcal{S}(D)$ be the set of positive integers for which there exists a binary quadratic form $f$ with integer ...
17
votes
4answers
450 views

In which cyclic cubic number fields does there exist this type of unit?

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$. Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/...
4
votes
0answers
101 views

On the parity of $|\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ \ j(j+1)\ \text{mod}\ p\,>\,k(k+1)\ \text{mod}\ p\}|$ with $p$ prime

Let $p=2n+1$ be an odd prime, and let $a_1<\ldots<a_{n}$ be all the quadratic residues mod $p$ among $1,\ldots,p-1$. For $a\in\mathbb Z$ let $\{a\}_p$ be the least nonnegative residue of $a$ ...