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

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Similar reduced integral matrices

Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example), but if necessary I'm willing to reduce to the case where $d$ is squarefree and $-...
GreginGre's user avatar
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1 vote
0 answers
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Height variation of abelian varieties within an isogeny class

Let $A$ be an abelian variety defined over a number field $K$ of dimension $g \geq 2$, and put $h_F(A)$ for the (stable) Faltings height of $A$. It is well-known from the seminal paper of Faltings ...
Stanley Yao Xiao's user avatar
2 votes
1 answer
211 views

On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$

Let $p$ be an odd prime. For $d\in\mathbb Z$ we define $$T(d,p):=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{0\le i,j\le(p-1)/2},$$ where $(\frac{\cdot}p)$ is the Legendre symbol. By (1.17) of my ...
Zhi-Wei Sun's user avatar
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5 votes
1 answer
419 views

To what extent does the $(\mathfrak{g},K_{\infty})$ module determines the automorphic representation?

In this question, we will follow the notations and definitions from the book "Automorphic Representations and L-Functions for the General Linear Group" by Goldfeld and Hundley. For simplicity, let's ...
Wenzhe's user avatar
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3 votes
3 answers
2k views

Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3)

Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?
Aza's user avatar
  • 31
4 votes
2 answers
1k views

Chebyshev's bias-conjecture and the Riemann Hypothesis

Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?
Dimitris Valianatos's user avatar
3 votes
2 answers
287 views

Algebraic points on a curve with small degree

Let $d \geq 2$ be a positive integer, and let $K_d$ denote the compositum of all fields of degree $d$ over $\mathbb{Q}$. Let $Y$ be an algebraic curve defined over the rationals and has genus $g \...
Stanley Yao Xiao's user avatar
0 votes
0 answers
696 views

On sets of coprime integers in intervals

Briefly, Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval? The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
Gerhard Paseman's user avatar
4 votes
0 answers
186 views

When does a continuous function's "Fourier series" converge pointwise almost everywhere to the function?

Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...
D_S's user avatar
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What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of compact support)?

Let $G$ be a linear algebraic group over a number field $k$. Let $\mathbb A$ denote the adeles of $k$, $\mathbb A_f$ the finite adeles, and $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. Here are ...
D_S's user avatar
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62 votes
5 answers
9k views

Jean Bourgain's relatively lesser known significant contributions

Jean Bourgain passed away on December 22, 2018. A great mathematician is no longer with us. Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions,...
-2 votes
1 answer
75 views

Approximation for $ \inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ by minimizing a distance

Under Goldbach's conjecture, let $r_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ and $k_{0}(n) : =\pi(n+r_{0}(n))-\pi(n-r_{0}(n)) $. The PNT implies that one can expect to have $ \dfrac{...
Sylvain JULIEN's user avatar
12 votes
4 answers
3k views

Prove that $\sum_{n = 1}^{p - 1} n^{p - 1} \equiv (p - 1)! + p \pmod {p^2}$ with $p$ being an odd prime

First, I have to admit that I have already asked the same question on MSE several days ago. If I am bending any rules, I apologize for that and moderator can delete or close this question without ...
Saša's user avatar
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1 answer
299 views

Proper Way To Compute An Upper Bound

I regard to the proof of Lemma 10 in "A remark on a conjecture of Chowla" by M. R. Murty, A. Vatwani, J. Ramanujan Math. Soc., 33, No. 2, 2018, 111-123, the authors used the average value $(\log x)^...
r. t.'s user avatar
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8 votes
1 answer
353 views

Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?

Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by $$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$ In 2004, R. Chapman [Acta ...
Zhi-Wei Sun's user avatar
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6 votes
2 answers
443 views

Name of a group-like structure

The late Vladimir Arnold, in Arnold, V., Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world, Bull. Braz. Math. Soc. (N.S.) 34, No. 1, ...
Thomas Sauvaget's user avatar
2 votes
0 answers
197 views

Point of smallest height on an algebraic curve

Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, and we suppose that $C(K) \ne \emptyset$ ($C$ may very well be defined over a proper subfield of $K$, but perhaps ...
Stanley Yao Xiao's user avatar
2 votes
1 answer
163 views

Approximate sequence of numbers

Let $n \in \mathbb N$ and $k_n \in \left\{0,..,n \right\}$ then we define the numbers $$x_{n,k_n} = \frac{k_n+n^2}{n^3+n^2}.$$ It is easy to see that these numbers satisfy $$x_{n,0} = \frac{1}{n+1} ...
T.Sell's user avatar
  • 21
10 votes
0 answers
252 views

How many partition values are expected to be prime?

Let $p(n)$ be the partition function. Let $P(N)$ count how many $1\leq n\leq N$ are such that $p(n)$ is prime. Are there any heuristics for how $P(N)$ should behave? A crude guess at how this ...
Thomas Bloom's user avatar
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14 votes
0 answers
595 views

Reverse Mathematics of Euclid's theorem

Euclid's theorem that there are infinitely many prime numbers has multiple proofs, ranging from Euclid's original theorem that constructs a new prime from a finite list of such, to Euler's proof that ...
David Roberts's user avatar
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9 votes
2 answers
970 views

Influential results by Swinnerton-Dyer

The conjecture of Birch and Swinnerton-Dyer had a tremendous influence on the development of arithmetic geometry. Which other results of Swinnerton-Dyer have had a lasting influence? [edit, in ...
8 votes
4 answers
780 views

Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$

Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...
T. Amdeberhan's user avatar
1 vote
0 answers
66 views

Probability distribution from standard domain (two primes) - IV

Pick a random pair $(a,b)\in\mathbb Z_n^2\setminus\{0,0\}$. Denote $N_2(a,b,n)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
Turbo's user avatar
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1 vote
1 answer
401 views

How large can $|\zeta(\sigma + it)|$ be for $\sigma<1/2$?

Let $\zeta$ be the Riemann zeta function. My question is: For fixed $\sigma<1/2$, how large can $|\zeta(\sigma+it)|$ be for $t\in \mathbb{R}$, even assuming zeta conjectures like the RH or the LH ?...
etihad10's user avatar
1 vote
0 answers
127 views

Universal elliptic curve over anticanonical tower

While I'm reading Scholze's paper "On torsion in the cohomology of locally symmetric varieties" he constructs the anticanonical tower passing through the construction of an integral model $X_{\infty}$ ...
rime's user avatar
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9 votes
1 answer
495 views

A question about Galois representations

Let $K$ be a number field and $(\rho,V)$, $(\rho',V')$ be two Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/K)$. Suppose that for some positive integer $n$ we have $\mathrm{Sym}^n\rho\...
User1930752648's user avatar
2 votes
1 answer
226 views

Has the "semidirect monoid of a semiring" been considered anywhere?

Given a semiring $S$, we get a monoid $M(S)$ as follows: The underlying set of $S$ is $S^2$ The identity element is $(0,1)$ The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...
goblin GONE's user avatar
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0 votes
1 answer
404 views

Reason Coppersmith fails here?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. $P$ has a binary expansion and so does $Q$. We can set the binary $0/1$ variables to be $x_1$ through $x_{\lceil\log P\rceil}$ and $...
Turbo's user avatar
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-3 votes
1 answer
265 views

Negative Dirichlet Pigeonhole Principle [closed]

From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|_\infty<\lceil\sqrt p\rceil$ holds where ...
Turbo's user avatar
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2 votes
1 answer
413 views

A complementary of the Collatz $3x+1$ problem

Let $\mathbb{N}_{\text{odd}}$ be the set of odd positive integers. For $x_0 \in \mathbb{N}_{\text{odd}}$ consider the set-valued sequence $\{A_n\}_{n=0}^{\infty}$ defined by the formula $$ A_0 = \{...
vassilis papanicolaou's user avatar
2 votes
1 answer
292 views

Homogeneous van der Waerden

The Erdős Discrepancy Problem is whether in any two-coloring of the naturals for any $C$ there is a sequence $d, 2d, \ldots nd$ such that the difference of red and blue numbers in it is more than $C$. ...
domotorp's user avatar
  • 18.3k
3 votes
2 answers
412 views

Prove that there exists a nonempty subset $ I$ of $ \{1,2,...,n\}$ such that $ \sum_{i\in I}{\frac {1}{b_i}}$ is an integer

Let $ a_1,a_2,...,a_n$ and $ b_1,b_2,...,b_n$ be positive integers such that any integer $ x$ satisfies at least one congruence $ x\equiv a_i\pmod {b_i}$ for some $ i$. Prove that there exists a ...
color's user avatar
  • 169
6 votes
1 answer
452 views

Equidistribution of $\{p_n^2α\}$

Let $p_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p_n \alpha\}$ is equidistributed. Is it known whether the sequence $\{p_n^2 \alpha \}$ is ...
zoidberg's user avatar
  • 200
2 votes
1 answer
275 views

Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?

The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that $$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes ...
OneTwoOne's user avatar
  • 105
7 votes
0 answers
205 views

Has this self-similar sequence the ratio $(\sqrt2+1)^2$?

This is inspired by a math.SE question, where an infinite sequence of pairwise distinct natural numbers $a_1=1, a_2, a_3, ...$ has been defined as follows: $a_n$ is the smallest number such that $s_n:=...
Wolfgang's user avatar
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4 votes
1 answer
473 views

A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (III)

As in Question 319254, for an odd prime $p$ and integers $c,d$ we define $$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right),$$ where $(\frac{\cdot}p)$ is the Legendre symbol. In my ...
Zhi-Wei Sun's user avatar
  • 14.4k
1 vote
0 answers
117 views

A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (II)

As in Question 319254, for an odd prime $p$ and integers $c,d$ we let $$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right).$$ If $p\equiv1\pmod4$, then we obviously have \begin{align}&\...
Zhi-Wei Sun's user avatar
  • 14.4k
8 votes
1 answer
168 views

On the existence of a particular type of finite measure on $\mathbb N$

Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
user521337's user avatar
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1 vote
0 answers
169 views

A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (I)

Let $p$ be an odd prime. Here I introduce the sum $$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right)$$ with $c,d\in\mathbb Z$, where $(\frac{\cdot}p)$ is the Legendre symbol. I have a ...
Zhi-Wei Sun's user avatar
  • 14.4k
1 vote
1 answer
139 views

Factoring with partial information on gaps

If $N=PQ$ is a semi-prime with $P=N^{\frac12 +\delta}$ and $Q=N^{\frac12-\delta}$ then if we know $\delta\in(0,\frac12)$ to a reasonable precision we can factor $N$ quickly. What precision (number of ...
Turbo's user avatar
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3 votes
1 answer
181 views

Divergence of a series related to Schinzel's hypothesis H

The Series Consider the series identity $$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$ $$R(n) = \left\...
Liam Eagen's user avatar
5 votes
1 answer
275 views

Counting primitive solutions to a diophantine inequality

This is a refinement (perhaps a simpler version) of a question I asked here before and couldn't get an answer for. Fix $\alpha \in (0,1]$ and a small constant $c>0$. For $x \in [0,1]$ and $N\in\...
Itay's user avatar
  • 539
6 votes
0 answers
247 views

Gaussian square-free moat

Is there a sequence $\{z_n\}_{n=1}^\infty$ of distinct square-free Gaussian integers with $$\sup_{n \geq 1} |z_{n+1} - z_n| < \infty ?$$ For the analogous problem with Gaussian primes instead, ...
Pablo's user avatar
  • 11.2k
6 votes
0 answers
213 views

Divisor bound for $r_2$ off the origin

If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
Rodrigo's user avatar
  • 1,235
13 votes
1 answer
1k views

Is there a substitution that relates every Fermat curve to an elliptic curve?

I asked this question on MSE but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level. A Fermat Curve of degree $n$ is the set of solutions to $x^...
YiFan's user avatar
  • 236
-1 votes
1 answer
240 views

On a certain representation of the Riemann zeta function

Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation $$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
OneTwoOne's user avatar
  • 105
3 votes
1 answer
106 views

Comparing the height of a curve and a singly branched cover

Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, having good reduction outside of a finite set $S$ of primes in $K$. A singly branched cover $C'$ of $C$ is a curve ...
Stanley Yao Xiao's user avatar
4 votes
0 answers
177 views

Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?

Prove or disprove: Claim: there exists a field $K$ and an integer $n$ such that $\pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $GL_n(K)$? Note that then $G_K \...
David Corwin's user avatar
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4 votes
0 answers
133 views

Plancherel measure and dimension

I would like to understand what the Plancherel measure is on the dual of a compact group, in more explicit terms than with an implicit definition. Indeed, to the extent of my knowledge it is defined ...
TheStudent's user avatar
15 votes
2 answers
645 views

A tantalizing Gamma quotient to challenge the Rohrlich-Lang Conjecture

The Rohrlich-Lang Conjecture for polynomial relations in Gamma values predicts that all polynomial relations between Gamma values over $\mathbb Q$ come from the functional equations satisfied by the ...
Wolfgang's user avatar
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