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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
votes
2answers
616 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)?
3
votes
2answers
170 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 \...
1
vote
0answers
120 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 ...
4
votes
0answers
112 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 ...
6
votes
0answers
95 views

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 ...
46
votes
5answers
4k views

Jean Bourgain's Relatively Lesser Known Significant Contributions

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, such as the proof of Vinogradov's conjecture ...
-2
votes
1answer
56 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{...
11
votes
3answers
355 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 ...
4
votes
1answer
244 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)^...
6
votes
0answers
206 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 ...
5
votes
2answers
401 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, ...
-4
votes
0answers
145 views

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

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. We define the determinant $D(d,p)$ by $$D(d,p):=\det\left[(i^2+dj^2)\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}...
2
votes
0answers
161 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 ...
2
votes
1answer
85 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} ...
9
votes
0answers
188 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 ...
9
votes
0answers
350 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 ...
9
votes
2answers
710 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 ...
7
votes
4answers
657 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 ...
-1
votes
0answers
48 views

Probability distribution from standard domain (multiple pairs single prime) - V

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. 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(...
1
vote
0answers
61 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,...
1
vote
1answer
258 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 ?...
1
vote
0answers
91 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 $X_{\infty}$ ...
0
votes
0answers
73 views

Probabilistic Howgrave-Graham Bounds with Coppersmith Technique?

Howgrave-Graham condition says roots of $$f(x_1,\dots,x_n)\equiv0\bmod R$$ at an $R\in\mathbb N$ are roots of $$f(x_1,\dots,x_n)=0$$ over $\mathbb Z$ if $$\|f(x_1X_1,\dots,x_nX_n)\|<\frac{R}{\sqrt{\...
9
votes
1answer
454 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\...
-2
votes
0answers
78 views

Are there proven cases of permutations of the Satake parameters being actually isometries ?

This is a follow-up to that rather old question of mine : Would a proof of Ramanujan Conjecture together with other known results about automorphic L-functions imply the Grand Riemann Hypothesis? I ...
1
vote
0answers
91 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 $a,A,b$...
0
votes
1answer
262 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 ...
-2
votes
1answer
216 views

Negative Dirichlet Pigeonhole Principle

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 ...
2
votes
1answer
198 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 = \{...
2
votes
1answer
252 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 $C$. ...
1
vote
2answers
257 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 ...
6
votes
1answer
173 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 ...
1
vote
1answer
164 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 ...
0
votes
0answers
27 views

Probability density from standard domain (Typical Box principle and Chinese Remainder Theorem) - III

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_2(a,b,n)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(...
-2
votes
0answers
97 views

On the sum $\sum_{x=0}^{(p-1)/2}(\frac{x^{4n}+cx^{2n}+d}p) $ with $p$ an odd prime

Let $p$ be an odd prime, and let $n$ be a positive integer. For $c,d\in\mathbb Z$ we define $$F_p^{(n)}(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^{4n}+cx^{2n}+d}p\right),$$ where $(\frac{\cdot}p)$ is ...
7
votes
0answers
137 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:=...
3
votes
1answer
315 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 ...
0
votes
0answers
81 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}&\...
8
votes
1answer
153 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: ...
1
vote
0answers
129 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 ...
1
vote
1answer
97 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 ...
3
votes
1answer
114 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\...
5
votes
1answer
164 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\...
5
votes
0answers
141 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, ...
5
votes
0answers
145 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 ...
9
votes
1answer
260 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^...
-1
votes
1answer
198 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 \...
3
votes
1answer
86 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 ...
4
votes
0answers
130 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 \...
4
votes
0answers
97 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 ...