# 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

10,459 questions

**4**

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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

**2**answers

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

**0**answers

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

**0**answers

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**

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**0**answers

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

**5**answers

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

**1**answer

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

**3**answers

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

**1**answer

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**

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**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**4**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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**

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**0**answers

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

**1**answer

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**

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**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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**

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**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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 ...