**1**

vote

**0**answers

44 views

### Relative Leopoldt defect

Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$.
Is there a bound of the Leopoldt defect of $M$ ...

**1**

vote

**0**answers

58 views

### The Linnik problem for dimension $2$

For $N$ an integer, let
$$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$
For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...

**2**

votes

**0**answers

48 views

### On the size of residue class

Let $n \in \mathbb{N}$ be a odd number. Let $S \subseteq \{1,3,5,7,...,n-2,n\}$ and $|S|$ is even number. Let $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...

**1**

vote

**0**answers

131 views

### Computational number theory

Suppose that $p$ is prime and $q$ is an even number divides $p-1$, such that $q<\frac{p-1}{q}$ and $u$ has order $q$ modulo $p$. Let $S$ be the subgroup of $Z^*_p$ consisting of the powers of $u$. ...

**-3**

votes

**0**answers

71 views

### Considering a matrix with integrer entries over $\mathbb{Z}/p \mathbb{Z}$, does it remain full rank? [on hold]

Suppose I have an $m \times n$ matrix $M$ with integer coefficients, and suppose it has full rank. Let $p$ be a prime and now consider the matrix $\bar{M}$ over $\mathbb{Z}/p \mathbb{Z}$. Is it true ...

**12**

votes

**1**answer

245 views

### Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?

Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...

**10**

votes

**1**answer

236 views

### Polylogarithm sheaves

In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{...

**4**

votes

**1**answer

151 views

### Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups"
https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf
contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...

**0**

votes

**0**answers

56 views

### On semi-complete ring K[X_1,X_2,…,X_∞]] and Popescu theorem

Let $P_n \colon= K[X_1,...,X_n]$ be a $n$-variables polynomial ring. We define 'semi-complete' polynomial ring $P_{\infty}$ by the following$\colon$
$P_{\infty} = K[X_1,...,X_\infty]] \colon = \...

**2**

votes

**1**answer

132 views

### Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map
$$
\pi:SL_n(R)\rightarrow SL_n(R/I)
$$
(In the original question I had put $GL_n$ instead of $SL_n$ ...

**4**

votes

**1**answer

127 views

### Hilbert modular forms twist-equivalent to their conjugates

Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$.
Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate ...

**1**

vote

**0**answers

98 views

### Combinatorial splitting in number rings

The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring.
Take an arbitrary non empty ...

**3**

votes

**3**answers

311 views

### Jacobi's theorem on sums of two squares (reference request)

One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...

**-4**

votes

**0**answers

77 views

### A basic Query regarding Riemann Zeta function [on hold]

The Euler Definition of Zeta is given as (extreme right):
Using only Euler's definition, how can any value of s (real/complex) lead to the function being 0. As, no matter what the denominator of ...

**14**

votes

**2**answers

340 views

### Can you use Chevalley‒Warning to prove existence of a solution?

Recall the Chevalley‒Warning theorem:
Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If
$$d_1 + \ldots + d_r < n,$$
then the ...

**2**

votes

**3**answers

267 views

### Geometry of numbers argument: counting integers with some linear condition

I am interested in the proof of the following result:
Suppose that $A > 1$, $\lambda \in \mathbb{R}$, and for $0 < Z \leq 1$, let $U(Z)$ be the number of integer solutions $v$ of
\begin{...

**11**

votes

**4**answers

640 views

### Smallest solution to $x^2 \equiv x\pmod{n}$

Given $n$, is it possible to upper bound the smallest $x > 1$ that satisfies the congruence $x^2 \equiv x\pmod{n}$? Obviously when $n$ is a prime power $x = n$, and we are in the worst situation. ...

**1**

vote

**0**answers

72 views

### Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...

**0**

votes

**0**answers

108 views

### Troost-Bourget identity $ N \sum_{d|N} 1 = \sum_{d| N} \sum_{l=1}^d \mathrm{gcd}(d,l) $ [on hold]

In the process of evaluating a "supersymmetric index", Bourget and Troost establish a rather elementary identity:
$$ \frac{N}{m} \sum_{d| N} \sum_{l=1}^{\mathrm{gcd}(d,m)} \mathrm{gcd}\left[ \mathrm{...

**-1**

votes

**0**answers

70 views

### Sum-free sets of powerful numbers

For $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ with distinct primes $p_i$, call $\alpha= (\alpha_1,\dots,\alpha_r)$ the type of $n$ and denote by $N_\alpha$ the set of all naturals of this type.
We ...

**-1**

votes

**0**answers

88 views

### A question about arithmetic progressions and prime numbers

"I took number $3$ and observed:
$3$ is an arithmetic progression of length one.
$3,5$ is an arithmetic progression of length two.
$3,5,7$ is an arithmetic progression of length three.
Then I took ...

**0**

votes

**1**answer

114 views

### How to count fixed-sized subsets of pairwise co-prime numbers less than a prime, satisfying an additional constraint?

In part of my research, I need to count (or find a sharp bound for) the number of possible ways to select $n$ distinct integers less than the prime $p$, say $r_1, r_2, …, r_n$, which are pairwise ...

**12**

votes

**2**answers

592 views

### Congruence equation and quadratic residue

The following observation makes me quite confused when I am trying to count the number of solutions of the equation:
$$\sum_{k=0}^{M}{M \choose k}^2x^k=0$$
on finite fields $\mathbb{F}_p$ with the ...

**-3**

votes

**1**answer

101 views

### Is a positive integer determined by its sequence of typical primality radii?

This question is a follow-up to About Goldbach's conjecture . Assuming the truth of Goldbach's conjecture, suppose $n$ and $m$ are two positive integers such that $N_{2}(n)=N_{2}(m)=:N$ and that ...

**-2**

votes

**0**answers

66 views

### On the existence of certain type of arithmetic progressions [closed]

I asked this question on math.stackexchange about 4 hours ago and did not receive neither a comment or an answer. Maybe I am a little impatient and should have waited for some time to see will I ...

**-1**

votes

**0**answers

48 views

### Solution to congruency modulo p^e [closed]

How many solution are there for z which satisfies z^2 = p^f (mod p^e). z is element of Zp^e, f is even integer less than e, e is positive integer, and p is prime.

**5**

votes

**1**answer

157 views

### A $p$-adic sum of reciprocals of powers

Let $p$ be a prime number and $k\geq 2$ an even integer. Consider the following $p$-adic integer:
$$
S_{p,k} := \lim_{r\to+\infty} \sum_{a=1}^{p^r} \big(\frac{p^r}{a}\big)^k
$$
Convergence is easy to ...

**1**

vote

**1**answer

139 views

### Complete subring of F_p[[X]]

Pointed out on famous disbelief, I know now that there is an embedding
$\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$
for any $n \leq \infty$. Then I would like to ask ...

**5**

votes

**0**answers

135 views

### Lifting points via étale morphism of adic spaces

This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism ...

**5**

votes

**1**answer

244 views

### Infinitely many primes coming from Euclid's proof

When teaching Euclid's classic proof of the infinitude of primes today, the following question appeared to me. Let $p_1,p_2,p_3,\ldots$ be the prime numbers, listed in increasing order. Set
$$k_n = ...

**2**

votes

**0**answers

30 views

### The Mahler measure of a binary form and the natural action of a matrix ring

Let $F(x,y) = a_0 \prod_{i=1}^d(x-\alpha_iy)$ be a binary form of degree $d \geq 2$ and nonzero discriminant $D(F)$. Define the Mahler measure $M(F)$ of $F$ by
$$M(F) = \left|a_0\right| \prod\limits_{...

**2**

votes

**0**answers

52 views

### An inequality involving the Mahler measure and the discriminant of a polynomial

Let $F(x,y) = a_0 \prod_{i=1}^d(x-\alpha_iy)$ be a binary form of degree $d \geq 2$ and nonzero discriminant $D(F)$. Define the Mahler measure $M(F)$ of $F$ by
$$M(F) = \left|a_0\right| \prod\limits_{...

**1**

vote

**0**answers

60 views

### Popescu-Neron Desingularization for K[[T_1,…,T_∞]]

Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$.
Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ ...

**15**

votes

**2**answers

523 views

### Why is 12 the smallest weight for which a cusp forms exists

I have already asked this question on MathStackExchange (here) but did not receive an answer, therefore I am trying it here:
On wikipedia (here) I have read the following:
Twelve is the smallest ...

**3**

votes

**2**answers

149 views

### $p$-simple integers from between $n$ and $n+p-1$

Let $\ p\ $ be an arbitrary prime. Then an integer $\ s\ $ is called $p$-simple $\ \Leftarrow:\Rightarrow\ s\ $ is not divisible by any prime $\ q<p.\ $
Could you prove my conjecture (or is it ...

**11**

votes

**1**answer

201 views

### A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups

Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A_4$ (the alternating group on four letters) such that all the decomposition groups are cyclic?
This question is motivated by the ...

**8**

votes

**0**answers

260 views

### Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational?

Is the number $$\sum_{p\text{ prime}}p^{-2}$$ known to be irrational?
The limit exists, since $$\sum_{p\text{ prime}}p^{-2}<\sum_{i=1}^{\infty}i^{-2}=\frac{\pi^{2}}{6}$$.

**1**

vote

**0**answers

52 views

### Coherence of subrings of K[[X,Y]]

Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$.
Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...

**0**

votes

**0**answers

55 views

### Congruent numbers and primorials

The first 10 primorials (2, 6, ... , 6469693230) are congruent numbers subject to the Birch Swinnerton-Dyer conjecture.
My question is - What is the first primorial not to be a congruent number (...

**5**

votes

**1**answer

603 views

### Ordinary primes vs supersingular primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. As shown by Serre, the set of supersingular primes for $E$ has density zero.
Is the analytic rank of $L(E,1)$ determined only by the ...

**1**

vote

**2**answers

137 views

### A multidimensional version of Hensel's lemma? (for more than one polynomial)

The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}_p[x]$ and $a \in \mathbb{Z}_p$ satisfy
$$
|f(a)|_p < | f'(a) |_p^2.
$$
Then there is a unique $\alpha \in \mathbb{Z}_p$...

**1**

vote

**0**answers

42 views

### Sieving for points not on a conic and quadratic residues

Let
$$\displaystyle F(x,y,z) = \sum_{\substack{0 \leq i_1, i_2, i_3 \leq 2 \\ i_1 + i_2 + i_3 = 2}} a_{i_1, i_2, i_3} x_1^{i_1} x_2^{i_2} x_3^{i_3}$$
be a ternary quadratic form with integer ...

**1**

vote

**1**answer

98 views

### Quadratic (and otherwise) squares, part II

This is a follow-up to this question: Quadratic squares
Consider now a polynomial in two variables $f(x, y).$ Are there bounds (upper or lower) for how many $x_0$ of height less than $N$ such that $f(...

**0**

votes

**1**answer

200 views

### Can we find primitive Pythagoras triplet (in integers), with two sides being as powerful numbers?

Actually this seemingly elementary question had very much surprised me, I though it is quite easy to find such triples described in question, but strangely after trying so many times, couldn't find ...

**1**

vote

**0**answers

36 views

### Representing sparse set as set of extremely good approximation

For $\alpha \in \mathbb R$ and $\varepsilon(n) > 0$, consider the set
$$
N(\alpha,\varepsilon) = \{ n \in \mathbb{N} \ : \ \lVert \alpha n \rVert < \varepsilon(n) \}
$$
(where $ \lVert x \rVert$...