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

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3
votes
1answer
110 views

The average number of a class of reduced, primitive, positive definite binary quadratic forms

Let $f(x,y) = ax^2 + 2bxy + cy^2 \in \mathbb{Z}[x,y]$ be a positive definite binary quadratic form with even discriminant. We say that $f$ is primitive if $\gcd(a,2b,c) = 1$ and it is reduced if $0 &...
1
vote
0answers
30 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
0answers
29 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 $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...
12
votes
1answer
196 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 ...
2
votes
1answer
124 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 $SL_n$ ...
10
votes
1answer
213 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{...
1
vote
0answers
96 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 $u$. ...
4
votes
1answer
139 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 ...
1
vote
0answers
96 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
0answers
68 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 ...
3
votes
3answers
299 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 ...
3
votes
0answers
114 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 ...
0
votes
0answers
54 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 = \...
14
votes
2answers
328 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 ...
41
votes
5answers
4k views

Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?

1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence? More formally, 2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA? Update (July 2010): So we have ...
2
votes
3answers
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{...
-4
votes
0answers
74 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 ...
30
votes
3answers
2k views

Is there a nice explanation for this curious fact about cyclic subgroups?

Here's something that I noticed that quite surprised me. Let $G$ be a finite abelian group. Consider the following expression. $$ \nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H| $$ It ...
9
votes
2answers
1k views

What is known about primes of the form x^2-2y^2?

David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...
4
votes
0answers
184 views

Zeros of polynomials modulo a non-prime

Suppose I have a set $S$ and I want to find a polynomial $p$ such that $p(s) = 0 \mod n$ if $s \in S$, and that it is non-zero modulo $n$ otherwise. In the literature such an $S$ is sometimes called ...
6
votes
1answer
372 views

Finite field “contour” sum

Let $\Bbb{F}_q$ be a finite field. Choose a non-square $\delta \in \Bbb{F}_q^*$ and form the quadratic extension $\Bbb{F}_q\big( \sqrt{\delta} \, \big)$. For an element $z \in \Bbb{F}_q\big( \sqrt{\...
11
votes
4answers
629 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. ...
0
votes
1answer
97 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 all possible ‎selections of $n$ distinct integers less than the prime $p$, say $r_1, r_2, …, r_n$, which are pairwise ...
12
votes
2answers
583 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 ...
2
votes
0answers
335 views

Counting points on an algebraic set over a finite field

Let $q=p^n$, for $p$ a prime. Let $C$ be an Artin–Schreier curve defined by $y^p-y=f(x)$ where $f(x) \in \mathbb{F}_{q}[x]$. Let $C_g$ be $y^p-y=f(x)$ where $x \in g(\mathbb{F}_{p^{n}})$ for some $...
1
vote
0answers
71 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
0answers
107 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{...
-3
votes
1answer
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 ...
5
votes
0answers
364 views

Unique quadratic subextension of a ray class field

Let $K_q$ denote the unique quadratic subextension of the ray class field over $\mathbb{Q}$ of conductor $q\times\infty$. Then $K_q$ should be $\mathbb{Q}(\sqrt{q})$ if $q$ if 1 mod 4 and $\mathbb{Q}(\...
-1
votes
0answers
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
0answers
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 ...
15
votes
3answers
711 views

History of the analytic class number formula

The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point $s=1$ (or, as I prefer, the leading Taylor coefficient at $s=0$). ...
6
votes
1answer
662 views

A general question about strictly non-palindromic numbers

For a definition, see the wikipedia page: http://en.wikipedia.org/wiki/Strictly_non-palindromic_number So according to the wikipedia page, under properties, all strictly non-palindromic numbers with ...
0
votes
0answers
203 views

Canonicity of Čech cohomology

For a topological space $X$, consider the Leray covering $U_\lambda$ (i.e. $\cap U_\lambda$ is sufficiently fine, e.g. affine for Zariski topology) of $X$. For a sheaf $F$ on $X,$ the cohomology $H^...
5
votes
1answer
156 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 ...
3
votes
0answers
101 views

Globalizing local field extensions with controlled ramification

Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the ...
1
vote
1answer
238 views

Some divisibility constraints in Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$. Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...
9
votes
1answer
391 views

Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...
-2
votes
0answers
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
0answers
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.
1
vote
1answer
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 ...
2
votes
0answers
51 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_{...
2
votes
0answers
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_{...
5
votes
0answers
134 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 ...
3
votes
2answers
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 ...
4
votes
1answer
716 views

The sum of reciprocals of odd numbers

I have a question that I've been thinking for a while now. Can you find a set of distinct positive odd integers $n_1, n_2, \ldots, n_k$ for some finite positive integer $k$ such that $\left(\frac{1}{...
21
votes
4answers
1k views

Hasse principle for rational times square

Does a Hasse principle hold for the property of being a rational times a square ? Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}_v$ of $\mathbb{K}$, $a$...
5
votes
1answer
243 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 = ...