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

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2
votes
1answer
149 views

Counting function for prime pair with bounded gaps between them

I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible. Yitang Zhang breakthrough result established that ...
2
votes
0answers
123 views

Is the twisted symmetric fifth power $L$-function holomorphic?

Let $\pi$ be a Maass cusp form for SL($2,\mathbb Z$). Let $\omega$ be a primitive Dirichlet character. Let us consider the $L-$ function $$L(s,Sym^5 \pi \times \omega)$$ or $L(s,Sym^6 \pi \times ...
6
votes
1answer
219 views

Compatible systems of $\ell$-adic representations

I am trying to understand the statement of the conjectures of Deligne on special values of certain $L$-functions, from his article titled, "Valuers de Fonctions L et periodes d'integrales" which ...
0
votes
0answers
66 views

Finite-index subgroups of the ideles

Let $k$ be a number field and denote by $J_k$ the idele group of $k$. Recall that the finite-index open subgroups of $J_k$ which contain $k^*$ are very important in class field theory. My question ...
13
votes
1answer
458 views

Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $

A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale. I would like ask about the much weaker statement forgetting ...
-1
votes
0answers
158 views

Straight line complexity of $k!a$ where $(a,p)=1$

In Qi Cheng's paper, an algorithm is provided to calculate $k!a$ at some random $a∈ℕ$ ($a$ is not input to algorithm, only $k$ is), what is the probability that given a prime $p$ such that ...
11
votes
1answer
273 views

Automorphisms of finite order in $Out(\widehat{F_2})$

Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime. Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...
5
votes
0answers
154 views

Examples of Rankin-Selberg L-functions from Eisenstein series

I've been digging for awhile to not much success, so I figure I would try here: I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...
-2
votes
0answers
104 views

question on collatz trajectories/pattern in form $2^n - 1$ [on hold]

I recently noticed this remarkable general "wedge" pattern in base 2 for Collatz iterates starting with values $2^n - 1$, displayed here for $n=20$. Has this been noticed or analyzed before? ...
-3
votes
0answers
76 views

How can I solve a diophantine equation with 3 variables? [on hold]

My equation looks like: $$ x + 17y + 289z = 3367 $$ where $$ 20 < x, y, z <100 $$ Is it possible to come up with a list of possible solutions? I do not know where to begin. I know I can't use ...
1
vote
1answer
286 views

Analytic Number Theory without Pigeonhole Principle [closed]

I don't know if this is an appropriate question for this website, but I will try my luck. I am an undergraduate student, and recently I became interested in analytic number theory. When I started ...
7
votes
1answer
343 views

What is the normal closure of $GL_2(\mathbb{Z})$ inside $GL_2(\mathbb{Z}_\ell)$?

This weird problem popped up in my research: Let $\ell$ be a prime. Is there a description of the smallest normal subgroup of $GL_2(\mathbb{Z}_\ell)$ containing $GL_2(\mathbb{Z})$? Is there a ...
7
votes
0answers
227 views

Residue class sufficiency sets for the Collatz conjecture

I have recently managed to show a sequence of sufficiency sets for the Collatz conjecture whose natural density approaches 0 (the set theoretic limit approaches the set $\{1\}$). It is an extension of ...
5
votes
1answer
192 views

Is $\liminf \frac{\sigma_{k}(n)}{n}$ finite for every $k$?

Can someone show me how to prove that $$\liminf_{n \to \infty} \frac{\sigma_{k}(n)}{n} < \infty$$ for every natural number $k$? Or is this problem open? Here, ...
6
votes
0answers
306 views

“Forthcoming paper” of Goldston-Graham-Pintz-Yıldırım

The above-named authors of [1] and its (significantly different) published version [2] write: In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...
2
votes
3answers
320 views

Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that $$ ...
3
votes
1answer
124 views

Given n and q, how to find p so q$\neq$n-th power (mod p)?

Reasonable exceptions allowed on $q$. Example solution: $n=2$. Suppose $q$ is odd. Let $p$ be so $pq\equiv -1$ (mod 8). Then $q\neq$ 2nd power (mod $p$) is the same as ...
10
votes
1answer
390 views

When is the image of an integral polynomial contained in the image of another?

Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$? For instance, this happens if $f=g\circ h$ for some ...
0
votes
2answers
203 views

Intersection of two lattices

Suppose that $\Lambda_1, \Lambda_2$ are two sub-lattices of $\mathbb{Z}^n$ of full rank, defined by congruence modulo a prime $p$. That is, there exist two vectors with integer entries $\mathbf{a}, ...
-2
votes
0answers
137 views

What would both Goldbach's conjecture and GRH tell us about the distribution of k-central numbers?

Assume Goldbach's conjecture. Then for all integer $n$ greater than 1, there exists a non negative integer $r$ such that both $n+r$ and $n-r$ are prime. I call such an $r$ a primality radius of $n$, ...
7
votes
2answers
279 views

Field of definition of dominant morphisms

Let $k$ be an algebraically closed field and $k_0$ a sub-field. Let $X,Y$ be two projective varieties defined over $k_0$. Suppose that that there exists a dominant morphism $f$ between $X_k=X\otimes ...
2
votes
0answers
83 views

Distribution of residue classes of totients of (univariate) polynomials

Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is ...
4
votes
1answer
164 views

Equidistribution of representations by a binary cubic form

Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of ...
12
votes
2answers
400 views

Number of nonzero terms in polynomial expansion (lower bounds)

Let $f(x) = a_1x^{z_1} + a_2x^{z_2} + \cdots + a_kx^{z_k}$ be a polynomial with coefficients $(a_1, \ldots, a_k) \in \mathbb{F}_q^*$ and $z_i$ are distinct positive integers. If I need to compute the ...
9
votes
2answers
419 views

Intuition behind Kronecker's congruence?

The modular polynomial is defined by$$\Phi_n(X, \tau) = \prod_{\tau} (X - j(\tau)),$$where $j$ is the elliptic modular function and $\tau$ is running through classes of imaginary quadratic integers of ...
14
votes
1answer
278 views

Why there are only finitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves with CM by $\mathcal{O}$?

For someone who does not have a very extensive knowledge of number theory, what is a good intuitive explanation as to why there are only finitely many $\overline{\mathbb{Q}}$ isomorphism classes of ...
0
votes
0answers
89 views

A question on integers relatively prime to their Euler totien function

For the problem I am working on I have realised that some of the proofs could be slightly simplified if a certain number theoretic question has a positive solution. Now, the problem is in finite group ...
0
votes
1answer
85 views

Computation of Hilbert symbol of order 4

We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for ...
13
votes
0answers
230 views

Result of Deuring, intuitive way to see it's true/quickest way to prove?

There is the following result of Deuring that goes as follows: Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary ...
8
votes
2answers
788 views
+50

Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$?

This question related to this question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$ for all positive integers $k$. Note. ...
4
votes
0answers
180 views

Unirationality over $\mathbb{Q}$

It is known that all smooth projective quartic hypersurfaces of suitably large dimension are unirational over $\overline{\mathbb{Q}}$. Are there any results regarding unirationality over $\mathbb{Q}$ ...
5
votes
1answer
260 views

Representations of the unit group in a ring of integers

Let $K/\mathbb{Q}$ be a finite extension of degree $d > 1$. Suppose that $\omega_1, \cdots, \omega_d$ is a basis for $K$ over $\mathbb{Q}$. Further, we assume that $\omega_1, \cdots, \omega_d \in ...
0
votes
0answers
40 views

Does such a morphism necessarily coincide with the degree?

Let $\mathcal{M}$ be the set of elements the Selberg class identical up to a twist (that is, we consider that $F\in\mathcal{M}$ and $F_{\theta}:s\mapsto F(s+i\theta)$ with $\theta\in\mathbb{R}$ are ...
9
votes
2answers
471 views

Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial

The starting point for this question is the following (false) statement $\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$ Given a polynomial function $p:\mathbb{N} \to ...
1
vote
0answers
67 views

Bounded discrepancy multiplicative functions

A rather specific question, concerning the second remark of Tao in ...
3
votes
1answer
131 views

Fourier Transform of Eisenstein Series - Sum of Divisors or Ramanujan Sums?

I am stuck on this computation of the Fourier coefficients of Eisenstein series. For $\Gamma = SL(2, \mathbb{Z})$ and $\Gamma_\infty = \left\{ \left( \begin{array}{cc} 1 & m \\ 0 & 1 ...
10
votes
0answers
202 views

Must the sum of the digits of $n^k$ decrease infinitely often, for $n,k\in\mathbb{N}$ and $n$ not a power of $10$?

This is a (rephrased) repost of a question I asked on MSE about 6 months ago, but didn't receive a definitive answer. Let $S(n)$ be the sum of the digits of $n$ (in base $10$), is it true that for ...
0
votes
0answers
54 views

Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...
12
votes
2answers
447 views

Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?

Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?
5
votes
0answers
184 views

Genus of $k(T)$ is $0$ without using Riemann-Roch

Let $F$ be a function field with one variable with total constant field $k$, and let $X$ be the set of all places of $F$. How do I show that the genus of $k(T)$ is $0$ without using Riemann-Roch? Is ...
2
votes
1answer
213 views

Best known zero-free region for Dirichlet $L$-functions in the $q$-aspect

It is classical that there is a $c > 0$ such that for all Dirichlet characters $\chi$ except for at most one exception, one has that $L(s,\chi)$ has no zeroes for $\sigma > 1 - \frac{c}{\log{q} ...
0
votes
0answers
97 views

Is there a rather natural space an automorphism of which is the Mellin transform?

Disclaimer: this question might be a little too vague and thus not suitable for this site despite the soft-question tag. If so, feel free to migrate it to MSE. I just read this and, trying to find ...
6
votes
1answer
258 views

Can one define “Ramanujan Summation” over algebraic number fields?

With some trepidation, I ask to "evaluate" badly divergent sums. Generalizing $\sum n = -\tfrac{1}{12}$ what would be the value of this sum over $\mathbb{Z}[i]$? $$\sum_{m,n \geq 0} (m+in) ...
0
votes
0answers
95 views

A (weak?) lower bound on primes in arithmetic progressions in short intervals

I was wondering if the following could be established by the methods that go into e.g. Linnik: $\textbf{Claim. } \text{Let $\chi$ be a nonprincipal quadratic character of conductor $q$, and (e.g.) $c ...
1
vote
1answer
94 views

On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit

Let $b$ be a fixed integer $\ge 2$ and $A$ a proper subset of $\{0, \ldots, b-1\}$. Then define $X$ to be the set of all positive integers whose base-$b$ representation consists only of digits from ...
7
votes
1answer
266 views

Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?

It is a result from additive-theory folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. ...
2
votes
2answers
209 views

Approximations to the Mertens function

The Mertens function $M(x)$ is the summatory Möbius function i.e. $$M(x) = \sum_{k=1}^{x} \mu (k)$$ The conjecture that $M(x) = \mathcal{O}\left(x^{\frac{1}{2} + \epsilon}\right)$ was shown to be ...
0
votes
0answers
101 views

Continued fractions and modular forms

Let $q=e^{2\pi it}$. If $u(t)$ is Ramanujan's octic continued fraction, is it true that the generator of the octahedral group can be expressed as a continued fraction of the form $$ ...
3
votes
0answers
161 views

Ramanujan conjecture and covariance of Kloosterman sums

There has been interest in moments and covariances/correlations of Kloosterman sums $S(m,n,c)=\sum_{ad=1\ (\text{mod}\ c)} e(\frac{ma+nd}{c})$ like $\sum_{m\in\mathbb F_c} S(m,n,c)^k$, ...
5
votes
0answers
134 views

Explicit bounds for exceptional zeros and/or $L(1,\chi)$ for real $\chi$

I would like to have an explicit upper bound (that is, one with explicit constants) for a possible real zero $\beta$ for $L(1,\chi)$ for real Dirichlet characters $\chi$. I need such a bound for real ...