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

Filter by
Sorted by
Tagged with
2 votes
0 answers
70 views

Lubin--Tate formal group construction in local class field theory using group cohomology

Let $K$ be a non-archimedean local field of characteristic 0. Fix a uniformiser $\pi$ and an algebraic closure $\bar{K}$. The theory of Lubin--Tate formal groups gives an explicit construction of the ...
  • 1,222
6 votes
1 answer
727 views

Ruling out the existence of a strange polynomial II

This is a refinement of my question asked earlier, which is answered beautifully in the negative by Thomas Browning. The example he gave was geometrically reducible. Now I want to ask the same ...
2 votes
0 answers
77 views

Explicit expression of automorphic representations as automorphic forms

Let‘s take $G=GL_n$ over a number field $F$ for example. It's already known that every irreducible automorphic representation $\pi$ is a irreducible component of a induced representation $I(G,P;\...
-9 votes
0 answers
76 views

Structure or specific non-structure on primes which would prove the Riemann Hypothesis [closed]

I am just an amateur. Usually proving something comes down to overcoming a finite number of obstructions. Typically a handful. I wonder if there could be specific structure or absence of specific ...
23 votes
1 answer
1k views

Ruling out the existence of a strange polynomial

Does there exist a polynomial $f \in \mathbb{Z}[x,y]$ such that $$\displaystyle f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$ and $$\displaystyle \liminf_{(x,y) \in \mathbb{R}^2} f(x,y) = -\...
9 votes
3 answers
359 views

Compilation of strategies to show that some constant is irrational

I'm looking into expanding my knowledge in ways to show that some constant is irrational. I'm gonna give some examples of irrationality proofs, and I'm interested in learning what strategies you guys ...
  • 467
4 votes
0 answers
104 views

About the structure of smooth automorphic forms

Recently I read Prof. Cogdell's notes: Lectures on L-functions, Converse Theorems, and Functoriality for $GL_n$. (Co) In chap.2.3, the conception of smooth automorphic forms is introdued. Explicitly, ...
0 votes
0 answers
69 views

What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?

Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$. Wieferich prime is Wieferich number with $n$ prime. It is an open problem if there are infinitely many Wieferich primes and ...
  • 23.7k
4 votes
1 answer
292 views

Is there any approach to solving statements about the natural numbers which are just true by chance?

I suspect that certain problems regarding the base 10 representation of natural numbers may be undecidable simply because there's no way to even start. Take any exponentially growing sequence like $16^...
2 votes
1 answer
264 views

The nontrivial zeros of the zeta function and the prime counting function

The truncated explicit formula has the shape \begin{equation}\label{1} \psi(x) =x-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\sum_{|\rho|\leq T}\frac{x^{\rho}}{\rho}+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}+...
  • 21
2 votes
0 answers
108 views

Expected error term in the distribution of Friedlander-Iwaniec primes

In 1998 John Friedlander and Henryk Iwaniec famously proved the asymptotic formula $$\displaystyle \mathop{\sum \sum}_{a^2 + b^4 \leq x}\Lambda(a^2 + b^4) = \frac{4x^{\frac{3}{4}}}{\pi} \int_0^1 (1 - ...
39 votes
2 answers
6k views

Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture

Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635 In that preprint, Kirti Joshi claims that he agrees with Scholze and ...
2 votes
0 answers
66 views

Closed form for coefficients related to excedance set of permutation

Working on suitable closed form for A329369, I discovered very useful coefficients, which have the following recurrence relation: $$T(0,1)=T(0,2)=1$$ $$T(n,1)=1, n>0$$ $$T(0,k)=0, k>2$$ $$T(2n+1,...
7 votes
1 answer
538 views

The $9$th tetration of $-\sqrt2$

Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and $$^{n+1}a=a^{^na}$$ for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...
2 votes
0 answers
93 views

Bounding number of solutions of a congruence

Let $d$ be a positive integer. Let $f(d,a)$ be the number of values of $x$ in $[1,d]$ such that $$x^{a}\equiv 1\pmod{d}.$$ I wanted to know if for some $0<\epsilon<1$, we can prove the following ...
2 votes
1 answer
186 views

Explicit bounds on number of primes of given size

How many prime numbers of $b$ bits are there? Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...
  • 456
4 votes
1 answer
186 views

Divisibility of special polynomials

Problem: classify all pairs $(k,P)$ such that $P(x)$ divides $$x^kP(x+1)+(x-1)^kP(x-1),$$ where $k\ge4$ is an integer, and $P$ a nonconstant monic polynomial with rational coefficients. I have found ...
  • 9,564
3 votes
1 answer
241 views

Can the twin-prime conjecture be related to the growth of $\phi_2(n)=n \prod\limits_{p>2\,\land\,p|n}\left(1-\frac{2}{p}\right)$?

This question is based on this Math StackExchange question by user buja and my related Math StackExchange question which was asked about a month ago and is still unanswered. I believe $$\phi_2(n)=n \...
1 vote
1 answer
142 views

How often does $-1$ have a square root in a local field?

Let $F$ be a nonarchimedean local field, say, charactersitic $0$. Is there any general theorem that tells when $\sqrt{-1}$ exists in $F$? How often does it happen?
  • 629
0 votes
0 answers
198 views

One question on perfect squares

Let $a_n$ be an integer sequence where $$ a_n=4n^6 + 4n^5 - 11n^4 + 7n^2 - 4n + 1 $$ Prove or disprove that no member of the sequence is a perfect square (except $a_0$ and $a_1$). My computer can't ...
1 vote
1 answer
144 views

Let $(a_0,a_1,\dotsc)$ be an infinite sequence of natural numbers such that $a_x\equiv a_y\pmod{m}$ iff $x\equiv y\pmod{m}$. Prove that $a_n=n$ [closed]

Let $(a_0,a_1,\dotsc)$ be an infinite sequence of natural numbers containing all the natural numbers. Assume that $a_x\equiv a_y\pmod{m}$ if and only if $x\equiv y\pmod{m}$, for all $x,y\in\mathbb{N}$ ...
  • 55
1 vote
0 answers
87 views

Closed form for $a(2^m(2^n-2^p-1))$

Let $q(n)$ be A007814, i.e., the number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $a(n)$ be A329369. Here $$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...
0 votes
0 answers
39 views

Extracting rational or integer coefficient values from multiple convolutions

Let us assume we have a convolution of g-copies of a summation and j-copies of another summation, and they are multiplied together, so we have a product of $g+j$ summations. And $g+j=u$. Series1: $ \...
  • 139
4 votes
0 answers
121 views

$L(1,\chi_d)$ for even d

This question concerns the asymptotic maximum of $L(1,\chi_d)/ \log \log d$ when $d$ is even. I computed it for even $d$ less than 7.5 million. The largest value is $1.230126$, occurring for $d= ...
3 votes
1 answer
276 views

Where odious numbers meet arithmetic progressions

Suggested by this problem: Do the sets of all odious / evel numbers meet every infinite arithmetic progression? A number is odious if it contains an odd number of digits $1$ in its binary ...
  • 21.8k
0 votes
1 answer
93 views

Asymptotic for a sum involving GCD and Euler totient function

Let $\varphi$ denote the Euler's totient function. Is there any reference in literature for the value of sum $$\sum_{\substack{r\le x\\ d\mid r}}\gcd(\phi(d),r)$$ where $d$ is some fixed positive ...
4 votes
0 answers
84 views

Asymptotics for a sum involving GCD and multiplicative order

Let $n$ be a positive integer and $\mathrm{ord}_{n}(a)$ be the least positive integer $d$ such that $n\mid a^{d}-1$. I wanted to know if for some choice of $y=y(x)$, one can obtain asymptotics for ...
-1 votes
0 answers
86 views

Solutions to an elliptic curve equation involving a prime [migrated]

Let $p$ be a prime and let $x$ be a positive integer. How do I prove that the equation $$y^2=px^3+p$$ has infinitely many integer solutions? I tried to tackle this problem using elliptic curves and ...
2 votes
1 answer
131 views

Voronoï summation for cusp forms with characters

In an attempt to solve an unrelated problem, I was led to the task of estimating/bounding from above sums of the form $$\sum_{m=1}^\infty\lambda(m)e\left(-\frac{am}{q}\right)h(m)$$ where $\sum_{m=1}^\...
  • 413
2 votes
0 answers
98 views

Relation between conductor of characters in a field of 3-adic numbers

Let $\chi$ be a non-trivial multiplicative character of the $3$-adic field $K:=\mathbb{Q}_3$. I would like to find out the relation between the conductor of $\chi$ and the conductor of $\chi^3$. Let ...
  • 21
21 votes
0 answers
6k views

Philosophy behind Zhang's 2022 preprint on the Landau–Siegel zero

Now that a week has passed since Zhang posted his preprint Discrete mean estimates and the Landau–Siegel zero on the arXiv, I'm wondering if someone can give a high-level overview of his strategy. In ...
  • 10.2k
0 votes
0 answers
126 views

About Diophantine equations

I am a self studying student and I am interesting with diophantine equations. I have the following questions: How can I know that a diophantine equation is solved or not? the second question is what ...
  • 19
1 vote
1 answer
227 views

How many ways to pick k integers with fixed sum and product

We are given $1\leq S \leq 10^9$ and $1 \leq P \leq 10^9$. We need to pick $k$ integers $x_1, x_2, \dotsc, x_k$ (all of which have to be $>1$) such that $\sum_k{x_k}=S$ and $\prod_k{x_k}=P$. What ...
2 votes
0 answers
238 views

An approximation for the prime counting function

NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses. SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, ...
2 votes
1 answer
99 views

Counting numerical semigroups by largest element of minimal generating set

For a given integer $n$, I am interested in the number of different numerical semigroups one can make with a generating set consisting only of integers in $[n]$. I have done some small examples. For $...
2 votes
0 answers
205 views

value of $L(1,\chi_{9529})$ and Littlewood's bound

Consider a quadratic field ${\mathbb Q}(\sqrt d)$ and its $L$-function, evaluated at 1. Littlewood proved a famous bound (under GRH) and it's known that for infinitely many $d$, half this bound is ...
1 vote
0 answers
236 views

Reductions of a system of equations at various primes

Let $f_1, \dots, f_n$ be a finite set of polynomials in the polynomial ring $Z[x_1, \dots, x_m]$. At a prime $p$, let $N_p$ be the number of solutions $x=(x_1, \dots, x_m)\in (\mathbb{Z}/p\mathbb{Z})^...
3 votes
0 answers
166 views

Reverse engineering an elliptic curve from its modular form?

Does there exist an algorithm or something of the sort to reverse-engineer a curve from its modular form (weight two eigenform with complex coefficients)? I am aware that sometimes there isn’t a ...
0 votes
0 answers
69 views

Lattice packing

Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number. Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...
  • 219
3 votes
0 answers
144 views

Generalisation of Sharifi's conjecture for Siegel varieties

I recently learned Sharifi's conjecture in this article by F.Takako and K.Kato. According to my understanding, this conjecture states that (the conjugation invariant) of the projective limit $\...
2 votes
0 answers
2k views

Contribution of Yitang Zhang latest results if correct to correlation conjecture of H. L. Montgomery?

There are some integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. One of them is the integral introduced by Selberg related to estimating the variance of primes in ...
2 votes
0 answers
102 views

Upper bound for the torsion subgroup of an elliptic curve over arbitrary number fields

Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn ...
1 vote
0 answers
53 views

Mordell-Weil rank in families of abelian surfaces

In the case of complex elliptic surfaces we can consider the elliptic modular surfaces, as Shioda did in his paper "On elliptic modular surfaces". The latter have Mordell-Weil rank 0. Is ...
8 votes
1 answer
1k views

For a non-square, is there a prime number for which it is a primitive root?

For a natural number not a perfect square, is there always at least a prime number for which it is a primitive root? Artin's conjecture on primitive roots is that there are infinitely many such primes....
5 votes
1 answer
217 views

Sum of three squares as class numbers and Waldspurger's formula

It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...
  • 1,411
2 votes
0 answers
71 views

Integers solutions of products of truncated Riemann zeta functions

Let $n \in \mathbb{N}$ be a positive integer. It is possible to prove that the equation $F_1(n)=m$ where $m \in \mathbb{Z}$ and $$ F_1(n)=(1+2+\ldots+n)\cdot\left(\frac{1}{1}+\frac{1}{2}+\dots+\frac{1}...
  • 1,251
1 vote
0 answers
71 views

CM-fields and ideal theoretic ray class group

Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers and $S$ a finite subset of the real places. Let $\mathfrak{m} \subset \mathcal{O}_K$ an ideal. The ideal theoretic ray class group of $\...
8 votes
1 answer
3k views

Did Zhang weaken the constant in his Landau-Siegel zero paper to get the current year?

Zhang 2022 proves a somewhat suspicious formula: $$L(1,\chi) \gg (\log D)^{-2022}$$ This raises the obvious-but-frivolous question: did he intentionally weaken the constant to get the current year?
0 votes
0 answers
90 views

One variable recurrence relation and two variable recurrence relation

Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $a(n)$ be A329369. Here $$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(2n-2^{...
9 votes
0 answers
85 views

Derived subgroups of 2-adic congruence subgroups of $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be a prime, and let $\Gamma_r$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^r\mathbb{Z}_p)$. Explicit formulas with formal ...

1
2 3 4 5
292