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

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4 votes
1 answer
290 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^...
7 votes
1 answer
520 views

Absolute convergence of Rankin–Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$. I heard that the rankin-selberg convolution L-function $L(s,\pi\...
9 votes
3 answers
357 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 ...
92 votes
15 answers
32k views

Collecting proofs that finite multiplicative subgroups of fields are cyclic

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
11 votes
1 answer
679 views

Numbers $k$ with $\{\binom nk:\ n\in\mathbb N\}$ dense in $\mathbb Z_p$ for any prime $p\le k$

Let $k$ be a positive integer and let $p$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:...
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}+...
5 votes
2 answers
245 views

Does the $p$-adic regulator depend on Weierstrass model?

I am a little confused on the $p$-adic regulator on elliptic curves and what happens when you switch to different Weierstrass models. Restrict to ell. curves over $\mathbb Q$ for simplicity. From my ...
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 ...
9 votes
2 answers
416 views

Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?

(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem) Let $f:\mathbb{R}\to [0,\infty)$ be such that (a) $\int_{\mathbb{R}} f(x) dx = 1$, (b) $\...
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 ...
35 votes
2 answers
11k views

What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi, I know this is a dangerous topic which could attract many cranks and nutters, but: According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...
6 votes
1 answer
726 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;\...
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}^\...
-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 ...
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, ...
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) = -\...
24 votes
1 answer
1k views

Parity of the multiplicative order of 2 modulo p

Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
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 ...
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 \...
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 - ...
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
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,...
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 ...
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 ...
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 ...
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 ...
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
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?
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 $x,y\in\mathbb{N}$ ...
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
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^{...
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: $ \...
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 ...
12 votes
1 answer
2k views

Squares in an arithmetic progression

Let $P(x;a,b) := \{an+b, 0\leq n \leq x \} $ denote an arithmetic progression. Further let $A(x;a,b)$ denote the number of elements of $P(x;a,b)$ that are squares. It's an old conjecture of Rudin ...
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 ...
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
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 $x > 0$, ...
4 votes
0 answers
341 views

Ramanujan's conjecture on modular forms and Riemann hypothesis

I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...
8 votes
3 answers
2k views

Maximal (non-abelian) extensions of number fields unramified everywhere

Hello! Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general ...
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
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 ...
1 vote
1 answer
130 views

Can $P(z)$ have a divisor in a given congruence class?

In the answer to this previous question , Noam D. Elkies proved that for any integer $x$, $x^3-x^2-2x+1$ can only have a divisors equal to $-1$, $0$, or $1$ modulo $7$. I would like to know what is ...
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 ...
8 votes
1 answer
459 views

Power sums and formal divisibility by the Euler totient function

For integers $s \geq 0$ and $n \geq 1$ let $\sigma_s(n) := 1^s + \dotsb + n^s$ and let \begin{equation} \sigma_s^*(n) \ : = \ \sum_{\substack{\scriptstyle 1 \, \leq \, m \, \leq \, n \\ \gcd(m,n) = 1}}...
3 votes
2 answers
384 views

Function or bounds for the number of solutions of $\sum_{i=0}^k \frac{1}{x_i} = 1$

Is there any result known for the number of different solutions of $1 = \sum\limits_{k=0}^n \frac{1}{x_k}$ in dependency of the length $n$ of this partition? All I know, up to now, is that there are ...
26 votes
2 answers
1k views

Primitive roots

If Artin's conjecture on primitive roots is true, then 2 generates $(\mathbb{Z}/p\mathbb{Z})^\times$ for infinitely many primes $p$. Can one at least show that $(\mathbb{Z}/p\mathbb{Z})^\times$ is ...
52 votes
2 answers
8k views

Walsh Fourier transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients. ##Special case: is Möbius nearly orthogonal to Morse ! Harold Calvin Marston Morse (24 March 1892 ...

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