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

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7
votes
0answers
88 views

Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
4
votes
1answer
184 views

Existence of a certain subset of natural numbers equidistributed modulo $m$ for every $m$

I was talking to a friend and the following set $S$ came up. Let $f$ be some real valued function tending to infinity. Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ ...
6
votes
3answers
399 views

Links between Geometric Group Theory and Number Theory

Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.
3
votes
1answer
181 views

Circle method on things other than the integers

The circle method is often used to estimate the number of solutions to the equation $$x_1 + x_2 + ... x_k = N$$ if for all $i$ $x_i\in A\subseteq\mathbb{N}_0$ and some subset of the nonnegative ...
4
votes
4answers
717 views

Advice for number theory library

Hi I just got a faculty position and it comes with a generous start up funds for "office supplies", which I must use or lose. What does a pure mathematician need? I have good computers already. I ...
3
votes
2answers
161 views

Interpolation of periods for a Hida family of modular forms

Let $\mathbf{f}$ be a Hida family of ordinary $p$-adic modular forms, and let $V(\mathbf{f})$ be the corresponding $\Lambda$-adic Galois representation (a quotient of the inverse limit $$ ...
9
votes
4answers
240 views

Is any quadric birational to a product of Brauer-Severi varieties?

Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let $$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$ be a non-singular ...
13
votes
1answer
594 views

How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...
5
votes
1answer
404 views

Has this strengthening of the PNT already been conjectured?

Suppose $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ is an arithmetic function that grows slower than the identity map. Has it already been conjectured that, under this general hypotheses, ...
1
vote
0answers
57 views

Do we know a lower bound for the number of critical zeros of the Riemann zeta-function with irrational imaginary part?

If I'm not mistaken, the imaginary parts of the critical zeros of the Riemann Zeta function are conjectured to be linearly independent over $\mathbb{Q}$, but I think we're very far from proving such a ...
-1
votes
1answer
96 views

notable inductive proofs relating to fractals

what are notable/ prominent inductive proofs relating to fractals? the motivation for this question is: fractals are very difficult mathematical objects to work with, and many ...
1
vote
1answer
122 views

Question on an arithmetic function with the sieve of Eratosthenes

I want to ask some question related with the sieve of Eratosthenes. The sieve of Eratosthenes: write it as $E_1(x) (=\pi(x)-\pi(\sqrt x)+1)$. Then we have an obvious result $$E_1(x)/x\ln^{-1}x = ...
11
votes
1answer
358 views

Normality of $\pi$ in base 16

It seems that in spite of the Bailey–Borwein–Plouffe formula it is still unknown whether $\pi$ is normal in base 16. What are the difficulties in using it for this purpose? In a comment to his answer ...
1
vote
0answers
130 views

Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three consecutive powerful numbers. Necessary condition for this is integer solution of $$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$ What are integer solutions ...
-4
votes
0answers
73 views

Where to include contact details in math paper? [on hold]

I recently submitted a paper to a math journal on a prime number patter using latex formatting, but they sent an email back saying that the contact details for the corresponding author should be in ...
1
vote
1answer
139 views

lattice in number field already a fractional ideal?

Let $K=\mathbb{Q}[\alpha]$ where $\alpha$ is integral over $\mathbb{Z}$ such that the Galois hull of $K$ can be embedded in $\mathbb{R}$. Let $S=\mathbb{Z}[\alpha]$. Let $x_1, \ldots , x_n$ be a ...
3
votes
1answer
107 views

Is every sufficiently dense well mixed set an additive basis?

Let $B \subset \mathbb{N}$ be a set of natural numbers such that $|B \cap [1,N]| \sim N^\gamma$, for some $\gamma > 0$ with the following property: For any pair of positive integers $k,n$ we ...
3
votes
0answers
91 views

“Almost” prime k-tuples in intervals

In this paper, Heath-Brown proves that there are $ \gg x(\log{x})^{-k}$ integers $n \leq x$ such that $$ \prod_{i=1}^{k} (a_{i}x + b_{i}) $$ is squarefree and that each term has a "small" number of ...
0
votes
1answer
60 views

Equivalent of Stirling-like numbers

let $b_{n,k}$ be the numbers defined formally by $$X^n=\sum_{k=0}^n b_{n,k}\binom{X}{k}$$ where $\binom{X}{n}=\frac{1}{n!}\prod_{k=0}^{n-1}(X-k)$. I am looking for an equivalent of $b_{n,k}$ when $k$ ...
2
votes
0answers
95 views

Is this a valid Hadamard product for $\frac{2\,\xi(s)-1}{s\,(s-1)}$?

This question builds on this MSE question: Take the well known equation: $$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}-1} + ...
2
votes
0answers
70 views

Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?

Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...
3
votes
1answer
138 views

What are the modular transformation properties of q-Pochhammer symbols?

Do q-Pochhammer symbols, defined as $(a;q) = \prod_{k=0}^{\infty} (1- a q^k)$ have known modular transformation properties? That is, if we write $q = q[z] = e^{2\pi i z}$, is there any reasonably ...
26
votes
1answer
2k views

Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes. Let $\mathrm{Li}(x)$ be the offset logarithmic ...
-9
votes
0answers
151 views

Fermat and the abc conjecture [on hold]

There exist finitely many triples of coprime $a+b=c$ such that $$|\frac{b}{c}|^n+|\frac{a}{b}|^n=b^n-a^n$$ where $n>2$ and $b^n-a^n=\mathfrak{Prime}.$ We know that it maybe true in this version ...
2
votes
1answer
320 views

On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski

In response to a comment posted under Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd ...
6
votes
1answer
328 views

The sum over zeros in the explicit formula for $\zeta(s)$

The explicit formula for $\zeta(s)$ is: $$ \psi(x)=x-\sum_{|\operatorname{Im}\rho|<T}\frac{x^\rho}{\rho}-\log(2\pi)-\log\left(1-\frac{1}{x^2}\right)+O\left(\frac{x\log^2T}{T}\right), $$ where ...
0
votes
0answers
76 views

Upper bound for $r_{0}(n)$ through probabilities

Assume Goldbach's conjecture. Then for every integer $n>1$ there exists a non negative integer $r$ such that both $n-r$ and $n+r$ are primes. For a given $N$, let's denote by $r_{0}(N)$ the ...
0
votes
1answer
49 views

Isometry group of an integer as of the corresponding $\Omega(n)$-parallelotope

Let $\prod_{i}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ and let's consider the $\Omega(n)$-parallelotope built with, for all $i\in I$, $a_{i}$ pairwise orthogonal vectors of ...
1
vote
3answers
206 views

Powers of $2$ and the products of initial odd primes

NOTATION: $O_x$ -- the product of all odd primes $\le x$. E.g. $O_7=3\cdot 5\cdot 7 = 105$. QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the ...
4
votes
1answer
200 views

Fermat surface known to have very few rational integer solutions

The motivation for this question is the Selmer curve, given by $$\displaystyle 3x^3 + 4y^3 + 5z^3 = 0.$$ One can show that this curve has no rational integer solutions, despite having a solution ...
1
vote
1answer
204 views

An infinite product: combinatorial interpretation

It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product ...
7
votes
1answer
321 views

Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...
4
votes
2answers
217 views

Reference for $p$-adic Hodge theory with coefficients

Let $K$ be a $p$-adic field and $L$ be a finite or infinite extension (maybe algebraic ?) of $\mathbb{Q}_p$. Is there a reference for $p$-Hodge theory for representations $\rho : Gal_K \rightarrow ...
2
votes
1answer
105 views

Properties of representations attached to p-adic modular forms

I found an old MOF post about representations attached to p-adic modular forms: Representations attached to p-adic modular forms and I have some follow up questions on the same topic. If we have a ...
9
votes
1answer
352 views

Roots of unity near 1 in Z / p Z

Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p. The ...
3
votes
0answers
278 views

Continued fraction representation of Zeta

A question at math.SE is asking for references. The fraction is quite nice! Check it out and post some references if you know of any. I found this at arxiv, but it doesn't apply to Zeta.
0
votes
0answers
105 views

Is there an accepted measure of the degree of coverage of different partial proofs of Fermat's last theorem? [closed]

For example, granting that, for co-prime $a,b$ (primitive cases), Fermat himself proved that $a^n+b^n=c^n$ has no solution for $n = 4$, and Euler next proved it for $n = 3$, one could say that FLT was ...
1
vote
1answer
199 views

Square-free integers not divisible by any “small” primes

I have two very related questions: If $f(N)$ is the number of square-free integers in the interval $[1, N]$, it is well known that $$f(N) \sim \frac{6}{\pi^{2}} N.$$ My first question is, if we ...
7
votes
1answer
152 views

Hasse principle and Brauer-Manin obstruction for forms of large degree

The Hasse principle is perhaps an at-first naive generalization of the Chinese remainder theorem; that if a linear equation can be solved modulo $p$ for any prime $p$, then it can be solved in the ...
0
votes
2answers
402 views

Yitang Zhang's paper [closed]

I just want make thing clear for myself. Others may have asked before in different ways. Does Yitang Zhang's paper prove that for any given length gap $g_n > N$ there is a prime $p_n$ for which ...
1
vote
2answers
251 views

How to prove $\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{f_x}(p) = 1} {\frac{1}{p}} = \ln 2$ for $p \le x$?

Let ${f_x}(m) = \sum\limits_{\left. p \right|m} {{f_x}(p)}$ be a strongly additive function on positive integer number $m$, where $p$ is a prime number. Set $${f_x}(p) = \left\{ ...
1
vote
0answers
33 views

Upper bounds for $f_k(n)$ (functions in the fast growing hierachy)

Mostly, for the functions in the fast growing hierachy, LOWER bounds are given like $f_k(n) > 2 \uparrow^{k-1} n$ but what abour (reasonable tight) UPPER bounds ? What are the best known UPPER ...
2
votes
0answers
119 views

Comparing numbers $a \uparrow^b c$ and $d \uparrow^e f$

Is there an efficient method to decide which of two numbers $a \uparrow^b c$ and $d \uparrow^e f$ is bigger ? The rules that hold in most of the cases are if b>e then $a \uparrow^b c$ is the ...
1
vote
2answers
130 views

Symmetric form for sum of reciprocals of primes equal an integer

Find all possible positive integers $m$ and $m$ primes ${{p}_{1}}<{{p}_{2}}<\cdots <{{p}_{m}}$ such that $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+\cdots ...
-3
votes
0answers
77 views

The sum of the reciprocals of the odd primes taken two at a time [closed]

From Steven R Finch, Mathematical Constants, page 95: The sum of the squared reciprocals of primes is 0.4522474200… Clearly, the sum of the reciprocals of the odd primes taken two at a time -- ...
0
votes
0answers
136 views

OEIS A182514 conjectures? [closed]

I have shown in sequence A182514 "Primes prime(n) such that (prime(n+1)/prime(n))^n > n." at https://oeis.org/A182514 that 1693182318746371 is a(6), by using maximal prime gaps. Now, I am wondering if ...
7
votes
1answer
203 views

$L^\times / K^\times$ torsion $\Rightarrow L = K$?

Let $L/K$ be an extension of fields of characteristic zero. I want to prove that if $L^\times/K^\times$ is a torsion group (i.e. for every element $\alpha \in L$, some power of $\alpha$ lies in $K$), ...
0
votes
1answer
170 views

Distinct primitive factorizations over integers of number fields

I am curious about the following. Let $K$ be a number field. For any $a \in \mathcal{O}_K$ in its ring of integers, let $N(a)$ be zero if there exist elements $b, c \in \mathcal{O}_K \setminus ...
1
vote
0answers
147 views

Growing rate of f(n) = $10 \rightarrow 10 \rightarrow … \rightarrow 10 \rightarrow 10$ with n 10's [closed]

I would like to have an estimate for a really fast growing function and a number. Define the following function f(n) = $10 \rightarrow 10 \rightarrow ... \rightarrow 10 \rightarrow 10$ with n 10's. ...
-7
votes
0answers
130 views

Firoozbakht QED? [closed]

Rewrite: The Firoozbakht's conjecture (1982) is equal to: $$(p_{n+1})^{n} < (p_n)^{n+1}.$$ Then the natural log is: $$n \ln(p_{n+1}) < (n+1)\ln(p_n).$$ Now, $$\ln(p_n) \leq \ln(n) + ...