# Tagged Questions

**7**

votes

**0**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**4**answers

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

**2**answers

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

**4**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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) + ...