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

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4
votes
0answers
33 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 ...
7
votes
2answers
257 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 ...
4
votes
1answer
314 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
46 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
86 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
113 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
336 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
113 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
69 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 ...
0
votes
0answers
96 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 ...
2
votes
1answer
85 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
71 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
58 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
78 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
67 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
131 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 ...
24
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
148 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 ...
1
vote
0answers
265 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
318 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
75 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
48 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
201 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
195 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
197 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
309 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 ...
3
votes
1answer
147 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
100 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
346 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
274 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
104 views

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

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
196 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
151 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
398 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
247 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
31 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
111 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
129 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
135 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
202 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
169 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
146 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
129 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) + ...
7
votes
3answers
202 views

Uniform bound for the number primes $p$ s.t. a polynomial has a root modulo $p$

It is well known that for a given polynomial $f \in \mathbb{Z}[x]$ the number of primes $p$ s.t. $f$ has a root modulo $p$ is infinite. In fact, one can even write down a formula for the density of ...
2
votes
1answer
239 views

Looking for paper: Weil's original 1952 “Sur les formules explicites de la théorie des nombres premiers”

I am looking for a source (preferably online) for Weil's original 1952 paper on the explicit formula. I am aware of an english translation available here, but would like to have access to the original ...
7
votes
1answer
291 views

Is there a p-adic Sato - Tate conjecture?

Take $E$ to be an elliptic curve over $\mathbb{Q}$, and consider the coefficients $a_E(p)$of the dirichlet series $L(E,s)$ so that $E(\mathbb{F}_p)=p+1-a_E(p)$. The Sato-Tate conjecture (now ...
2
votes
0answers
86 views

Domains with prime ideal theorems

Let $D$ be a domain, and for prime ideals $\frak P$ of $D$ the norm is $N({\frak P}):=|D/{\frak P}|$. The prime ideal counting function of $D$ is given by $\pi_D(x)=\#\{{\frak P}\in{\rm ...
3
votes
1answer
132 views

Congruences among primes modulo which a given polynomial has roots

Suppose $f(x)\in\mathbf Z[x]$ is nonconstant. I would like to know if either of the following statements is true. If $a$ and $b$ are coprime integers (probably with some additional restriction), ...
4
votes
1answer
108 views

Asymptotic expansion of the Mordell integral

my question concerns the Mordell integral $$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$ which frequently occurs in ...