Tagged Questions
Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
1answer
68 views
Congruence for the number of points in the elliptic curve $y^2 = x^3+b \pmod{p}$
Let $E$ be the elliptic curve $y^2=x^3+1$ and $p \equiv 1 \pmod{3}$ a prime. Computing the number of points mod $p$ of $E$ using the naive method gives:
$$ \#E(\mathbb F_p) = 1+ \sum_{x=0}^{p-1} ...
1
vote
1answer
104 views
A convergence issue
Disclaimer: This could be a stupid question and could have a very simple answer which I am unable to see.
Edited
Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space ...
0
votes
0answers
58 views
Does Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?
Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...
-3
votes
0answers
45 views
Product of Positive Intever Divisors of 6^16 equals 6^k [on hold]
Product of Positive Intever Divisors of 6^16 equals 6^k
How would I find K?
Don't give me the answer, just how to get it
Thanks
9
votes
0answers
152 views
Lifting Abelian Varieties to p-adic fields
Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
4
votes
1answer
215 views
Existence of a certain subset of natural numbers equidistributed modulo $m$ for every $m$
I was talking to a friend and the following set $S$ came up.
Let $f$ be some real valued function tending to infinity.
Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ ...
6
votes
3answers
441 views
Links between Geometric Group Theory and Number Theory
Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.
3
votes
1answer
200 views
Circle method on things other than the integers
The circle method is often used to estimate the number of solutions to the equation $$x_1 + x_2 + ... x_k = N$$ if for all $i$ $x_i\in A\subseteq\mathbb{N}_0$ and some subset of the nonnegative ...
4
votes
4answers
815 views
Advice for number theory library
Hi I just got a faculty position and it comes with a generous start up funds for "office supplies", which I must use or lose. What does a pure mathematician need? I have good computers already. I ...
3
votes
2answers
176 views
Interpolation of periods for a Hida family of modular forms
Let $\mathbf{f}$ be a Hida family of ordinary $p$-adic modular forms, and let $V(\mathbf{f})$ be the corresponding $\Lambda$-adic Galois representation (a quotient of the inverse limit
$$ ...
9
votes
4answers
276 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 ...
14
votes
1answer
767 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
417 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
63 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
100 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
126 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
$$E_1(x)/x\ln^{-1}x = ...
11
votes
1answer
362 views
Normality of $\pi$ in base 16
It seems that in spite of the Bailey–Borwein–Plouffe formula it is still unknown whether $\pi$ is normal in base 16. What are the difficulties in using it for this purpose?
In a comment to his answer ...
1
vote
0answers
136 views
Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$
According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions ...
-4
votes
0answers
73 views
Where to include contact details in math paper? [on hold]
I recently submitted a paper to a math journal on a prime number patter using latex formatting, but they sent an email back saying that the contact details for the corresponding author should be in ...
1
vote
1answer
142 views
lattice in number field already a fractional ideal?
Let $K=\mathbb{Q}[\alpha]$ where $\alpha$ is integral over $\mathbb{Z}$ such that the Galois hull of $K$ can be embedded in $\mathbb{R}$. Let $S=\mathbb{Z}[\alpha]$. Let $x_1, \ldots , x_n$ be a ...
3
votes
1answer
109 views
Is every sufficiently dense well mixed set an additive basis?
Let $B \subset \mathbb{N}$ be a set of natural numbers such that $|B \cap [1,N]| \sim N^\gamma$, for some $\gamma > 0$ with the following property:
For any pair of positive integers $k,n$ we ...
3
votes
0answers
101 views
“Almost” prime k-tuples in intervals
In this paper, Heath-Brown proves that there are $ \gg x(\log{x})^{-k}$ integers $n \leq x$ such that $$ \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}$$ where $\binom{X}{n}=\frac{1}{n!}\prod_{k=0}^{n-1}(X-k)$.
I am looking for an equivalent of $b_{n,k}$ when $k$ ...
2
votes
0answers
98 views
Is this a valid Hadamard product for $\frac{2\,\xi(s)-1}{s\,(s-1)}$?
This question builds on this MSE question:
Take the well known equation:
$$\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
71 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
145 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
153 views
Fermat and the abc conjecture [closed]
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
387 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
333 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
77 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
51 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
209 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
204 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
208 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
322 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
226 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
107 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
353 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 ...
5
votes
0answers
396 views
+50
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
107 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
202 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
154 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
404 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
253 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
122 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
132 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 ...
7
votes
1answer
204 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 ...
