All Questions
Tagged with reference-request nt.number-theory
1,408 questions
7
votes
1
answer
683
views
The Gauss Circle Problem asymptotic in dimension
The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\...
6
votes
2
answers
1k
views
$\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?
There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples:
Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ ...
- 1,776
13
votes
1
answer
2k
views
For a proof of the three-square theorem without using Dirichlet's theorem on primes in arithmetic progressions
The three-square theorem states that $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three squares if and only if it is not of the form $4^k(8m+7)$ ($k,m\in\mathbb N$). This was first proved by ...
- 15.6k
13
votes
2
answers
586
views
How does the Bernstein-Zelevinsky construction of irreducibles from supercuspidals parallel the representations of the Weil-Deligne group?
In the Corvallis article Number Theoretic Background, here is what John Tate has to say on the local Langlands correspondence for a $p$-adic field $F$:
So, granting a correspondence between ...
- 6,180
3
votes
0
answers
139
views
L-functions for the Weil group over short exact sequences
Let $(\rho,V)$ be a continuous finite dimensional representation of the Weil group $W_F$ over a local field $F$. If $V$ decomposes as a direct sum $V_1 \oplus V_2$ of representations, then
$$L(s,\...
- 6,180
19
votes
1
answer
1k
views
Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$
Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\...
- 6,180
4
votes
0
answers
313
views
Four-distance problem
A Problem due to Steinhaus is the following: do there exists a point in the plane at rational distance from the corners of the unit square?
Please give me a reference if this problem is solved.
- 367
0
votes
1
answer
322
views
On the largest prime factor of $1+n^k$
For every positive integer $n>1$ , let $f(n)$ denote the largest prime factor of $n$. How fast does $f(1+n^k)$ grow with respect to $k$ ? Is it true that $f(1+n^k) > 2k, \forall n >2, \forall ...
user111492
0
votes
0
answers
142
views
Mobius function on values of an irreducible quadratic polynomial
Are there infinitely many integers $n$ for which $n^2 + 1$ is square-free, and has an even number of (necessarily distinct) prime factors ?
- 11.3k
3
votes
1
answer
188
views
Reference request: denominators of lonely runner numbers
The lonely runner conjecture states that for $M=\{m_1,...,m_n\}$ a set of distinct positive integers the quantity
$$
\kappa(M):=\sup_{t \in \mathbb{R}} \min_i ||tm_i||
$$
satisfies $\kappa(M) \geq \...
- 2,753
14
votes
0
answers
481
views
If $ab^2$ is a sum of three squares, then so is $a$. How to see it quickly?
Here $a, b$ are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way?
- 109k
3
votes
0
answers
201
views
Growth Rate of the Square-Free Part
In the course of considering this Diophantine equation, I convinced myself that the following question is interesting:
If $n$ is large, must it be the case that the square-free part of $2^n-1$ is ...
- 153
4
votes
0
answers
190
views
Restricted Iwasawa theory
Let $p$ be a prime number, let $K$ be a number field, and let $L$ be a Galois extension of $K$ such that the Galois group $\Gamma$ of $L$ over $K$ is (continuously) isomorphic to the (additive) group $...
- 11.3k
8
votes
2
answers
354
views
Let $f \in \mathbb{Z}[x]$. Does $\bar{f}$ have as many roots in $\mathbb{F}_p$ as $f$ has in $\mathbb{C}$ for infinitely many primes $p$?
Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}_p[x].$ Let $\rho_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}_p$, and let $\rho$ be the number ...
1
vote
1
answer
306
views
Basic question regarding notation of summation over primitive characters
This seems like a very standard notation in analytic number theory, and I see it a lot. But I was confused with it and I would greatly appreciate any clarification.
When one writes sum of the shape
$$...
- 3,625
3
votes
0
answers
154
views
Is there a name for sequences of integers reduced to their lowest prime divisors?
When trying to obtain the value of Jacobsthal's function for some $n$; to find the largest sequence of consecutive numbers that are all coprime to $n$, one approach (and the only direct approach that ...
- 161
6
votes
2
answers
782
views
Books building up to the Gross-Zagier formula
I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of ...
- 169
0
votes
1
answer
308
views
Have you seen this prime distribution before?
The basic question is : has this system been considered before, and how do I find it? References to the literature would be most welcome, but I am asking for reasonable search terms. I will try the ...
- 13k
13
votes
1
answer
760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
- 25.4k
2
votes
1
answer
191
views
Synchronised $\beta$-shifts
I have been reading some papers recently, in particular, Blanchard's paper $\beta$-expansions and symbolic dynamics which state that a $\beta$-shift $S_{\beta}$ is a synchronised shift if and only if ...
3
votes
1
answer
330
views
Growth Rate of Number of Divisors of Highly Composite Numbers
Let $H(k)$ be the $k^{th}$ highly composite number (HCN).
What is an asymptotic estimate for the size of $H(k)$ in terms of $k.$
What is an asymptotic estimate for the size of $d(H(k))$ where $d(n)$ ...
- 10.4k
4
votes
1
answer
292
views
References on Erdos conjecture on arithmetic progressions
Erdos conjectured that any set $ A $ of positive integers such that $ \sum_{n\in A}\dfrac{1}{n} $ diverges contains arbitrary long arithmetic progressions. The celebrated Green-Tao theorem is a ...
- 6,984
8
votes
2
answers
1k
views
How are such sets of natural numbers called?
I heard about this problem an year ago, but I just can't remember the name.
The problem goes like this: study the sets
$\{a_1,a_2,\dotsc,a_m\}\subseteq\mathbb{N}$ such that if $1\leq i<j\leq m$,...
- 103
3
votes
1
answer
632
views
Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$?
This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even ...
- 128k
2
votes
1
answer
241
views
2-parts of class numbers of binary quadratic forms for non-fundamental discriminants
I need a formula for the 2-adic valuation of the number of proper equivalence classes of primitive positive definite binary quadratic forms of discriminant $-D$, call it $h_0(-D)$. I'm sure the answer ...
- 189
7
votes
3
answers
530
views
Lower bound for the fractional part of $(4/3)^n$
My son, who is 16, is doing some independent research. A lower bound depending on $n$ for $\left\{ \left( \frac{4}{3} \right)^n \right\}=\left( \frac{4}{3} \right)^n-\left\lfloor \left(\frac{4}{3} \...
- 5,450
3
votes
0
answers
198
views
On the equation $x^3+y^3+z^3-2xyz=N$
The following question was asked at MSE without any solution:
Show that the equation $x^3+y^3+z^3-2xyz=1$ have infinitely many integer solutions $(x,y,z)$.
A more general question was also ...
- 3,153
11
votes
1
answer
328
views
Critical points of Dirichlet L functions
Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal
character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached ...
- 1,411
4
votes
0
answers
104
views
Asymptotics of the dimensions of the plus and minus space of Atkin-Lehner
Let $S_2^{+}(p)$ be the space of newforms of level $p$ that have Atkin-Lehner eigenvalue +1, and $S_2^{-}(p)$ be the space that have Atkin-Lehner eigenvalue $-1$. What is known about the asymptotics ...
- 2,429
6
votes
1
answer
287
views
Number of solutions for the inequality with square roots
Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality
$$|\sqrt{n_1}+\sqrt{n_2}-\sqrt{n_3}-\sqrt{n_4}|<\delta\sqrt{M},$$
where $...
- 7,193
3
votes
1
answer
189
views
The Golay-Rudin-Shapiro sequence as “Hankel transform”
Let ${\left( {{a_n}} \right)_{n \geqslant 0}}$ be a sequence of real numbers and ${H_n} = \det \left( {{a_{i + j}}} \right)_{i,j = 0}^{n - 1}$ the $n-$th Hankel determinant. The sequence ${\left( {{...
- 5,622
8
votes
1
answer
811
views
Primes of the form $x^2 + y^2 + 1$
There are infinitely many primes of the form $x^2+y^2+1$, as proved by Bredihin. Motohashi improved the result by showing that there were $\gg x/\log^2 x$ such primes up to $x$. But we expect $\Theta(...
- 9,114
13
votes
1
answer
358
views
Cartography of the duals of GL, PGL, SL, etc
A short version of this question could be
What are the duals of $PGL(2,\mathbf{Q}_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?
I should obviously add some precisions.
there are different ...
- 5,424
2
votes
1
answer
334
views
Intuition behind the proof of key step in Minkowski's second inequality on successive minima
I recently knew of this note in which Prof. M. Henk presents a proof of Minkowski's second inequality on successive minima which is (purportedly) based on ideas in Minkowski's original proof. Let me ...
- 8,842
6
votes
1
answer
714
views
Best estimate of the Mertens function without assuming the Riemann Hypothesis
I'm searching the best known upper bound for the Mertens function, but without assuming the Riemann hypothesis.
Landau, in 1901, have proved that $M(x)= O(x \exp(-c\sqrt{\ln x})$, but I am unable to ...
- 173
2
votes
2
answers
509
views
Question about Zeta Function of Singular Plane Curve
I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes).
I ...
- 53
1
vote
2
answers
259
views
Does the set of Diophantine $m$-tuples has full measure?
We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies the Diophantine condition of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\...
- 798
8
votes
1
answer
247
views
Origin of definitions of ramified Hecke operators
Consider a classical space $M_k(N)$ of elliptic modular forms of weight $k$ for $\Gamma_0(N)$. The definition of an unramified Hecke operator $T_{p^m}$ in terms of double cosets is the disjoint union ...
- 6,039
1
vote
1
answer
199
views
Simultaneous Diophantine Condition and Growth Rate of Convergents Denominators
Let $\omega=(\omega_1,\ldots,\omega_{m})$ be an $m$-tuple of real numbers. Let $|\omega|_{m}:=\sup\limits_{1 \leq j \leq m}|\omega_j|_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\...
- 798
4
votes
0
answers
115
views
Relations between an projective variety and galois cohomology
Let $f_1, \cdots, f_k$ be homogeneous polynomials over $\mathbb{Q}[x_0, \cdots, x_n]$. They define an projective variety $X$ over $\mathbb{P}^n(\mathbb{C})$, namely their set of zeros $$X = Z(f_1, \...
- 423
3
votes
0
answers
254
views
Looking for Coleman's paper "The Gross-Koblitz formula"
I am looking for a copy of the following paper by Robert Coleman:
The Gross-Koblitz formula.
Galois representations and arithmetic algebraic geometry
(Kyoto, 1985/Tokyo, 1986), 21–52,
Adv. Stud. ...
- 3,107
1
vote
0
answers
115
views
Properties of the function $\chi_{s,k}$
Let $\chi_{s,k}$ be the characteristic function of integers $n$ which are expressed as sum of $s$ positive $k$-th powers i.e $\chi_{s,k}(n)=1$ if and only if $n=a_1^k+\cdots+a_s^k.$ Examples of this ...
- 1,257
1
vote
1
answer
241
views
Is it possible to approximate irrational by fractions with denominator and numerator odd? [closed]
Suppose $\alpha$ is a positive irrational, and $\epsilon$ is an arbitrary positive real, are there $m,n$(non-negative integers) such that $$|\alpha-(2m+1)/(2n+1)|<\epsilon/(2n+1)?$$
If they exist, ...
- 201
5
votes
0
answers
317
views
Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$
where $k$ is unknown, $f_{k}...
- 12.3k
8
votes
2
answers
564
views
Distribution of primitive roots, as p varies
For a prime number $p$, let $\Phi(p)$ be the subset of $\{ 1, 2, \ldots, p-1 \}$ consisting of primitive roots modulo $p$. (Thus $\# \Phi(p) = \phi(p-1)$, where $\phi$ denotes the totient.)
I am ...
- 13.3k
2
votes
1
answer
217
views
Diagonalising a symmetric matrix with polynomial entries
Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x_1, \ldots, x_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^...
- 3,625
9
votes
2
answers
2k
views
Any simple concrete proof of Faltings theorem?
Are there simple proofs of some concrete special cases of Faltings's theorem? Any help would be appreciated.
- 3,094
7
votes
3
answers
926
views
Lefschetz fixed-point theorem for the Frobenius map
Where can one find a proof of Lefschetz fixed-point theorem for the Frobenius map on elliptic curves over algebraic closures of $F_{p}$ ?
This could immediately follow if their coholomogies (for the ...
- 305
9
votes
1
answer
419
views
Infinitely many solutions to $a^4+b^4+c^4=18$ over $\mathbb{Z}[i]$
We got infinitely many solutions to $a^4+b^4+c^4=18$ over
$\mathbb{Z}[i],i^2=-1$. Probably we can get infinitely many solutions
to $a^5+b^5+c^5=N$ over $\mathbb{Z}[\alpha]$ for algebraic $\alpha$.
...
- 25.4k
0
votes
0
answers
257
views
Hercules and the Hydra with time constraints
The game of Hercules vs. the Hydra can be put in terms of a single number in hereditarily-factorized form. For example, if the Hydra is $2^{19^3} \cdot 5^{11^7}$, Hercules must choose between two ...
- 2,302