All Questions
Tagged with reference-request nt.number-theory
1,408 questions
6
votes
4
answers
896
views
Mathematical induction vis-à-vis primes
One of the most used proof-techniques is mathematical induction, and one of the oldest subjects is the study of prime numbers. Thanks to Euclid, we can consider the primes as a infinite monotone ...
- 12.9k
1
vote
1
answer
180
views
+50
On a probabilistic integer factorization algorithm given bounds for one prime factor
We got a probabilistic integer factorization algorithm and experimental evidence with large
integers given bounds for one factor.
Let $D \ge 2$ be real number and let $p,q$ be primes and $N=pq$.
...
- 25.4k
9
votes
1
answer
650
views
Sum of three squares as class numbers and Waldspurger's formula
It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...
CommunityBot
- 1
8
votes
1
answer
410
views
An unpublished note by Bloch-Kato on p-divisible groups and Dieudonné crystals
I wonder if anyone could find the following unpublished paper of Bloch-Kato:
Spencer Bloch and Kazuya Kato, $p$-divisible groups and Dieudonné crystals, unpublished.
A similar question is here ...
- 188k
1
vote
0
answers
265
views
Are there connections between Calabi-Yau manifolds and number theory?
I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
- 63.9k
0
votes
0
answers
140
views
State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
2
votes
1
answer
213
views
Representing positive integers $n$ by binary forms $n=ax^2+by^2$, $a\geq 0$, $b\geq 0$
In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$
can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?
This is related to ...
- 105k
8
votes
1
answer
673
views
Infinite series and sum of two squares
Consider the following infinite sequence $a(n)$ generated by
$$\sum_{n\geq0} a(n)q^n
=\frac{\sum_{k\geq0}F(2k+1)q^{\binom{k+1}2}}{\sum_{k\geq0} q^{\binom{k+1}2}}$$
where the $F(2k+1)$ are the odd ...
- 79.9k
1
vote
2
answers
295
views
Possible refinements of the large sieve inequality
Let $a_n$, $1\leq n\leq N$, be complex numbers, and set $S(\alpha)=\sum\limits_{n=1}^{N}a_ne(n\alpha)$, where $e(\alpha)=\exp(2i\pi\alpha)$. Then, Selberg's large sieve inequality says that
$$\sum\...
- 341
-3
votes
1
answer
194
views
Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
- 1,381
1
vote
0
answers
45
views
Proper Pisot n-tuples
Recall that x is a Pisot number if it is real and x>1, while all of its conjugates have magnitude less than 1. Then $\{(x)^k\}$ (where $\{\cdot\}$ is the fractional part of x) approaches 0 ...
- 3,319
10
votes
0
answers
351
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 24.2k
2
votes
5
answers
611
views
A good introduction to the study of the Thue Equation
Hi,
I am interested in studying the Thue equation, where we are concerned with a binary form $F(x,y) = a_0 x^r + a_1 x^{r-1}y + \cdots + a_r y^r$ and solutions of the form
$$F(x,y) = h$$
for some ...
- 704
42
votes
4
answers
4k
views
Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
3
votes
1
answer
439
views
Chinese remainder theorem for target interval
Given $n$ pairwise coprime natural numbers $m_{1}, \dots, m_{n}$ with remainders $y_{i}$, for all $i \leq n$. Furthermore, we have a target interval $I := \left[ a, b \right]$, with $1 \leq a < b \...
CommunityBot
- 1
1
vote
1
answer
241
views
The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.
is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?
if $a$ is ...
- 651
3
votes
1
answer
121
views
Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 105k
5
votes
1
answer
310
views
Is there a statement in Presburger arithmetic about primes this simple heuristic fails for?
I came up with the following conjecture while thinking about ways to formulate some heuristics about primes:
Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary ...
- 16.6k
6
votes
1
answer
835
views
Beauty of some numbers discovered by Ramanujan
I am a graduate PhD student and my topic is analytic number theory. I am also a mathematics teacher. I am planning to give a course to pupils in high school that motivates them to study arithmetic and ...
- 19.6k
2
votes
1
answer
186
views
On local Galois deformation rings
Let $p,\ell$ be two different primes. Let $K$ be a finite field extension of $\mathbb{Q}_{\ell}$ and $ \bar{\rho}:G_{K}\to {\rm GL}_{n}(\mathbb{F}_p) $ be a continuous mod $p$ representation of the ...
- 190
1
vote
0
answers
150
views
What are alternative mathematical definitions of observers beyond Bennett and Hoffman's framework?
Motivation:
This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested.
In the study of information ...
- 3,064
12
votes
2
answers
1k
views
Dihedral extensions and the Ankeny–Artin–Chowla conjecture
Jensen and Yui (Polynomials with $D_p$ as Galois group
J. Number Theory 15, 347–375 (1982)) proved that if $p = 4n+1$
is a regular prime, then there is no normal extension of the
rationals with Galois ...
- 12.9k
42
votes
5
answers
14k
views
The unproved formulas of Ramanujan
Are there any formulas due to Ramanujan that have still not been proved—or disproved?
If so, what are they?
I believe this conjecture is due to Ramanujan and still open: if $x$ is a real number and $2^...
- 2,858
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
6
votes
1
answer
407
views
Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 10.7k
2
votes
1
answer
431
views
Shadows of partitions of lcm
$\DeclareMathOperator\lcm{lcm}$Fix an integer $n\geq1$. Denote the least common multiple $L_n=\lcm(1,2,\dots,n)$.
QUESTION. Is the following true? For each integer partition $\lambda=(\lambda_1,\...
- 23.7k
7
votes
2
answers
1k
views
Selberg class definition and Riemann hypothesis
Looking at the Selberg class definition on Wikipedia, under "Comment on definition", there is this paragraph:
"The condition that the real part of $\mu_i$ be non-negative is because ...
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 105k
55
votes
2
answers
3k
views
Is it known? A sum over lattice parallelograms of area one is equal to $\pi$
I recently discovered a formula, my proof is really a high school proof in three lines.
$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
- 5,063
3
votes
0
answers
148
views
Solutions of a quadratic Diophantine equation over algebraic integers
The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring.
Precisely, let $\...
- 287
1
vote
1
answer
175
views
Original text about Riesel numbers?
Riesel numbers appeared in the answer to my question
Solving $2^{x+1} m -1=p^y$ for prime $p$ and natural $x,y$.
The original reference
Riesel, Hans (1956). "Några stora primtal". Elementa. ...
- 105k
5
votes
0
answers
156
views
Reference Request: Completeness of the space of all Whittaker models(a lemma in JPSS1981)
$\DeclareMathOperator\GL{GL}$There is a lemma in the proofs of local converse theorem stated as
Suppose $F$ is a non-archimedean local field, $\psi$ is a non-trivial addtive character on $F$. $N_n$ ...
- 53
1
vote
1
answer
222
views
Sum over three squares
Let $x$ be a sufficiently large number. Is there an explicit or asymptotic formula for the following sum
$$\sum_{\substack{n\leq x\\ n=a^2+b^2+c^2}} 1.$$ Any reference would be helpful.
- 105k
0
votes
0
answers
101
views
Identities for Prime Coefficients of Certain Cusp Forms
While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients.
Let $a(n)$ be the Fourier coefficients ...
2
votes
2
answers
363
views
Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
- 11.1k
3
votes
3
answers
383
views
On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 50.6k
11
votes
4
answers
707
views
Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?
Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
- 14.6k
5
votes
0
answers
261
views
Primes generated by cyclotomic polynomials
Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=...
CommunityBot
- 1
8
votes
1
answer
465
views
Smooth numbers in short intervals
Let $\psi(x,y)$ be the number of positive integers up to $x$ which are $y$-smooth, that is, integers whose prime factors are at most of size $y$. There has been, for a few decades now, a lot of ...
- 14.6k
1
vote
0
answers
25
views
Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
- 111
6
votes
2
answers
804
views
Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the ...
- 44.4k
2
votes
1
answer
191
views
Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\sum_{\substack{n\le x\\ n\equiv a\pmod k}}d_r^{\ell}(n)\ll\frac{x}{k}\left(\frac{\phi(k)}{k}\log x\right)^{r^{\ell}-1}.$$
Here, $d_r(n)...
- 651
7
votes
1
answer
707
views
How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
CommunityBot
- 1
3
votes
0
answers
79
views
Reference request for unitary Shimura varieties
Let $K$ be an imaginary quadratic field and let $X$ be the Shimura variety associated to the unitary group $U(m,n)$ over $K$ (after a suitable choice of PEL datum).
Is there a reference that explains ...
- 1,949
4
votes
1
answer
165
views
Reference Request: on an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields
The 1978 preprint by S.Kato 'On an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields' is cited by papers involving unramified computations of local ...
- 188k
0
votes
0
answers
108
views
looking for reference for two elliptic curves with equal formal group
I am looking for a reference.
In this post, @Chris Wurthrich made the following comment:
If the formal group laws (probably upon particular choice of coordinates) of two elliptic curves over any ring ...
- 18.4k
2
votes
1
answer
101
views
Bound on number of extensions of Q unramified outside a fixed prime
Are there any known asymptotic bounds on the number of degree $d$ extensions of $\mathbb{Q}$ unramified outside a fixed prime $p$?
- 26.9k
4
votes
0
answers
168
views
Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 328
0
votes
2
answers
215
views
Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
- 127
32
votes
2
answers
3k
views
The Erdős–Turán conjecture or the Erdős conjecture?
This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
Conjecture: If $A\subset \mathbb{N}$ and $$\...
CommunityBot
- 1