All Questions
Tagged with reference-request nt.number-theory
1,408 questions
1
vote
1
answer
179
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
1
vote
0
answers
264
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 ...
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 ...
- 43.2k
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 ...
- 14k
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 ...
- 680
-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 ...
- 107
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
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
1
vote
1
answer
241
views
The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.
is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?
if $a$ is ...
- 133
0
votes
0
answers
140
views
State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
1
vote
0
answers
150
views
What are alternative mathematical definitions of observers beyond Bennett and Hoffman's framework?
Motivation:
This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested.
In the study of information ...
- 3,064
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
6
votes
1
answer
407
views
Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 2,907
2
votes
1
answer
431
views
Shadows of partitions of lcm
$\DeclareMathOperator\lcm{lcm}$Fix an integer $n\geq1$. Denote the least common multiple $L_n=\lcm(1,2,\dots,n)$.
QUESTION. Is the following true? For each integer partition $\lambda=(\lambda_1,\...
- 43.2k
1
vote
1
answer
222
views
Sum over three squares
Let $x$ be a sufficiently large number. Is there an explicit or asymptotic formula for the following sum
$$\sum_{\substack{n\leq x\\ n=a^2+b^2+c^2}} 1.$$ Any reference would be helpful.
- 1,257
5
votes
0
answers
156
views
Reference Request: Completeness of the space of all Whittaker models(a lemma in JPSS1981)
$\DeclareMathOperator\GL{GL}$There is a lemma in the proofs of local converse theorem stated as
Suppose $F$ is a non-archimedean local field, $\psi$ is a non-trivial addtive character on $F$. $N_n$ ...
- 53
0
votes
0
answers
101
views
Identities for Prime Coefficients of Certain Cusp Forms
While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients.
Let $a(n)$ be the Fourier coefficients ...
3
votes
0
answers
148
views
Solutions of a quadratic Diophantine equation over algebraic integers
The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring.
Precisely, let $\...
- 287
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
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)|...
user536681
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.
$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
- 5,063
3
votes
3
answers
383
views
On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 87
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
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)=...
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 ...
- 667
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)...
user536681
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
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 ...
- 195
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 $p$?
- 2,907
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 ...
- 43
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 ...
- 11.1k
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:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
2
votes
0
answers
179
views
A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound
In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$
I ...
user344548
1
vote
1
answer
142
views
Algorithm for computing isogeny class of elliptic curve
Is there an algorithm for computing the entire isogeny class of a given elliptic curve $E/\mathbb{Q}$?
References/ideas are welcome. Thanks!
- 2,907
3
votes
0
answers
148
views
Casimir eigenvalues of p-adic automorphic representations
In the context of p-adic local Langlands correspondence:
Is it possible to define Casimir eigenvalues for p-adic automorphic representations? If a local representation arises from a global Galois ...
- 2,907
0
votes
0
answers
81
views
Computing elliptic periods from modular form
How are the periods of a modular elliptic curve computed as path integrals of its associated normalized weight 2 cusp form on the modular curve? Please provide specific paths for both periods and cite ...
- 2,907
8
votes
0
answers
208
views
Elliptic curves of rank 1 over number fields
I am interested what is known about the following statement:
For every number field $K$, there exists an elliptic curve $E$ defined over $K$ with algebraic rank equal to $1$.
Is this statement known ...
- 502
3
votes
2
answers
342
views
Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism
I'm looking for a reference that provides a concise statement of the Beilinson-Bloch conjecture, specifically formulated in terms of an isomorphism under the Beilinson regulator map.
More precisely, I'...
- 2,907
12
votes
1
answer
238
views
Number of planes generated by integer vectors
For fixed dimension $d$ and large $R$ consider all non-zero integer vectors in the ball $B(0,R)\subset \mathbb{R} ^d$ of radius $R$ centered at the origin. The number of such vectors grows as $c_d\...
- 109k
8
votes
2
answers
178
views
Distribution of traces and max entries of words of fixed length in $\operatorname{SL}_2(\mathbb{N})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{\mathsf{tr}}$$\SL_2(\mathbb{N})$ is a free monoid on the generators
$$
L=\begin{pmatrix}1&0\\1&1\end{pmatrix},\quad R=\begin{pmatrix}1&1\...
- 609
2
votes
0
answers
102
views
Division based recurrence with negative coefficients, e.g. $F(n)= -F(\lfloor n/2\rfloor) - F(\lfloor n/3\rfloor)$
A famous problem of Erdos dealt with the division-based recurrence $a_n = a_{\lfloor n/2\rfloor}+a_{\lfloor n/3\rfloor}+a_{\lfloor n/6\rfloor}$ with $a_0=1$ (and was about the limit $\lim_{n\to\infty} ...
- 833
1
vote
0
answers
98
views
Are there known effective bounds on the number of semisimple Galois representations?
In continuation to my question here, are there known effective bounds on the total number of semisimple $p$-adic Galois representations unramified outside a finite set of primes $S$, of dimension $d$, ...
- 2,907
71
votes
8
answers
12k
views
Possible new series for $\pi$
In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \...
- 82.7k
0
votes
1
answer
223
views
Reference for Faltings' proof on finiteness of semisimple $d$-dimensional $p$-adic Galois representations
I'm looking for a reference to Faltings' proof concerning the finiteness of $d$-dimensional semisimple $p$-adic Galois representations. Specifically, the result states that there are only finitely ...
- 2,907
2
votes
0
answers
146
views
Reference for accelerated sum to compute the Meissel-Mertens constant
The Meissel-Mertens constant
$$ B_1 = \lim_{n \to \infty} \left(\sum_{p \leq n} \frac{1}{p} - \log\log n\right) $$
has the series representation
$$
\begin{equation} \tag{1}
B_1 = \gamma + \sum_{n=2}^{...
- 451
16
votes
1
answer
357
views
Galois cohomology for non-Galois extensions
If $L/K$ is a Galois extension with group $G$ then we can consider $H^*(G;L^\times)$. This is useful in algebraic number theory, and there are many results about it.
Now let $L/K$ be a finite ...
- 56.9k
16
votes
3
answers
4k
views
Is it known that the Collatz-like sequence with 7n+1 diverges to infinity starting with 7?
In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
- 541
5
votes
0
answers
174
views
Effective Hecke Equidistribution
In 1918 and 1920 Hecke introduced his L-functions attached to his Grössencharakteren (Hecke characters) and proved they are equidistributed in a sense to made precise momentarily. One can identify ...
- 177
15
votes
3
answers
1k
views
Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?
So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...
- 2,689
10
votes
0
answers
350
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 ...
- 2,836
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. ...
- 14k