Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
15,637
questions
0
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0
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115
views
Does there exist an $L$-function for any subset of $\mathbb{N}$?
Consider the following prime sum:
\begin{aligned}
\sum _{p}{\frac {\cos(x\log p)}{p^{1/2}}}
\end{aligned}
whose spikes appear at the Riemann $\zeta$ zeros as shown here.
Taking these detected spikes (...
- 1,873
0
votes
0
answers
162
views
Is the integer factorization into prime numbers normally distributed?
Let $P_1(n) := 1$ if $n=1$ and $\max_{q|n, \text{ }q\text{ prime}} q$ otherwise, denote the largest prime divisor of $n$.
Let us define some rooted trees $T_{n,m}$ for $1 \le m \le n$ by:
$T_{n,m}$ ...
- 2,126
3
votes
0
answers
46
views
$R$-recursion for the A249833 (similar to A235129)
Let $a(n)$ be A249833 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A(x) = 1 + \int A(x) + (A(x))^2\log A(x)\,dx
$$
The sequence begins with
$$
1, 1, 2, 7, ...
- 3,018
2
votes
0
answers
100
views
Computing the Dieudonné module of $\mu_p$ from Fontaine's Witt Covector
In Groupes $p$-divisibles sur les corps locaux, Fontaine introduced a uniform construction of Dieudonné modules through the definition of the Witt covector. Consider a perfect field $k$ of ...
- 135
0
votes
0
answers
173
views
On fifth powers forming a Sidon set
We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct.
Erdős conjectured ...
- 661
0
votes
0
answers
98
views
What is the logic behind the Extended Euclidean Algorithm procedure? [closed]
Thank you beforehand for reading my question.
In the terms that I'd want to understand the Extended version of the Euclidean Algorithm, I understand the Euclidean Algorithm as follows:
You find the ...
- 1
0
votes
1
answer
179
views
Is there any use of logarithmic derivatives of modular forms?
Does taking the logarithmic derivative of a modular form have any uses, such as identifying patterns in its coefficients or possible zeros of its corresponding L function?
4
votes
1
answer
240
views
Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?
Let $K$ be a number field. One has the following exact sequence relating the unit group and ideal class group $\text{cl}(K)$:
$$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$
...
- 141
4
votes
0
answers
76
views
Buchi's conditional proof of the non-existence of finite algorithm to decide solubility of system of diagonal quadratic form equations in integers
I am doing some literature review regarding Buchi's problem. In particular, I am reading the relevant section in this survey paper by Mazur (Questions of Decidability and Undecidability in Number ...
- 25k
4
votes
0
answers
108
views
Taking integer values of a sequence of Beurling primes
Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
- 41
0
votes
0
answers
131
views
what is the current best estimation for the upper bound of the exponential sum for an arbitrary irrational number $\alpha$
I would like to know what the current best estimation for the upper bound of the exponential sum
$$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\...
- 543
2
votes
1
answer
187
views
'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
- 113
2
votes
0
answers
84
views
$R$-recursion for the A235129
Let $a(n)$ be A235129 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A'(x) = 1 + A(x)\exp(A(x))
$$
The sequence begins with
$$
1, 1, 3, 12, 64, 424, 3358, ...
- 3,018
6
votes
0
answers
381
views
+50
Genus of a number field
I'm reading Algebraic Number Theory by Neukirch. In chapter 3, he defines the genus of a number field as
$$ g = \log \frac{ |\mu (K)| \sqrt{|d_K|}}{2^{r} (2\pi)^{s}} $$ where $|\mu(K)|$ is its ...
-6
votes
0
answers
125
views
How is research on Modular Forms usually conducted? [closed]
I have just started working on modular forms through an introductory course and let led me to think how is research on them conducted. For example, what exactly do you research about modular forms, is ...
4
votes
0
answers
121
views
Sequence of digits of powers of two
Elementary number theory tells us a lot about the final digits of the powers of two, and ergodic theory (more specifically the theory of equidistribution of points in the orbit of an irrational ...
- 19.1k
2
votes
0
answers
159
views
Can all modular forms be written as Eta Quotients?
I have been going through a couple of introductory courses in modular forms and am quite curious whether all modular forms can be written as eta quotients of the Dedekind eta function?
5
votes
0
answers
209
views
Video abstracts for mathematical papers
I recently published a video abstract of a manuscript of mine (number theory), finding that more people are interested in its content than when I uploaded the preprint on arXiv.
Now, my main question ...
- 883
5
votes
2
answers
391
views
On the number of distinct prime factors of $p^2+p+1$
Is it true that, for each positive integer $c$, there exists a prime number $p$ such that $p^2+p+1$ is divisible by at least $c$ distinct primes?
- 886
3
votes
0
answers
81
views
Is there a closed form for the Rudin-Shapiro sequence?
The Rudin-Shapiro sequence is defined as follows:
Let $a_n=\sum\epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2...$ are the digits in the binary expansion of $n$. $WS(n)$, the $n$th term of the ...
- 131
4
votes
2
answers
537
views
Computing hypergeometric function at 1
I'm looking to compute
$${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr
1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$
for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if ...
- 213
3
votes
0
answers
82
views
A question on the averages of Kloosterman sums
Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is,
For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound
$$\sum_{...
- 553
8
votes
1
answer
586
views
Does this partial sum over primes spike at all zeta zeros?
Below is a plot of $\exp \sum _p^x -\frac{\cos \left(x \log \
p\right)}{\sqrt{p}}$, where $p$ runs over the primes, and the $x$-values of the Riemann $\zeta$ zeros are marked with dashed lines:
Below ...
- 1,873
0
votes
0
answers
57
views
Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?
What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$,
every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...
- 19.1k
2
votes
0
answers
109
views
Can K$_3$ of finite fields be related to Teichmüller cocycles?
This is sort of a blind shot, but...
For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$.
To simplify matters, let $R$ be a finite field $\mathbb ...
- 17.3k
8
votes
1
answer
580
views
Arithmetic sequences and Artin's conjecture
(Sorry if this is a naive question; it is not my area!)
Consider the following strengthening of Artin's conjecture on primitive roots (and Dirichlet's theorem) for the case of $n=2$: every arithmetic ...
-1
votes
0
answers
359
views
Are these finite semirings known?
I am trying to prove the properties below, and by doing this, I hope to find a way to speed up the computation of the below defined addition and multiplication. I am also interested if these finite ...
- 2,126
3
votes
0
answers
85
views
Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operators" by Coleman and Iovita
Let $G$ be an abelian variety with good reduction over a finite extension $K$ of $\mathbb{Q}_p$. In equation (2.4) on page 179 of my edition of "The Frobenius and monodromy operators for curves ...
- 376
-4
votes
0
answers
189
views
EC primes the heuristic says something different from the reality [closed]
Ec-primes are primes of the form $(2^n-1)\cdot 10^d+2^{n-1}-1=ec(n)$, where d Is the Number of decimal digits of $2^{n-1}$.
Up to $n=10^5$, there are 30 primes of this type.
A rough heuristic says 18 ...
- 1
2
votes
2
answers
242
views
$L^1$ norm for a product of cosines
Let $k$ be an integer and consider the function
$$
f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t).
$$
I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
- 167
2
votes
0
answers
41
views
Simultaneous computation of the three Weber modular functions
Recall that the three classical Weber modular functions are defined by
$f(\tau)=e^{-\pi i/24}\eta((\tau+1)/2)/\eta(\tau)$,
$f_1(\tau)=\eta(\tau/2)/\eta(\tau)$, and
$f_2(\tau)=\sqrt{2}\eta(2\tau)/\eta(\...
- 11.1k
5
votes
0
answers
469
views
Theorem 7.11 in Scholze's $p$-adic Hodge Theory
I was trying to understand the statement and proof of Theorem 7.11 in Scholze's paper "$p$-adic Hodge Theory for Rigid-Analytic Varieties". I'll reproduce part of the statement below:
Let $...
- 51
2
votes
2
answers
569
views
Can any Hurwitz zeta function be written as an Euler product?
I am attempting to understand the behavior of Hurwitz zeta functions and for what $a$ do they have an analytic continuation. Is it possible to write any Hurwitz zeta as an Euler product or are there ...
1
vote
1
answer
95
views
Existence of tamely ramified tower of extension over $\mathbb{Q}_p$
Let $p$ be a prime. There exist following containment : $$\mathbb{Q}_p \subset \mathbb{Q}_p^{\rm nr} \subset \mathbb{Q}_p^{\rm tr} \subset \overline{\mathbb{Q}}_p$$
Here $\mathbb{Q}_p^{\rm nr}$ and $\...
- 133
6
votes
2
answers
601
views
Number of divisors which are at most $n$
I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
- 375
4
votes
0
answers
240
views
An algebraic version of the implicit function theorem for integers
$ \def \x {\boldsymbol x}
\def \a {\boldsymbol a}
\def \Z {\mathbb Z} $
The famous version of the implicit function theorem (IFT) starts with the assumption of continuous differentiability on the ...
- 345
5
votes
1
answer
339
views
Closed-form for the number of partitions of $n$ avoiding the partition $(4,3,1)$
Let $a(n)$ be A309099 i.e. the number of partitions of $n$ avoiding the partition $(4,3,1)$.
We say a partition $\alpha$ contains $\mu$ provided that one can delete rows and columns from (the Ferrers ...
- 3,018
9
votes
1
answer
361
views
How fast can elliptic curve rank grow in towers of number fields?
Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in ...
- 2,645
0
votes
0
answers
110
views
Roots in indefinite lattice of K3 surfaces
Anyone who likes $K3$ surfaces cares about lattices of the form $$ (2d)\cdot y^2 - 2x \cdot z$$ (namely the mukai pairing on $H^*_{alg}(K3)$ of picard $1$ with polarization $d$).
Inside we have ...
- 255
2
votes
1
answer
119
views
Cusp forms of weight 2 and level $\Gamma_0(p)$ where $p < 11$
Using Hida theory, we can prove that there is a cusp form of weight 2 and level $\Gamma_0(11)$. Are there ways to prove that there is no cusp forms of weight 2 and level $\Gamma_0(p)$ where $p < 11$...
- 133
2
votes
1
answer
181
views
Generating function over primes in an arithmetic progression
Given a newform $\sum_{n=1}^{\infty}a(n)q^n$. Is the generating function
$$
\sum_{p\equiv a\pmod{m}}a(p)q^p
$$
over the primes $p\equiv a\pmod{m}$ still a modular form? Any help is highly appreciated! ...
- 197
-1
votes
0
answers
98
views
Is it possible to write any Dirichlet series as a sum of reciprocals of arithmetic progressions? Where the Dirichlet character is primitive and real?
I was wondering since Dirichlet series can often be written as a subtraction or addition of sums of reciprocals of sequences, could you propose such a definition of a Dirichlet series, for any ...
2
votes
0
answers
57
views
Elementary recursion for the A258173
Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$.
A Dyck path of semilength $n$ is a $(x,y)$-...
- 3,018
2
votes
0
answers
128
views
Zeta zeros and prolate wave operators
Recently, Connes, Consani and Moscovici in https://arxiv.org/abs/2310.18423 have blended two of their results on zeta zeros and the prolate wave operators, which, they say, "suggests the ...
- 21
0
votes
1
answer
238
views
Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
- 2,126
1
vote
0
answers
64
views
Is there an upper bound on the number of partitions of a finite set of primes into 3 sets the products of 2 of which sum to the product of the third?
Is there an upper bound on the number of partitions of a finite set $S$
of prime numbers into 3 sets $A$, $B$ and $C$ for which the following holds?:
$$
\prod_{p \in A} p \ + \ \prod_{p \in B} p \ = ...
- 19.3k
5
votes
2
answers
374
views
When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension?
When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension?
For example, does $[\mathbb{Q}(\sqrt[n]{2},\sqrt[m]{3}):\mathbb{Q}]=mn$ hold true? Are there more ...
- 51
1
vote
1
answer
118
views
Unramified composition for every extension
Let $K$ be a number field and $S$ be a finite set of primes. Is it possible to construct a finite extension $M$ of $K$ such that $LM/M$ is unramified at (the primes above) $S$ for all degree $n$ ...
- 13
4
votes
1
answer
408
views
Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
- 191
3
votes
1
answer
552
views
binomial coefficients are integers because numerator and denominator form pairs?
I've heard of a claim that when calculating the binomial formula with integer input:
$\mathrm{Bin}(n,k):=\prod^k_{i=1}\frac{n+1-i}{i}\in \mathbb{N}\ (\forall n,k\in\mathbb N)$
each denominator divides ...
- 187