All Questions
Tagged with reference-request nt.number-theory
388 questions with no upvoted or accepted answers
3
votes
0
answers
163
views
Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$
Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding
$$
\int_0^T L\left(\tfrac{1}{2} + it, f \...
- 1,570
3
votes
0
answers
168
views
Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
3
votes
0
answers
158
views
Conics over number fields
I am looking for a reference for the following fact.
Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such ...
- 21.3k
3
votes
0
answers
293
views
multiplicity of automorphic representation of unitary similitude group
Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its ...
- 66
3
votes
0
answers
454
views
What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$
What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,...
- 3,094
3
votes
0
answers
680
views
Birch/Swinnerton-Dyer "Notes on Elliptic Curves II"
I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168).
...
- 521
3
votes
1
answer
340
views
Proof of continued fraction identity of subfactorial
This question is part of a wider conjecture I have formed with someone which has its roots in Raayoni et al. (2019). The subfactorial function can be written as $$!n=\frac{n!}e+\frac{(-1)^n}{n+2-\...
- 1,459
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
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
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
$$
\begin{equation} \tag{1}
B_1 = \gamma + \sum_{n=2}^{...
- 451
2
votes
0
answers
278
views
On $(k,\ell)$-sumfree sets
Call a set $\mathcal S \subset \mathbb N$ to be $(k,\ell)$-sumfree if there are no non-trivial solutions to the equation
$$x_1+\dots +x_k = y_1+\dots +y_\ell$$
in the set (for distinct $x_i$'s and $...
- 791
2
votes
0
answers
129
views
Imaginary quadratic fields with prime class number
Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$.
In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write,
"Since $h_K = p$, there ...
- 707
2
votes
0
answers
221
views
Squares whose differences are squares
EDIT. I've just noticed a thread from 2011 in the "Related" column on the right (click me), where a closely related question is being discussed (the main difference seems to be that, in ...
- 10.5k
2
votes
0
answers
489
views
Are these finite semirings known?
I am trying to prove the properties below, and by doing this, I hope to find a way to speed up the computation of the below defined addition and multiplication. I am also interested if these finite ...
- 3,064
2
votes
0
answers
286
views
Is Sturm's theorem able to do these?
$\newcommand{\Ord}{\operatorname{Ord}}$Let $p$ be a positive integer and $F(q)=\sum A(m)q^m$ be a formal power series with integer coefficients. Then $\Ord_p(F(q))$ is defined by
$$\Ord_p(F(q)):=\min\{...
- 43.2k
2
votes
0
answers
103
views
On equidistribution of primes in positive characteristic
In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
- 675
2
votes
0
answers
115
views
Semiprimes of the form n!+1
It is a well-known conjecture that there exist infinitely many primes of the form $n!+1$.
This problem is somewhat similar to the famous Landau's problem about the infinitude of the primes of the form ...
- 21
2
votes
0
answers
228
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 43.2k
2
votes
0
answers
109
views
Action of Galois group on the lattice of a Drinfeld module - a reference?
What is a reference for the following construction?
Let $K$ be a finite extension of $\Bbb F_q(\theta)$ and $M$ a Drinfeld module of rank $r$ defined over $K$. Its lattice $L(M)$ is a free $\Bbb F_q[\...
2
votes
0
answers
79
views
Which sets of natural numbers are "lambda-analytic"?
Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define
$$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$
for all real numbers $x \in ...
- 13.3k
2
votes
0
answers
110
views
Asking for a generating function for an arithmetic sequence
For fixed integer $n\geq1$, let $c_m(n)$ be the number of divisors $d$ of $m$ such that $n<d\leq 2n$. Here is an experimental generating function for which I ask:
QUESTION. Is this true?
$$\sum_{m\...
- 43.2k
2
votes
0
answers
197
views
Mumford's computation of the determinant of cohomology of a relative curve
In Integral Grothendieck-Riemann-Roch theorem, Pappas mentions that Mumford computed the determinant of cohomology of $f:X\to S$ a relative curve integrally, and thus proved an integral version of GRR ...
- 2,044
2
votes
0
answers
182
views
On the relative class number of a cyclotomic extension
Let $\Bbb Z[\zeta_p]$ denote the cyclotomic integers where $p$ is a prime and let $h_1 = h_1(p)$ denote its relative class number.
Question: Is it known whether there are infinitely many primes $p$ ...
- 18.8k
2
votes
0
answers
490
views
On quasi-modular forms with integer Fourier coefficients
It is well-known that the ring $M$ of modular forms has the structure $M=\mathbb{C}[E_4,E_6]$, where $E_k$ are the Eisenstein series.
It is also known that one can define the concept of quasi-modular ...
- 6,123
2
votes
0
answers
156
views
A question on terminology for sequences satisfying $\gcd(a_m,a_n)=a_{\gcd(m,n)}$
How do you refer to those sequences $\{a_{n}\}_{n \in \mathbb{Z}^{+}}$ of integers that satisfy the condition $\text{gcd}(a_{m}, a_{n}) = a_{\text{gcd}(m,n)}$ for every $(m,n) \in \mathbb{Z}^{+} \...
- 87
2
votes
0
answers
1k
views
Advanced texts on analytic number theory?
So a friend of mine is very interested in analytic number theory, and is looking for resources past the basic level.
He has studied analytic number theory from several books, among them are Hardy’s ...
2
votes
0
answers
150
views
Closeness of a rational approximation
What is
$$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N}
|2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$
where $\mathbb N:=\{1,2,\dots\}$?
In other words, I would like to ...
- 128k
2
votes
0
answers
115
views
Reference request: "A result of Siegel" related to Ramanujan-Nagell type equations
Wikipedia refers to the Diophantine equation
$ x^2 + D = AB^n $
as an "equation of Ramanujan–Nagell type". It also says that "A result of Siegel implies that the number of solutions in ...
2
votes
0
answers
93
views
Showing that it is not possible that for every $q_j$ from a finite set of odd primes, it holds that $2+\prod_{k \neq j} q_k $ is divisible by $q_j$
This a repost of a question which was asked at MathStackExchange, but got no answer so far, so I am trying here.
Let $n\ge 1$ and let $Q= \{q_1,\dotsc, q_n\}$ be a set of $n$ odd primes, all different ...
- 505
2
votes
0
answers
245
views
Ambiguity about the exact definition of coefficients of modular forms
You can see the parts after my questions in the boxes. I received the answer to my first question in the comments.
I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
2
votes
0
answers
80
views
Set of integer non-negative matrices with positive diagonals
This is essentially a reference request/name inquiry. Is there a name for the set $M_k$ formed by $k$ by by $k$ matrices with non-negative integer entries and positive values on the diagonal? Related, ...
- 6,969
2
votes
0
answers
140
views
Integers with exactly three factor pairs whose successors are relatively prime
I am interested in the following problem, and will appreciate pointers around how it can be solved – partially or fully – and/or indicators around whether it is even tractable:
Characterize $N \in \...
- 7,831
2
votes
0
answers
161
views
Monotonicity of the cycle index polynomial under restriction
The cycle index (polynomial) of the symmetric group $\mathfrak{S}_n$ is given by the formula:
$$Z(\mathfrak{S}_n)(x_1,\dots,x_n)=\sum_{1j_1+2j_2+\cdots+nj_n=n}\prod_{k=1}^n\frac{x_k^{j_k}}{k^{j_k}j_k!}...
- 43.2k
2
votes
0
answers
480
views
About derived divided power envelope
Assume $A$ is a $\mathbb{Z}_{(p)}$-algebra with ideal $I$ and $A,A/I$ are $p$-torsionfree.
In this survey, Akhil Mathew defines the derived divided power envelope $LD_I(A)$ in Construction 7.15, after ...
- 121
2
votes
0
answers
157
views
On hypergeometric functions over finite fields
Let $\mathbb{F}_q$ be a finite field of $q$ elements. Let $A,B,C,\cdots$ denote the multiplicative characters over $\mathbb{F}_q$, and let $\overline{A}$ denote the inverse of $A$, i.e., $A(x)\...
- 93
2
votes
0
answers
161
views
Has there been much research on the Iwasawa theory of bi-quadratic fields?
The simplest non-trivial example of $\mathbb{Z}_p$-extensions happen over imaginary quadratic fields and I've seen a good amount of research done here (from my understanding still a lot to learn). I ...
- 707
2
votes
0
answers
110
views
Spectral decomposition of idele class group $L^2(\mathbb{I}_F/ F^{\times})$
Let $F$ be a number field. Let $\mathbb{A}_F$ be the ring of adeles. The group of units of $\mathbb{A}_F$ is called the group of ideles $\mathbb{I}_F=\mathbb{A}_F^{\times}= GL_1(\mathbb{A}_F)$. The ...
- 421
2
votes
0
answers
98
views
Sublattices in the standard integral symplectic lattice
Let $V$ denote $\mathbb{Z}^{2g}$ with its standard integral symplectic form $\omega = \sum_{i=0}^{g-1}dx_{2i} \wedge dx_{2i+1}$ (or, the homology lattice of a genus $g$ surface with its intersection ...
- 427
2
votes
0
answers
145
views
On some rational points on an elliptic curve over finite field
Let $p\equiv3\pmod4$ be a prime. We consider the elliptic curve $E$ over the finite field $\mathbb{F}_p$
(in affine coordinates) defined by
$$y^2=x^3+x.$$
Clearly the discriminant of $E$ is $-2^6$. ...
user125345
2
votes
0
answers
491
views
Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
- 739
2
votes
0
answers
186
views
Dyadic models in number theory and "spillover"
In a classic blog post, Tao discusses the appearance of "dyadic models" in various guises in various areas of math. The number-theoretic version of the idea is to study polynomials over a ...
2
votes
0
answers
65
views
Request for resources or techniques for bounding the infinity norm of an infinite product convolved with a simple function
I'm attempting to bound an expression of the form.
$$
\lVert(\prod_{i=1}^{\infty} \phi_i) * s \rVert_{\infty}
$$
Where $\phi_i$ are bounded periodic step functions which can be replaced by smoothed ...
- 41
2
votes
0
answers
313
views
On the Chowla and twin prime conjectures
I'm reading https://terrytao.wordpress.com/tag/chowlas-conjecture/ and at some point it is mentioned that, the twin prime conjecture is a variant of Chowla's conjecture that $\sum_{n\leq x} \lambda(n)\...
- 1,019
2
votes
0
answers
97
views
Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?
Nowadays there are many papers on the number theory using heuristics.
I have read some of them.
But I have no clear understanding of the Bayesian Probability(subjective probability).
The concept of ...
- 1,013
2
votes
0
answers
341
views
Dirichlet series of powers of the prime omega function
Let $\omega(n)$ denote the number of distinct prime factors of a positive integer $n$.
I was wondering what is known about the dirichlet series
$$\sum_{n=1}^{\infty}\frac{\omega(n)^k}{n^s},$$
in ...
- 195
2
votes
0
answers
212
views
show that sequence $\{(-1)^n\Upsilon_n\}$ is convergent and strictly decreasing
Edit: Few years ago, I have posted my claim on $\Upsilon$ function regarding prime number but recently I have observed, last observation turns false that's way, (by putting $\Upsilon$ value in ...
- 599
2
votes
0
answers
125
views
How many integers below $n$ can be expressed as a sum of $k$ $m$th powers?
For $m,k \geq 2$, let $C_{m,k}(n)$ denote the number of positive integers less than or equal to $n$ which can be expressed as a sum of $k$ $m$th powers.
I am interested in the asymptotic behavior of ...
- 163
2
votes
0
answers
147
views
Well-known estimate for $L(s,\chi)$ for $\sigma=\text{Re}s\geq 1/2$
This is a very short question.
Let $s=\sigma+it$ be a complex number with $\sigma \geq 1/2$.
In the paper 'Jutila, Matti. "On the Mean Value of $L(1/2, \chi)$ FW Real Characters." Analysis 1.2 (...
- 663
2
votes
0
answers
160
views
Where can I find a copy of this paper of Chowla and Vijayaraghavan?
Does anyone know where I can find a copy of Chowla and Vijayaraghavan's paper, ''On the largest prime divisors of numbers''?
The relevant literature say it was published in the Journal of the Indian ...
- 1,019
2
votes
0
answers
76
views
Is there an estimate available for a sum of the form $\sum_{\mathbf{x} \equiv \mathbf{a} (H) } \mu^2(x_1 x_2)$
I am interested in a sum of the shape
$$
\sum_{ \substack{ 1 \leq x_1, x_2 \leq B\\
\mathbf{x} \equiv \mathbf{a} (H) } } \mu^2(x_1 x_2).
$$
I figured it must have been considered before, but I have ...
- 3,625