All Questions
Tagged with reference-request nt.number-theory
1,408 questions
0
votes
0
answers
121
views
Quadratic residue problem involving prime divisors of a polynomial
Let $n$ be a square-free natural number, and let $f\in\mathbb{Z}[x]$ be monic and irreducible of degree $\geq2$. I am trying to determine whether there always exists a prime $p$, $p\nmid n$, ...
- 1
10
votes
1
answer
550
views
Igusa's $\chi_{10}$ and Borcherds products
Igusa defined a genus 2 Siegel modular form $\chi_{10}$, which vanishes on the Humbert surface $G_{1}$ (the image of a "degenerate" Hilbert modular surface, the product of modular curves, ...
- 3,309
2
votes
1
answer
110
views
Equations for $H_1(M)$ and $T$-Tate module of Anderson t-motive $M$ are equivalent: a reference?
What is a reference for the following construction?
Let $M$ be an Anderson t-motive of rank $r$ dimension $n$, i.e. a module over the Anderson ring $\mathbb{C}_\infty[T]\{\tau\}$ satisfying some ...
3
votes
2
answers
459
views
Short sequence beats long sequence
I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, ...
- 43.2k
5
votes
1
answer
788
views
Is there a reference for these types of cubic identities?
I'm looking at the following generalizations for sums of two cubes.
$u^3+v^3=(u+v)(u^2-uv+v^2).$
$u^3+(u+r)^3+v^3+(v+r)^3=(u+v+r)(2u^2-2uv+2v^2+ru+rv+2r^2).$
$u^3+(u+q)^3+(u+r)^3+(u+q+r)^3+v^3+(v+q)^3+...
- 1,109
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[\...
37
votes
1
answer
1k
views
What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite ...
- 2,959
10
votes
1
answer
1k
views
A generalisation of theorem of Landau on sum of two squares?
Let
$r(B)$ be the number of integers $1 \leq n \leq B$ such that $n = x^2 + y^2$ for some $x, y \in \mathbb{Z}.$
Then it is a known theorem of Landau that
$$
r(B) \sim C \frac{B}{\sqrt{\log B}}
$$
...
- 3,625
5
votes
1
answer
340
views
About an asymptotic behavior in number theory
Where can I read about the asymptotic behavior (with $N$ tending to infinity) of the sum of the fractional parts obtained from dividing $N$ by all prime numbers up to $N$ divided by the number of ...
2
votes
1
answer
289
views
Known upper bounds for $\sum_\limits{p\equiv a \pmod q, p\leq x}\frac{1}{p}$
I am trying to deduce an upper bound for $\sum_\limits{p\equiv a \pmod q, p\leq x}\frac{1}{p}$ where $p$ is a prime and $(a,q)=1$. Using Theorem 1 of Carl Pomerance's article on amicable numbers (...
- 309
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
7
votes
2
answers
816
views
Well known applications of Roth's theorem
Roth's theorem in Diophantine approximation (1955) is a well known milestone. It has been generalised in the case of number fields for simultaneous approximations considering several places.
It is an ...
- 321
2
votes
2
answers
1k
views
Using Kolmogorov complexity to measure difficulty of problems?
We call the natural number $n$ a partition number $\iff$
$$
\exists d | n: \gcd\left(d,\frac{n}{d}\right)=1 \text{ and } \Omega(d) = \Omega\left(\frac{n}{d}\right)\;,
$$
where $\Omega$ counts the ...
- 3,064
1
vote
1
answer
393
views
Steuerwald's theorem
Background:
The perfect numbers are the positive integers $n$ such that $$\sigma(n)=2n,$$ where $\sigma(n)$ is the sum of divisors function.
The function $\sigma(n)$ is multiplicative and satisfies $$\...
user178594
2
votes
0
answers
110
views
Asking for a generating function for an arithmetic sequence
For fixed integer $n\geq1$, let $c_m(n)$ be the number of divisors $d$ of $m$ such that $n<d\leq 2n$. Here is an experimental generating function for which I ask:
QUESTION. Is this true?
$$\sum_{m\...
- 43.2k
3
votes
0
answers
142
views
Reference for formula expressing products of two Fibonacci numbers in Zeckendorf-basis
It is well-known folklore that every natural integer has a unique Zeckendorf expansion as a
sum over a finite set of Fibonacci numbers containing no pair of consecutive Fibonacci numbers.
It is easy ...
- 17.9k
4
votes
1
answer
95
views
Limiting values of particular functions
Let's define the functions
$$A_n(q)=\sum_{k=0}^n(-1)^k\cdot\frac{(1+q)q^k}{1+q^{2k+1}}\cdot\frac{2k+1}{n+k+1}\binom{2n}{n-k}.$$
I'm interested in the following:
QUESTION. Let $n\geq1$ be integers. ...
- 43.2k
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
7
votes
2
answers
615
views
Reference request for recurrence relation of division polynomials
The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...
- 265
2
votes
0
answers
197
views
Mumford's computation of the determinant of cohomology of a relative curve
In Integral Grothendieck-Riemann-Roch theorem, Pappas mentions that Mumford computed the determinant of cohomology of $f:X\to S$ a relative curve integrally, and thus proved an integral version of GRR ...
- 2,044
3
votes
1
answer
216
views
Reference request: Serre's Groupes discrets
I'm reading some articles and at some point they both reference:
J-P. Serre: Groupes discrets (in collaboration with H. Bass),
Collège de France, 1969
However I have trouble finding this reference. ...
- 2,224
12
votes
2
answers
2k
views
What is the Perrin-Riou logarithm (or regulator)?
Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this workshop at the CRM in ...
- 3,309
4
votes
1
answer
601
views
Reference for a proof of Euclid's Theorem for the infinitude of primes
I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:
Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of ...
- 17.9k
3
votes
1
answer
206
views
The growth of certain continued fractions
I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ ...
- 1,990
7
votes
1
answer
723
views
Alternate algorithms for Chinese remainder theorem
I was teaching Discrete this semester and set the students loose on a system of linear congruences. One of them came up with this solution. Say $$ x \equiv 1 \textrm{ mod } 3 $$ $$ x \equiv 3 \textrm{ ...
- 525
7
votes
0
answers
265
views
"Reference Request" for a lecture note by C. Skinner: Galois Representations, Iwasawa Theory, and Special Values of $L$-functions
This was originally posted on math.stackexchange as https://math.stackexchange.com/questions/4589793, where I was suggested to move it here.
I'm searching a lecture note by C. Skinner named "...
- 71
10
votes
1
answer
755
views
The $9$th tetration of $-\sqrt2$
Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and
$$^{n+1}a=a^{^na}$$
for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...
- 128k
2
votes
1
answer
112
views
Counting numerical semigroups by largest element of minimal generating set
For a given integer $n$, I am interested in the number of different numerical semigroups one can make with a generating set consisting only of integers in $[n]$.
I have done some small examples. For $...
- 807
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 ...
- 2,215
1
vote
1
answer
364
views
Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
- 17
3
votes
2
answers
711
views
Binomial coefficient congruence modulo $p^n$
I am interested in the following congruence
$$\binom{ap^n}{bp^n}\equiv \binom{a}{b}\pmod{p^n}$$
I am aware that by some reference in a book the above it should actually hold modulo $p^{3n}$; the ...
- 847
1
vote
0
answers
98
views
Reference request for a result in additive combinatorics
Let $p$ be a prime number and $[p-1]=\{1, 2, \ldots, p-1\}$.
The following proposition is proved: (but I cannot find out where)
Proposition: The non-empty subset sums of $[p-1]$ are equally ...
- 3,866
6
votes
1
answer
183
views
Mean value of the divisor function over Piatetski-Shapiro sequences
Let $c>1$, $c\not\in\mathbb{Z}$ and consider the sum
$$
\sum_{n\leq x} \tau(\lfloor n^c \rfloor),
$$
where $\tau(n)$ is the number of divisors of $n$. I'm almost certain I've seen an evaluation of ...
- 1,990
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
1
answer
305
views
Level spacing statistics for primes
In the preprint "Level Spacing Statistics for Primes", we have found some patterns of prime spacings, which may provide new insights on the distribution of primes:
We would like to know ...
- 149
10
votes
1
answer
669
views
The paper behind an olympiad problem
In IMO Shortlist 2013, there is a number theory problem:
Determine whether there exists an infinite sequence of nonzero digits $a_1,a_2,a_3,...$ and a positive integer $N$ such that for every integer $...
- 1,462
3
votes
0
answers
115
views
Counting monomials modulo prime numbers
The present quest emanates from this study by R. Stanley, including his recent MO question. Define the product (polynomials after full expansion)
$$I_n(x)=\prod_{i=1}^n(1+x^{F_{i+1}})$$
based on the ...
- 43.2k
2
votes
2
answers
308
views
Reference for zero sum estimates of Dirichlet L functions
Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$.
I am reading a paper by Ihara and Murty where they use following estimate :
$\...
- 41
5
votes
2
answers
717
views
Sum of many squares modulo $n$
Let $n$ be a positive integer and $0 \leq i < n$. Define
$$
N(i) = \# \left\{ (x_1,\dots, x_s) \in [1, n]^s: x_1^2 +\dots + x_s^2 \equiv i \mod n \right\}.
$$
I am looking for a reference for ...
- 579
3
votes
1
answer
182
views
$l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$
Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "G.Laumon - Fonctions zeta des variétés de Siegel" (Lemma 4.1) that ...
- 1,270
3
votes
1
answer
228
views
What fraction of the values of a quadratic polynomial can be prime?
I have an explicit, monic quadratic polynomial $P(x)$ and an integer $m$. Can I bound the number of prime values in $P(0), P(1), \ldots, P(m)$? A reference would be appreciated, if available. An ...
- 9,114
1
vote
1
answer
177
views
Distribution of $\alpha n^2/q$ modulo $1$?
Let $0 \neq \alpha \in [0,1]$ and $q$ a positive integer.
Let $||.||$ denote the distance to the closest integer and define
$$
N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \...
- 3,625
1
vote
1
answer
153
views
Number of distinct near-squares primes dividing an odd perfect number
I'm curious about if the following question is in the literature or what work can be done about it.
Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
1
vote
0
answers
87
views
Doubly log-concave or doubly log-convex
Suppose $(a_k)_{k\geq0}$ is a sequence of real numbers. Consider the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$.
We say $(a_k)_k$ is log-concave (resp. log-convex) provided $\mathcal{L}a_k\geq0$ (...
- 43.2k
1
vote
1
answer
220
views
Gaussian at $q=\pm1$, log-concave polynomials, Catalan numbers
Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ with $[0]_q!:=1$ and the Gaussian polynomials $\binom{n}k_q=\frac{[n]_q!}{[k]_q!\,\cdot\,[n-k]_q!}$. Adopt the convention that $\binom{n}k_q=0$ whenever $k&...
- 43.2k
5
votes
1
answer
386
views
Is there a simple expression for $\sum_{k =1}^{(p-3)/2} \frac{1\cdot 3\cdots (2k-1)}{2\cdot 4 \cdots 2k\cdot(2k+1)} \bmod p$?
Let $p \equiv 1 \pmod 4$ be a prime and $E_n$ denote the $n$-th Euler number. While investigating $E_{p-1} \pmod{p^2}$ I have encountered this summation (modulo $p$)
\begin{align*}
\sum_{k =1}^{\frac{...
- 707
1
vote
2
answers
349
views
Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$
There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...
- 7,182
3
votes
1
answer
188
views
Density of numbers with a prime factor satisfying a congruence
Let $S$ be the set of integers with at least one prime factor in the arithmetic progression $km+d$, $(m, d)=1$. I am looking for results on the density of $S$. I found this post which talked about the ...
- 31
7
votes
2
answers
570
views
Finite generation of motivic cohomology of number fields
Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups
$$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$
...
- 1,790
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 ...
- 1,199