All Questions
Tagged with reference-request nt.number-theory
1,409 questions
5
votes
1
answer
389
views
Results in an article by Siegel
Studying the Eisenstein cocycle by Sczech, I noticed that to understand its connection with the values at negative integers with zeta functions it is necessary to understand the resuts by Siegel in
...
- 3,107
0
votes
0
answers
155
views
On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$
The Dedekind psi function is defined for a positive integer $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$
with the definition $\psi(1)=1$. See ...
11
votes
2
answers
1k
views
Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...
- 151k
6
votes
2
answers
366
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Provoking involutions further
Let $\mathfrak{S}_n$ denote the permutation group, and $I_0(n)=\sum_{j\geq0}\binom{n}{2j}\frac{(2j)!}{2^jj!}$ stand for involutions see A000085 for more interpretations. There is also these numbers $...
- 43.2k
9
votes
1
answer
317
views
Counting monomials in cyclotomic polynomials
Let $\Phi_n(x)$ denote the $n$-th cyclotomic polynomial. There are numerous properties and utilities of these polynomials. My interest is more basic and in the spirit of
Tewodros Amdeberhan and ...
- 43.2k
3
votes
1
answer
487
views
Is this line of thought (using linear algebra to get number theoretic results) already being pursued in the literature?
Let $Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where $n = \prod_{i=1}^r p_i^{\alpha_i}$ and $p_i$ is the $i$-th prime, $\alpha_i \ge 0$, $e_i$ is the $i$-th standard basis vector. For example $6 = 2\...
user6671
9
votes
1
answer
318
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A weak form of the Erdős-Turán conjecture
This question is motivated by the answer of Gowers to the question Erdos Conjecture on arithmetic progressions.
Question. (1)-Suppose $A \subset \mathbb{N}$ is such that
Lim$_n$ $log(n) \cdot |A \...
- 32.2k
3
votes
2
answers
505
views
On odd perfect numbers and a GCD
(Note: This question is closely related to this other one in MSE.)
Let $N = q^k n^2$ be an odd perfect number.
From this paper in NNTDM, we have the equation
$$i(q) := \frac{\sigma(n^2)}{q^k}=\frac{...
2
votes
0
answers
97
views
Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?
Nowadays there are many papers on the number theory using heuristics.
I have read some of them.
But I have no clear understanding of the Bayesian Probability(subjective probability).
The concept of ...
- 1,013
17
votes
1
answer
3k
views
Is $x^{n}-x-1$ irreducible?
Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.
- 11.3k
10
votes
2
answers
1k
views
Algebraic independence of exponentials
First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution ...
- 3,738
2
votes
1
answer
154
views
Non-asymptotic results in probabilistic number theory
I'm a beginner. When I searched for results in probabilistic number theory most of the results were asymptotic in nature. Are there any results like with probability 1-$\epsilon$ (w.h.p) some property ...
- 25
3
votes
1
answer
137
views
Subexponential algorithms that apply only one of factoring and discrete logarithm?
Shor (quantum polynomial), Number Field Sieve (subexponential), Pollard rho (square root) all have both factoring and discrete logarithm over $\mathbb F_p^*$ variants.
What are the subexponential ...
- 13.9k
7
votes
0
answers
174
views
A diagonal generating function for Fibonacci: Part II
In my earlier MO question, I mentioned although we have for the Fibonacci numbers that
$$F_n=[x^n]\left(\frac1{1-x-x^2}\right),$$
is there a function $F(x)$ such that $F_n=[x^n]\left(F(x)\right)^n$?
...
- 43.2k
7
votes
4
answers
2k
views
Interactions of number theoretic conjectures and other fields of mathematics
There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...
7
votes
1
answer
1k
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Can every finite graph be represented by an arithmetic sequence of natural numbers?
(This is a follow-up to my previous questions Natural models of graphs?.)
Erdös in The Representation of a Graph by Set Intersections (1966) states:
Theorem. Let $G$ be an arbitrary
graph. Then there ...
- 9,720
3
votes
0
answers
73
views
Reference: Asymptotic bit-complexity of algebraic operations and transcendental functions
This question is a reference request. Does anyone know of a reference that lists the asymptotic bit-complexity of algebraic operations and transcendental functions implemented on a Turing machine that ...
- 131
2
votes
0
answers
341
views
Dirichlet series of powers of the prime omega function
Let $\omega(n)$ denote the number of distinct prime factors of a positive integer $n$.
I was wondering what is known about the dirichlet series
$$\sum_{n=1}^{\infty}\frac{\omega(n)^k}{n^s},$$
in ...
- 195
17
votes
1
answer
2k
views
Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
I am wondering if there is a multi-dimensional analog of the
Birch/Swinnerton-Dyer (BSD) conjecture.
The recent famous result inching toward resolution of that conjecture is:
Bhargava, Manjul, and ...
- 151k
11
votes
1
answer
328
views
Critical points of Dirichlet L functions
Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal
character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached ...
- 1,411
7
votes
1
answer
1k
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The Correlation of the Möbius Function and Dirichlet Characters
Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$
In other words
$$\phi_{\chi}(n)=\sum_{d|...
- 11.4k
14
votes
4
answers
4k
views
Deligne's letter to Piatetskii-Shapiro from 1973
Could anyone point me to a place where I could find Deligne's letter to Piatetskii-Shapiro from 1973? It is cited for example in Berkovich's "Vanishing cycles for formal schemes II".
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 ...
- 32.6k
4
votes
1
answer
530
views
Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$
Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
EDT 1: A possible answer is Analysis of the ...
- 12.3k
-1
votes
1
answer
142
views
If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is
Let $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer.
In a previous post I asked if $p_n(a,b)$ was a ...
- 3,153
5
votes
1
answer
525
views
Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$
Let $G$ be a compact abelian group. The unitary characters of $G$ form an orthonormal basis of $L^2(G)$, so every square integrable function $f: G \rightarrow \mathbb C$ admits a Fourier expansion
$$...
- 6,180
8
votes
1
answer
890
views
Weisinger's thesis
I am currently reading Atkin and Li's paper on Twists of newforms and Atkin-Lehner pseudo eigenvalues and one of the references there is to Weisinger's thesis:
Weisinger J., Some results on classical ...
- 363
2
votes
0
answers
65
views
Request for resources or techniques for bounding the infinity norm of an infinite product convolved with a simple function
I'm attempting to bound an expression of the form.
$$
\lVert(\prod_{i=1}^{\infty} \phi_i) * s \rVert_{\infty}
$$
Where $\phi_i$ are bounded periodic step functions which can be replaced by smoothed ...
- 41
3
votes
0
answers
145
views
Gauss-Manin and Hilbert modular forms
There is a geometric formulation of Hilbert modular forms (HMFs) that parallels that for classical modular forms (sections of a line bundle over the moduli space of Hilbert-Blumenthal Abelian ...
- 949
5
votes
4
answers
2k
views
Good books on arithmetic functions?
As I was studying the Möbius $\mu$ function and Gram series,
I got myself some pretty nice books:
Ribenboim - The New Book of Prime Number Records
Apostol - Introduction to Analytic Number Theory
...
- 731
15
votes
1
answer
1k
views
A p-adic analogue for a formula of Riemann?
This might be naive question but I was wondering whether a p-adic analogue of the following (shockingly) beautiful formula $$\zeta(s)\Gamma(s) = \int_0^\infty \frac{t^{s-1}}{e^t-1} dt$$ (vaild for $\...
- 2,029
0
votes
0
answers
91
views
Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 739
-1
votes
1
answer
125
views
BV analogue with well-factorable function. (Primes in arithmetic progression)
Is this special case known?
For $\lambda(q)$ -- well-factorable function and $q|P(z)$, $\pi(x;q,a)$ $a=1$.
$\displaystyle \sum_{q\leq x^{1-\epsilon}} \lambda(q) ( \pi (x;q,1)-\frac{\pi(x)}{\varphi (q)}...
- 150
12
votes
2
answers
764
views
Minimal possible cardinality of a $(a_1, ..., a_k)$-distributable multiset
Suppose we have a multiset $M$ of positive rational numbers. Sum of $M$ equals $1$. We'll call this multiset $n$-distributable for some $n\in \mathbb{N}$, if there exists a partition $M_1 \sqcup ... \...
- 381
5
votes
0
answers
1k
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Differential Galois number theory
Following https://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...
- 6,982
7
votes
3
answers
2k
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If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark:
One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...
- 10.5k
6
votes
2
answers
782
views
Books building up to the Gross-Zagier formula
I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of ...
- 169
20
votes
1
answer
2k
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When complex conjugation lies in the center of a Galois group
Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...
- 1,499
12
votes
1
answer
472
views
Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations
Let $\pi$ be a generic irreducible admissible representation of $GL_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The conductor of $\pi$ ...
- 1,453
8
votes
3
answers
2k
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Numerical evaluation of the Petersson product of elliptic modular forms
It is known how to compute the Fourier expansion of elliptic modular forms using modular symbols, and it is known how to get numerical evaluations of $L$-functions of various type ; it's possible to ...
3
votes
1
answer
330
views
Turan Inequalities
A real entire function
$$\psi(x)=\sum_{k=0}^{\infty} \gamma_k\frac{x^k}{k!}$$
is said to be in the Laguerre-Polya class, denoted $\psi(x) \in \mathcal{LP}$, if it can be represented in the form
\...
- 73
16
votes
1
answer
1k
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On (a generalization of) the Gauss Circle Problem
Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
- 5,169
9
votes
2
answers
286
views
What is the density of integers of the form $a^2+nb^2$?
Landau proved that the mean density of integers of the form $a^2+b^2$ up to $x$ is $K\frac{x}{\sqrt{\log x}} (1+o(1))$, where $K$ is an explicit constant. One proof is based on the fact that a prime $...
- 3,362
4
votes
2
answers
1k
views
Fricke groups and Fricke curves
The congruence subgroup $\Gamma_{0}(n) \subset PSL_{2}(\mathbb{Z})$ is normalized by the Fricke involution $F_n: z \mapsto -1/nz$ and so we may form the Fricke modular group $\Gamma_{0}^{+}(n)\langle \...
- 1,859
0
votes
1
answer
305
views
Redundancy of the Cantor enumeration of the rationals
What is the cardinality of the set of values corresponding to the first $n$ rationals generated in Cantor's enumeration scheme for proving their countability?
Edit:
following the suggestion of Todd, ...
- 13.2k
0
votes
0
answers
759
views
On sets of coprime integers in intervals
Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
- 13k
2
votes
5
answers
611
views
A good introduction to the study of the Thue Equation
Hi,
I am interested in studying the Thue equation, where we are concerned with a binary form $F(x,y) = a_0 x^r + a_1 x^{r-1}y + \cdots + a_r y^r$ and solutions of the form
$$F(x,y) = h$$
for some ...
- 26.9k
7
votes
2
answers
2k
views
Basics on anabelian geometry and Grothendieck's section conjecture
Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask ...
- 601
0
votes
1
answer
132
views
Different solution of power Diophantine equation based on constant term
Let us define a power Diophantine equation by 2 algebraic functions $f,g$ (having different degree) and by integers $k, l >0$ where, there are finite solutions for $f(x)+k=g(y)$, but there exists $...
- 267
1
vote
2
answers
259
views
Does the set of Diophantine $m$-tuples has full measure?
We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies the Diophantine condition of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\...
- 798