All Questions
Tagged with reference-request nt.number-theory
1,408 questions
14
votes
7
answers
3k
views
A special type of generating function for Fibonacci
Notation. Let $[x^n]G(x)$ be the coefficient of $x^n$ in the Taylor series of $G(x)$.
Consider the sequence of central binomial coefficients $\binom{2n}n$. Then there two ways to recover them:
$$\...
- 43.2k
13
votes
2
answers
1k
views
Question about a lesser-known "class number formula" of Gauss
My question refers to article 301 of section 5 of Gauss's D. A. - there Gauss gives an asymptotic formula for the mean number of classes of forms with positive discriminant ($D>0$):
$$h(D) = \frac{...
- 2,099
-4
votes
3
answers
524
views
Relation between elliptic curve and Fermat's last thereom
I am looking for a elaborate explanation how the elliptic curve $E (a, b) := y^2=x(x-a)(x-b)$ is associated with the solution of $a^n+b^n=c^n$.
In 1969 Hellegouarch performed the elliptic curves $E (a,...
5
votes
2
answers
589
views
On Exercise 2.5.10 in Ram. M. Murty's book, "Problems in Analytic Number Theory."
I have just been told about this result, available as Exercise 2.5.10 in Ram. M. Murty's book, "Problems in Analytic Number Theory (2nd edition)". It says:
Let $\alpha>0$. Suppose $a_n \ll n^{\...
- 1,019
3
votes
1
answer
369
views
Is there a clear criterion or rule about when one can use the heuristic given by Cramér random model for prime numbers?
I don't know if I have misunderstood the information from the section about the heuristic related to Cramér's conjecture and the known facts about Maier's theorem, from the Wikipedia Cramér's ...
5
votes
1
answer
355
views
Research work on $ax^n-by^m=1$
I am looking for results on the equation $$ax^n-by^m=1 \tag 1 $$ where $\gcd(m,n)=1$ and $a,b,n,m$ are constants.
I found literature for $ax^n-by^n=1$ (R. A. Mollin, D. T. Walker) but couldn't ...
1
vote
0
answers
234
views
Why is the $\mathbb{Z}_p$-corank of $\operatorname{Sel}_{p^\infty}(E/\mathbb{Q})$ finite?
I'm interested on the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$. I read that the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group $\operatorname{Sel}_{p^\infty}(E)\doteq\...
user105226
11
votes
1
answer
699
views
Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?
Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion
$$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$
and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...
- 6,180
0
votes
1
answer
410
views
Rankin-Selberg convolution and product of degrees as of Christmas 2019
Almost 5 years ago (time flies), I asked in Rankin-Selberg convolution and product of degrees whether the Rankin-Selberg convolution of two automorphic representations of respectively $\operatorname{...
- 6,984
8
votes
1
answer
1k
views
Quick reference for general Weyl's inequality in number theory
I would like a reference for the result here. Having that $t$ there makes me happy. I would prefer not to have to, in my paper, run through and (not trivially but not too greatly) alter the proof of ...
- 1,354
4
votes
1
answer
507
views
A weaker version of the Brocard's Conjecture
Brocard's conjecture states that: If $p_{k}$ and $p_{k+1}$ are consecutive prime numbers greater than $2$, then between $p_{k}²$ and $p_{k+1}²$ there are at least four prime numbers.
I know that is ...
- 1,197
2
votes
2
answers
253
views
Approximation of a square with an irrational arithmetic progression
Let $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ be irrational. Does the arithmetic progression $(n\alpha )_{n\in\mathbb{N}}$ becomes arbitrarily close to squares?
More precisely, what can be said ...
- 3,574
0
votes
1
answer
224
views
Counting multiples in short intervals
Has anyone seen a problem like this in the literature? There are likely more generalized versions in sieve theory, which I am willing to tackle, but I would prefer a more elementary approach if ...
- 13k
9
votes
1
answer
731
views
Tamagawa numbers
Let $K$ be a finite extension of $\mathbb{Q}_p$ with absolute Galois group $G_K$. Let $A$ be an abelian variety defined over $K$. The (geometric) Tamagawa number is defined as the order of the ...
- 323
8
votes
1
answer
595
views
Why was the factor $\frac12$ introduced in the Riemann $\xi$ function?
The factor $\frac12$ in the Riemann $\xi$ function:
$$\xi(s)=\frac12 s(s-1)\,\pi^{-s/2}\,\Gamma(s/2)\,\zeta(s)$$
was introduced by Riemann, however appears to be redundant. Once he had arrived at:
...
- 4,169
10
votes
1
answer
555
views
Sidon sets of $\mathbb{Z}/p\mathbb{Z}$
A set $S \subseteq \mathbb{Z}/p\mathbb{Z}$ is called a Sidon set if given $a, b, c, d \in S$ and $a+ b = c+ d$, then $\{a, b\} = \{c,d\}$. I was interested in knowing about the largest possible Sidon ...
- 3,625
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
1
vote
1
answer
291
views
Show that, If $a-1\mid S(a-1,2m)$ and $a-1>2m+1$ then $(f(a,2m))_a\in X_a$
Define $X_a$ be the set as, namely$\{ x=(\ \underbrace{ 1\ 1\cdots\ 1\ 1}_{\text{$n$ terms}}\ \ 0 \ \ \underbrace{ \alpha_t\ \alpha_{t-1} \cdots \alpha_1 \ \alpha_0}_{\text{$k$ terms, k=t+1}})_a \mid\ ...
- 599
0
votes
1
answer
178
views
Size of parities in counting partitions into odd parts
Let $p_{odd}(n)$ be the number of partitions of $n$ into odd parts (see here). For instance, one has the generating function
$$\prod_{k\geq1}\frac1{1-q^{2k-1}}.$$
QUESTION. What is the size of this ...
- 43.2k
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
7
votes
2
answers
505
views
A good reference to the Gauss result on the structure of the multiplicative group of a residue ring
I need a good reference (desirably some textbook in Number Theory) to the following known result, attributed to Gauss in Wikipedia.
Theorem (Gauss). Let $p$ be a prime number, $k\in\mathbb N$ and $\...
- 42k
2
votes
1
answer
1k
views
Sum of the digits in base $p+1$
Definition
Let $W$ be the function , defined as $W(a,b)=r$
given $a,b\in \mathbb{Z_+}$ and $a>1$
Take $m$ to be the integer s.t. $a^{m+1} \ge b > a^{m}$, i.e. $m = \...
- 599
3
votes
2
answers
546
views
Approximation of $\sum_{\rho}\frac{1}{|\rho|^2}$, over the non-trivial zeros of the Ramanujan's zeta function
I would like to know if it in the literature an approximation for
$$\sum_{\rho}\frac{1}{|\rho|^2}\tag{1}$$
where the sum is over all of the non-trivial zeros of the Ramanujan's zeta function (also ...
11
votes
0
answers
269
views
Proving a group with two generators is not free that uses the Brahamagupta-Pell equation
Hello I encountered the following while reading a set of notes on free groups. It's not a homework question.
"Does there exist a rational number $\alpha$ with $0 <|\alpha| < 2$ such that the ...
- 111
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
3
votes
0
answers
85
views
Kronecker limit formula for antiperiodic boundary conditions
The celebrated Kronecker limit formula gives the $\zeta$-reguralized determinant of the Laplacian on the torus $\mathbb{R}/(\mathbb{Z}\omega_1+\mathbb{Z}\omega_2)$ in terms of Dedekind eta function of ...
- 8,992
3
votes
1
answer
309
views
How to estimate the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
Let $\tau(n)$ be the number of positive divisors of $n\in \mathbb{N}$.
Is it possible to get some good estimate for the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
I know that the sum is $\mathcal O(x^2)$...
- 3,866
0
votes
1
answer
266
views
Alternative reference to Davenport's Analytic Methods for geometry of numbers?
I was wondering if someone would be willing to suggest an alternative reference to Davenport's book Analytic Methods for Diophantine Equations and Diophantine Equations. I like the book but I would ...
user147650
5
votes
1
answer
351
views
Divisibility of certain polynomials
Consider the finite sums
$$F_n(q)=\sum_{k=1}^nq^{\binom{k}2}$$
with exponents the triangular numbers $\binom{k}2$. When $n$ is odd, it appears that $F_n(q)$ does not factorize over $\mathbb{Z}[q]$. On ...
- 43.2k
1
vote
0
answers
112
views
The $p$-adic valuation of powers of consecutive integers
Let $n > 0, K > 0$ integers and, for $i \in \{1,...,n\}$, let $k_i$ and $l_i$ be integers such that $k_i + l_i = K$. Assume that for some $i,j \in \{1,...,n\}$ we have $k_i \neq k_j$.
Claim: ...
- 365
3
votes
1
answer
405
views
Lower bound on Carmichael Function
What is the tightest lower bound currently known for the Carmichael function?
I imagine it must grow much more slowly than the Euler's totient function which according to here is bounded as
$$ \phi(...
- 2,689
5
votes
2
answers
941
views
$B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$
Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression
$$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho ...
- 20.2k
5
votes
1
answer
268
views
Is it a known property of positive integers $n> 2 $ that one must have $n < \mathrm{rad}(n(n-1)(n-2))$?
Let $P(n)$ be the statement that
$$n < \mathrm{rad}(n(n-1)(n-2)),$$
where $\mathrm{rad}$ is the radical of an integer, that is defined as
$$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ ...
- 1,776
4
votes
2
answers
593
views
Squares in Lucas sequences
Good night, everyone!
According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
- 87
-2
votes
1
answer
396
views
Published articles in journals about the Firoozbakht's conjecture, whose main goal or focus is the study of this conjecture
I would like to know what articles are in the literature about the known as Firoozbakht's conjecture, see the Wikipedia Firoozbakht's conjecture.
Question. What articles have been published in ...
6
votes
1
answer
361
views
Decay of matrix coefficients of non-tempered representation
A theorem of Cowling--Haagerup--Howe gives an effective decay rate of the matrix coefficients of a tempered representation $\pi$ of a semi-simple algebraic $G$ in terms of Harish-Chandra $\Xi$ ...
- 1,692
0
votes
0
answers
133
views
What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?
It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...
6
votes
2
answers
453
views
Reduction to Lie algebra version of fundamental lemma?
Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration.
For the purposes of the trace formula, one actually needs the fundamental ...
- 2,516
13
votes
3
answers
811
views
Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function A004125 from the OEIS.
Example. We'...
9
votes
0
answers
887
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k
21
votes
2
answers
2k
views
Applications of number theory in dynamical systems
I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics.
...
- 760
3
votes
0
answers
164
views
Using the Hilbert symbol to find nice field extensions
Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
- 2,516
1
vote
1
answer
218
views
Chinese Remainder Theorem for Remainder Intervals
Given $n$ natural numbers $m_1,\dots,m_n$ and $n$ remainder intervals $[a_1,b_1],\dots,[a_n,b_n]$ holding $a_i < b_i$ for all $i\leq n$ the task is to search for the smallest natural number $x$ ...
- 11
4
votes
1
answer
226
views
Simple trace formula with different spectral footprint?
A standard idea when dealing with the Arthur-Selberg trace formula (or a relative trace formula, for that matter) is to impose local conditions on the test function $f=\prod_vf_v$ to obtain a simple ...
- 2,516
3
votes
1
answer
280
views
Reference request for Euler products in positive characteristic
Let $K$ be a global field with ring of integers $O$, and let $f$ some integer valued function whose domain is the set of ideals of $O$ (e.g. $f(I)=|O:I|$). Extending the ordinary definition from the ...
- 993
8
votes
1
answer
408
views
Max order of an isogeny class of rational elliptic curves is 8?
I am looking for a reference for the proof of the following question following Theorem 5 in Mazur's Rational Isogenies of Prime Degree.
Theorem 5 There is a constant $C$ such that every elliptic ...
- 163
0
votes
0
answers
83
views
Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?
I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
4
votes
1
answer
448
views
The sign of an interesting sum involving a Dirichlet character
Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ).
For example for $q=5$ we have
\begin{equation}
\begin{aligned}
\chi_{5,1}&=(1, 1, 1, 1, 0),\\
...
- 603
7
votes
1
answer
465
views
A theorem by Harald Cramér?
In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:
Suppose $\{X_n\}_{n=2}^\infty$ is a sequence of ...
- 2,618
4
votes
1
answer
463
views
Density of twin square-free numbers
It is well-known how to compute the density of square-free numbers, to get
$$ \lim_{N\to\infty} \frac{\#\{ n \leq N : n \text{ square-free}\}}{N} = \frac{6}{\pi^2}.$$
What is the density of numbers ...
- 2,175