All Questions
Tagged with reference-request nt.number-theory
1,408 questions
7
votes
3
answers
933
views
In search of an alternative proof of a series expansion for $\log 2$
We all know the series expansion
$$\log 2=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}n. \tag1$$
I also am able to use the method of Wilf-Zeilberger to the effect that
$$\log 2=3\sum_{n=1}^{\infty}\frac{(-1)^{...
- 43.2k
4
votes
2
answers
579
views
Hardy-Littlewood circle method for non-diagonal quadratic forms
In short, the question is for any references describing how to use the Hardy-Littlewood circle method to find an asymptotic for the number of solutions to $F(x_1, ..., x_s) = k$ for $(x_1, ..., x_s) \...
- 813
3
votes
1
answer
329
views
Fully explicit Linnik's Theorem
Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < ...
- 1,663
3
votes
1
answer
251
views
"Radical" Catalan numbers?
Let $C_n=\frac1{n+1}\binom{2n}n$ be the well-known Catalan numbers. Here is a curiosity.
QUESTION. Are there infinitely many $C_n$ that are "radical", i.e. that are square-free?
- 43.2k
2
votes
1
answer
236
views
Divisibility of (finite) power sum of integers
Consider the power sum
$$S_a(b)=1^{2b}+2^{2b}+\cdots+(3a-2)^{2b}.$$
Let $\nu_3(x)$ denote the $3$-adic valuation of $x$.
QUESTION 1. (milder) Is this true?
$$\nu_3\left(\frac{S_a(b)}{S_a(1)}\right)=0....
- 43.2k
3
votes
0
answers
179
views
Generalizing an estimate of Jutila
I'm working on a problem right now in which I need an upper bound for an exponential sum of the form
$$
\tag{1}
\sum_{N < n \leq 2N} \tau_3(n) e(f(n)),
$$
where $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ ...
- 1,990
9
votes
1
answer
574
views
Innovations in number theory leading to breakthroughs in statistical mechanics
Might there be a good reference on the interaction of number theory with statistical physics? I am particularly interested in innovations in number theory that have led to breakthroughs in statistical ...
- 3,871
6
votes
1
answer
636
views
The history and original paper of the Rosser–Iwaniec sieve
I'm trying to find Rosser's original paper where he introduces his eponymous sieve. I've already found https://arxiv.org/pdf/math/0505521 (where the reference isn't given, but where it is indicated ...
- 559
5
votes
0
answers
524
views
Generalization of Weil Conjectures
is there a reference in English, besides Deligne's original publication: "La conjecture de Weil: II", not synthetic but complete that deals with the original argument of the generalization ...
- 51
6
votes
1
answer
2k
views
Sum of square roots of natural numbers
Recently, I've encountered the following question:
Assume that $n_{1}, \ldots, n_{k}$ are (not necessary distinct) natural numbers. If
$$ (\sum_{i = 1}^{k}\sqrt{n_{i}}) \in \mathbb{N},$$ can we ...
1
vote
1
answer
200
views
Inequality for $3$-adic valuation
This should probably be not that hard, but I would like to see a nifty way of proving it.
Consider the double-indexed sequence given by
$$f(n,k)=\binom{2n + 2k}{n + k}\binom{n + k}{n - k}3^k.$$
...
- 43.2k
3
votes
1
answer
444
views
Details about the $\mod p$ reduction map
Let $N$ be a natural number and let $\Gamma_1(N)$ be the congruence subgroup of $SL_2(\mathbb{Z})$. Let $M(N)$ denote the space of all integer weight holomorphic modular forms for $\Gamma_1(N)$ whose ...
- 589
3
votes
0
answers
219
views
On partial sums of the Ramanujan sums
Let $n$ be a positive integer and $c_{m}(n)$ denote the $m^{th}$ Ramanujan sum at $n$. What is the best known estimate for $\sum_{m=1}^{N} c_{m}(n)$?
- 31
6
votes
1
answer
302
views
A 3rd formula for the central Delannoy numbers?
There are several in the literature proving the two alternative formulas for the (diagonal) Delannoy numbers; namely that
$$d_n=\sum_{k=0}^n\binom{n}k\binom{n+k}k=\sum_{k=0}^n\binom{n}k^22^k.$$
Each ...
- 43.2k
9
votes
2
answers
709
views
Egyptian number theory
Might there be a good historical reference on Egyptian number theory ($ \sim 2000$ B.C.)? The following online reference by a professor at the UCLA indicates that they were aware of the Pythagorean ...
- 3,871
5
votes
2
answers
307
views
Modulo $3$ calculations for a binomial-sum sequence
Introduce the sequence (this is A047781 on OEIS)
$$t_n=\sum_{k=0}^{n-1}\binom{n-1}k\binom{n+k}k$$
and denote the set $T(ij)=\{n\in\mathbb{N}: \text{the ternary digits of $n$ contain $i$ or $j$ only}\}$...
- 43.2k
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
8
votes
2
answers
1k
views
Question about functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $x$ is prime whenever $f(x)$ is prime
Let $f: \mathbb{\mathbb{Z}^+} \to \mathbb{Z^+}$ be a function and suppose
$(\star)$ For all integers $x \geq 3$, if $f(x)$ is prime, then $x$ is prime.
A trivial example of such a function is the ...
- 707
6
votes
2
answers
547
views
2-adic valuation of a certain binomial sum
Consider the sequence (of rational numbers) given by
$$a_n=\sum_{k=1}^n\binom{n}k\frac{k}{n+k}.$$
Let $s(n)$ be the sum of binary digits of $n$, i.e. the total number of $1$'s.
QUESTION. Is it true ...
- 43.2k
3
votes
0
answers
97
views
Study of relative class number of 'non-abelian' CM field by using L-functions
I'm currently interested in finding good upper bounds for the relative class numbers of non-abelian CM-fields.
So I'm looking for some references to learn the techniques that can be useful.
So far, I ...
- 1,013
6
votes
2
answers
804
views
Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the ...
- 25.4k
7
votes
1
answer
303
views
on a strange character sum
Recently while studying cubic residues modulo a prime $p$ with $p\equiv1\pmod 3$, I met the following character sum:
$$\sum_{0\le x\le p-1}\left(\frac{x}{p}\right)\left(\frac{x+1}{p}\right)\left(\frac{...
- 93
1
vote
0
answers
85
views
Reference request for "Divisibility of certain arithmetic functions" by Serre
The paper mentioned in the title can be found here. My problem is that it is in french and my French is only as good as Google Translate. Is there any english translations out there or is there any ...
- 589
5
votes
1
answer
365
views
Power of $2$ dividing a specialized Mittag-Leffler polynomial
While studying the so-called Mittag-Leffler Polynomials, denoted $M_n(x)$, I was looking into the sequence $\frac1{n!}M_n(n)$ which takes the following form
$$a_n:=\sum_{k=1}^n\binom{n-1}{k-1}\binom{n}...
- 43.2k
3
votes
1
answer
247
views
Explicit bounds on number of squarefree numbers coprime to a certain number
We know that the number of squarefree integers $\le x$ that are coprime to $A$ is
$$
Q_A(x) = x \prod_{p|A} \left(1-\frac{1}{p}\right) \prod_{p \nmid A} \left(1-\frac{1}{p^2}\right) + O(\sqrt{x}).
$$
...
- 301
9
votes
2
answers
629
views
How to read the paper of Arthur on trace formula on general reductive groups
My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily.
I would like ...
- 93
3
votes
1
answer
207
views
Partity of partitions with distinct parts of parts $>1$
This question is motivated by my earlier (unanswered) MO post.
The number of partitions into distinct parts is generated by $\sum_{n\geq0}Q(n)x^n=\prod_{k\geq1}(1+x^k)$. Focusing on parity of ...
- 43.2k
9
votes
0
answers
358
views
Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$
Consider the generating function of "partitions with distinct parts"
$$\sum_nQ(n)x^n=\prod_k(1+x^k).$$
It's known that
$$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}...
- 43.2k
14
votes
1
answer
285
views
Lower bounds for class number of function fields with fixed $q$, growing $g$
Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-...
- 156k
2
votes
0
answers
161
views
Has there been much research on the Iwasawa theory of bi-quadratic fields?
The simplest non-trivial example of $\mathbb{Z}_p$-extensions happen over imaginary quadratic fields and I've seen a good amount of research done here (from my understanding still a lot to learn). I ...
- 707
4
votes
0
answers
294
views
Modular forms on $\Gamma(N)$
I'm wondering where I can find a good reference about what is known about modular forms (especially cuspidal eigenforms) of full principal level $\Gamma(N)$, in terms of their Hecke theory, old/...
- 2,044
1
vote
2
answers
287
views
In search of a combinatorial proof for a multinomial sum
There is this sequence listed on OEIS - named Domb numbers. I'm curious about
QUESTION. Is there a direct combinatorial proof for the identity
$$\sum_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}
=...
- 43.2k
8
votes
0
answers
481
views
Formal degree of discrete series representations
Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \...
- 5,424
4
votes
1
answer
139
views
A close reative of "Inflated" Eulerian polynomials
I came across this post Coefficients of the Inflated Eulerian Polynomial by AULI-GRAHAM-SAVAGE. In particular, the polynomials related to descents interested me
$$P_n(x)=\sum_{\pi\in\mathfrak{S}_n}x^{...
- 43.2k
7
votes
1
answer
707
views
How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
- 707
11
votes
4
answers
707
views
Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?
Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
- 20.2k
9
votes
3
answers
658
views
Vinogradov-Korobov for Dirichlet L-functions?
Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
- 20.2k
1
vote
1
answer
183
views
A binomial convolution of Catalan numbers vs "utterly odd numbers"
An integer is called utterly odd if the terminal string of $1$’s in its binary representation has odd length. A number $2^{k+1}m+(2^k-1)$ where $m\geq0$ (every non-negative integer has this form) is ...
- 43.2k
23
votes
1
answer
3k
views
A list of proofs of the Hasse–Minkowski theorem
I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
7
votes
2
answers
191
views
Character sums concerning $a^x-1$
Let $p$ be a prime number and $\mathbb{F}_p$ be a finite field with $p$ elements. Let $\chi$ be a multiplicative character from $\mathbb{F}_p^{\times}$ to $\mathbb{C}$, where $\mathbb{F}_p^{\times}=\{...
user125345
4
votes
1
answer
205
views
Multiplicative set of positive algebraic integers
Let $S$ be a set of algebraic integers such that:
$\mathbb{N}_{\ge 1} \subseteq S \subset \mathbb{R}_{\ge 1}$,
$\alpha, \beta \in S \Rightarrow \alpha \beta \in S$,
$\alpha, \beta \in S \Rightarrow ...
20
votes
2
answers
4k
views
information-theoretic derivation of the prime number theorem
Motivation:
While going through a couple interesting papers on the Physics of the Riemann Hypothesis [1] and the Minimum Description Length Principle [2], a derivation(not a proof) of the Prime Number ...
- 3,871
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
4
votes
1
answer
165
views
De Bruijn's sequence is odd iff $n=2^m−1$: Part II
Assume $a\in\mathbb{N}$. Among the families of sequences studied by Nicolaas de Bruijn (Asymptotic Methods in Analysis, 1958), let's focus on the (modified)
$$\hat{S}(2a,n)=\frac1{n+1}\sum_{k=0}^{2n}(-...
- 43.2k
8
votes
2
answers
395
views
De Bruijn's sequence is odd iff $n=2^m-1$: Part I
Among the families of sequences studied by Nicolaas de Bruijn (Asymptotic Methods in Analysis, 1958), let's focus on the (modified)
$$\hat{S}(4,n)=\frac1{n+1}\sum_{k=0}^{2n}(-1)^{n+k}\binom{2n}k^4.$$
...
- 43.2k
26
votes
1
answer
2k
views
The "stubborn" solutions to sums of three cubes
It is conjectured (see [1]) that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to
$$
a^3+b^3+c^3=k.
$$
Numerical investigations of this conjecture show that ...
- 7,193
13
votes
1
answer
601
views
Congruences for "colored partitions" a la Ramanujan
Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by
$$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$
The numbers $p_t(n)$ can be regarded as enumerating ...
- 43.2k
3
votes
0
answers
195
views
Congruence for the polynomials $(t+1)^n$
An interesting polynomial congruence is given by
$$A_n(t^m)\equiv \left(\frac{1+t+\cdots+t^{m-1}}m\right)^{n+1}A_n(t) \qquad \mod (t-1)^{n+1}, \tag1$$
where $A_n(t)$ are the Eulerian polynomials with ...
- 43.2k
9
votes
0
answers
251
views
Exponential sums over integers with a fixed number of prime divisors
Are there bounds in the literature on sums of the form
$$\sum_{\omega(n)= k} e(\alpha n) \;\;\;\;\;\text{or}\;\;\;\;\; \sum_{\Omega(n)=k} e(\alpha n)$$
for $\alpha$ on minor arcs (i.e., not very close ...
- 20.2k
11
votes
1
answer
498
views
Siegel--Walfisz for number fields
For a number field $K$, we write $\Delta_K$ for its absolute discriminant. I was hoping for a Siegel--Walfisz type theorem of the following type:
Let $A > 0$. Then for every $X > 0$, every ...
- 502