All Questions
Tagged with reference-request nt.number-theory
1,408 questions
12
votes
1
answer
406
views
Looking for a "clever" argument for a $q$-series identity
Consider the below $q$-series identity. One of the things I like about this expansion is how nicely the difference on the left hand side factors to the right hand side of the equation.
$$\prod_{k\geq1}...
- 43.2k
1
vote
0
answers
203
views
Generalizing "partition into odd parts=partition into distinct parts"?
The number of partitions into distinct parts is known to agree with the number of partitions with odd parts. For instance, this follows from
$$\prod_{k=1}^{\infty}(1+q^k)=\prod_{n=1}^{\infty}\frac1{1-...
- 43.2k
5
votes
3
answers
300
views
Closed formula for $(-1)$-Baxter sequences
The number of the so-called Baxter permutations of length $n$ is computed by
$$a_n=\frac1{\binom{n+1}1\binom{n+1}2}\sum_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}.$$
There has also been a ...
- 43.2k
4
votes
1
answer
246
views
Is $C_n$ infinitely log-convex?
A sequence $a_n$ is called log-convex if $\mathcal{L}(a_n):=a_{n+1}a_{n-1}-a_n^2\geq0$ for all $n$; it is infinitely log-convex provided that all the iterates $\mathcal{L}^k(a_n)$ are still log-convex,...
- 43.2k
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
9
votes
2
answers
1k
views
Extracting constant terms: is there a direct way?
$\DeclareMathOperator\CT{CT}$
Let $\CT_t(f(t))$ denote the constant term of the Laurent polynomial of $f(t)$.
Define the two functions $F(x_1,\dots,x_n)$ and $G(y)$ by
$$F:=\prod_{i=1}^nx_i^{-1}(1-x_i)...
- 43.2k
3
votes
1
answer
159
views
Literature on analogous arithmetic function of logarithm function
In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-Mangoldt function. I am looking for an analogous ...
- 356
6
votes
0
answers
456
views
Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)
For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as
${\displaystyle \eta (q)
=q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$
By an $\eta$-quotient ...
0
votes
1
answer
195
views
Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?
Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$.
Also $\lambda_n$ is given as a sum over the non ...
user469908
2
votes
0
answers
245
views
Ambiguity about the exact definition of coefficients of modular forms
You can see the parts after my questions in the boxes. I received the answer to my first question in the comments.
I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
15
votes
4
answers
3k
views
Collecting alternative proofs for the oddity of Catalan
Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
- 43.2k
1
vote
0
answers
255
views
Globalization of a local field
I am reading the paper ''Endoscopic classification of representations of quasi-split unitary groups'' by Chung Pang Mok, and cannot come up with the proof of theorem 7.2.1.
Here is the statement.
...
user163485
3
votes
1
answer
111
views
Asymptotic growth of ternary partitions of integers $3n$
Consider the binary partitions of $2n$ in powers of $2$, denoted by $b(2n)$, with the generating function
$$\sum_{n\geq0}b(2n)\,x^n=\frac1{1-x}\prod_{k\geq0}\frac1{1-x^{2^n}}.$$
A result of De Bruijn ...
- 43.2k
13
votes
2
answers
803
views
Two interpretations of a sequence: an opportunity for combinatorics
The sequence that is addressed here is resourced from the most useful site OEIS, listed as A014153, with a generating function
$$\frac1{(1-x)^2}\prod_{k=1}^{\infty}\frac1{1-x^k}.$$
In particular, look ...
- 43.2k
7
votes
1
answer
293
views
On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series
The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series:
\begin{align*}
\int_{-\infty}^{\...
- 7,609
3
votes
1
answer
280
views
Congruence residues of integer partitions
Consider the number of integer partitions $p(n)$ of $n$ whose (product) generating function reads
$$\sum_{n\geq0}p(n)\,x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
There are many congruences for $p(n)$ ...
- 43.2k
4
votes
1
answer
539
views
A (mild?) question on the number of monomials
Let $[n]_q=\frac{1-q^n}{1-q}$ with $[0]_q=0$. Recall the $q$-factorials $[n]_q!=[1]_q[2]_q\cdots[n]_q$ (with $[0]_q!=1$) and the $q$-binomials
$$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\,[n-k]_q!}.$$
Now, ...
- 43.2k
2
votes
0
answers
80
views
Set of integer non-negative matrices with positive diagonals
This is essentially a reference request/name inquiry. Is there a name for the set $M_k$ formed by $k$ by by $k$ matrices with non-negative integer entries and positive values on the diagonal? Related, ...
- 6,969
5
votes
1
answer
264
views
Analogue of the second Hardy-Littlewood conjecture for numbers of divisors?
Let $f(n)$ denote the proposition "There exists some $k>1$ such that
$$
\sum_{m=k}^{k+n-1}\tau(m) < \sum_{m=1}^n\tau(m)
$$
where $\tau(m)$ is the number of the divisors of $m$." (This ...
- 9,114
2
votes
0
answers
140
views
Integers with exactly three factor pairs whose successors are relatively prime
I am interested in the following problem, and will appreciate pointers around how it can be solved – partially or fully – and/or indicators around whether it is even tractable:
Characterize $N \in \...
- 7,831
7
votes
0
answers
307
views
Number of rational points over finite fields mod $q$ is birational invariant
I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...
- 988
4
votes
1
answer
298
views
Enumeration of dominated Dyck paths
Using horizontal steps $(1,0)$ and vertical steps $(0,-1)$, consider the lattice paths starting from $(0,q)$ and reaching $(p,0)$ with $p$ horizontal and $q$ vertical steps. The set of such paths $\...
- 43.2k
1
vote
1
answer
155
views
Log-concavity of sequence related to overpartitions
The number $p_1(n)$ of overpartitions of $n$ is generated by
$$\sum_{n\geq0}p_1(n)\,q^n=\prod_{k=1}^{\infty}\frac{1+q^k}{1-q^k}.$$
Let $t\in\mathbb{N}$. Now, extend this to construct a family of ...
- 43.2k
3
votes
0
answers
115
views
p-adic density of the image of a polynomial
Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. Recently, a user of MO proved that the limit
$$\delta_p(P) := \lim_{n \to \infty} \frac{|\{P(x) \bmod p^n : x = 1,\...
- 453
3
votes
1
answer
492
views
Where can I find a rigorous proof of this statement in the literature? : $\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0$
In Wolfram MathWorld site at Moebius Function topic
there is identity number 10, which states that $\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0$.
Could you help me find a rigorous proof of this statement ...
- 39
25
votes
1
answer
911
views
Reference request for a proof of the two-square Theorem
One can show (see below for a sketch of a proof) that every odd prime number $p$
can be written in exactly $(p+1)/2$ different ways as
$$p=a\cdot b+c\cdot d$$
with $a,b,c,d\in\mathbb N$ satisfying $\...
- 17.9k
3
votes
0
answers
119
views
Furtwängler's family of irreducible polynomials
In the question Examples of nice families of irreducible polynomials over Z, user trew mentions a family of irreducible polynomials over the integers of the following form:
$$ p(x) = x^4 \prod_{i=1}^{...
- 125
2
votes
0
answers
161
views
Monotonicity of the cycle index polynomial under restriction
The cycle index (polynomial) of the symmetric group $\mathfrak{S}_n$ is given by the formula:
$$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
3
votes
2
answers
335
views
Reference where the Siegel-Walfisz theorem for the Möbius function is proved
Let $A>0$ and $q\leq (\log N)^A$. Then there exists a constant $c$ depending on $A$ such that $\displaystyle \sum_{n\equiv a \bmod q; n\leq N}\mu(n)\ll N\exp(-c\sqrt{\log N})$. I know this result ...
user147650
3
votes
1
answer
206
views
asymptotic growth of a sum involving partitions
Let $\lambda\vdash n$ denote the integer partition of $n$. Define the product $\mathcal{N}(\lambda)=\lambda_1\lambda_2\cdots\lambda_r$ when $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_r>0)...
- 43.2k
9
votes
1
answer
430
views
$2$-adic valuations: a tale of two $q$-series
Let $\nu_p(n)$ denote the $p$-adic valuation of $n$, i.e. the highest power of $p$ dividing $n$.
Consider the following two $q$-series formed by infinite products
$$\prod_{n\geq1}\left(\frac{1+q^n}{1-...
- 43.2k
0
votes
0
answers
171
views
Total sum of characters over partitions with distinct parts
In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
- 43.2k
1
vote
0
answers
139
views
Alternative Mersenne numbers
Let $\ b\in\mathbb Z,\ $ and $\ |b|>1.\ $ Call
$$ M_b(n)\ :=\ \frac{b^n-1}{b-1} $$
to be $n$-th Mersenne number mod $b$. The necessary condition for $\ M_b(n)\ $ to be a prime is that $\ n\ $ is a ...
- 4,786
1
vote
0
answers
52
views
Two types of the Germain prime siblings
Let $\ p\ $ and $\ q:=2\cdot p+1\ $ be primes — they are called Germain prime siblings. Such a pair belongs to the first type
$\ \Leftarrow:\Rightarrow\ \frac{q^2-1}8\equiv\pm1\mod8,\ $ and to the ...
- 4,786
4
votes
1
answer
700
views
Total sum of characters of the symmetric group $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.2k
3
votes
1
answer
270
views
Distance formula for continued fractions
In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated:
$$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...
- 406
0
votes
0
answers
138
views
A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem
I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
18
votes
1
answer
1k
views
Distinct integer roots for a degree 7+ polynomial and its derivative
Question: Is there a polynomial $f \in \mathbb{Z}[x]$ with $\deg(f) \geq 7$ such that
all roots of $f$ are distinct integers; and
all roots of $f'$ are distinct integers?
Background:
I asked a ...
- 7,831
1
vote
1
answer
147
views
Name for an easy combinatorial game
What is the name of the following combinatorial game:
Two players, moving in turn.
Positions: $0,1,2,\ldots$.
Moves: $n\longmapsto n-1$ or $n\longmapsto \lfloor n/2\rfloor$
if $n>0$.
No move for $0$...
- 17.9k
3
votes
0
answers
150
views
When is the Fermat Catalan surface a rational surface?
Related to Fermat Catalan conjecture and scholar.google.com didn't return any results.
Define the Fermat Catalan surface
$$ S_{m,n,k}: x^m+y^n=z^k$$
Where $\frac1m+\frac1n+\frac1k < 1$.
Q1 When is ...
- 25.4k
3
votes
1
answer
324
views
Regarding the Weierstrass $\wp$-function of the hexagonal lattice
Playing with the Weierstrass $\wp$-function of the hexagonal (or triangular) lattice $\mathbb{T}$,
$$
\wp'(z)^2 = 4 \wp(z)^3 - 1,
$$
I noticed that the zeros of $\wp'(z) + \sqrt{3}$ are
$$
\frac{\...
3
votes
0
answers
154
views
Reference request for the following results
I am looking for references on the following results. In what follows $\pi(x)$ denotes the prime counting function.
Result 1. For all real $k>1$ there exists $x^k_0 \in \mathbb{R}$ such that for ...
- 31
8
votes
0
answers
367
views
References for Yoichi Miyaoka's work around Fermat's Last Theorem
Apparently, Yoichi Miyaoka made a serious attempt to prove FLT in 1988. See the following question.
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last ...
29
votes
2
answers
4k
views
Closed formula for a certain infinite series
I came across this problem while doing some simplifications.
So, I like to ask
QUESTION. Is there a closed formula for the evaluation of this series?
$$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{...
- 43.2k
7
votes
1
answer
474
views
Fibonacci embedded in Catalan?
Given a partition $\lambda$ and its Young diagram $\pmb{Y}_{\lambda}$, we say $\lambda$ is a $(t,s)$-core partition provided that neither $t$ nor $s$ is a hook length in $\pmb{Y}_{\lambda}$. We now ...
- 43.2k
6
votes
0
answers
149
views
Dickson's conjecture for Beatty sequences
A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided ...
- 1,990
2
votes
1
answer
76
views
Reference request: Given a non-degenerate integral quadratic lattice $L,q$ over a PID, the quotient $L^*/L$ is given by SNF of $q$
Let $R$ be a PID with field of fraction $K$.
Let $L$ be a lattice with non-degenerate quadratic form $q:L\times L \to R$.
Let
$$
L^* = \{x \in L\otimes K \text{ s.t. } q(x,l) \in R \text{ for all } l \...
- 887
5
votes
1
answer
435
views
Limit on a certain double sum
While working with multi-zeta functions, I encountered the below (experimental) value for a certain evaluation (in a limit sense). Notice first this well-known fact in context
$$\sum_{n,m\geq1}\frac1{...
- 43.2k
2
votes
0
answers
480
views
About derived divided power envelope
Assume $A$ is a $\mathbb{Z}_{(p)}$-algebra with ideal $I$ and $A,A/I$ are $p$-torsionfree.
In this survey, Akhil Mathew defines the derived divided power envelope $LD_I(A)$ in Construction 7.15, after ...
- 121
0
votes
0
answers
96
views
A way to bound $\sum_{1 \leq n \leq X} \min ( \| \alpha n \|^{-1} , X/n)$?
Let $\alpha$ be a real number and $|| \cdot ||$ be the distance to
the nearest integer.
I want to find a non-trivial upper bound for
$$
\sum_{1 \leq n \leq X} \min ( || \alpha n ||^{-1} , X/n),
$$
...
- 3,625