All Questions
Tagged with reference-request nt.number-theory
1,408 questions
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
$$\prod_{n\geq1}\left(\frac{1+q^n}{1-...
- 43.2k
2
votes
0
answers
490
views
On quasi-modular forms with integer Fourier coefficients
It is well-known that the ring $M$ of modular forms has the structure $M=\mathbb{C}[E_4,E_6]$, where $E_k$ are the Eisenstein series.
It is also known that one can define the concept of quasi-modular ...
- 6,133
6
votes
2
answers
547
views
2-adic valuation of a certain binomial sum
Consider the sequence (of rational numbers) given by
$$a_n=\sum_{k=1}^n\binom{n}k\frac{k}{n+k}.$$
Let $s(n)$ be the sum of binary digits of $n$, i.e. the total number of $1$'s.
QUESTION. Is it true ...
- 43.2k
6
votes
1
answer
183
views
Mean value of the divisor function over Piatetski-Shapiro sequences
Let $c>1$, $c\not\in\mathbb{Z}$ and consider the sum
$$
\sum_{n\leq x} \tau(\lfloor n^c \rfloor),
$$
where $\tau(n)$ is the number of divisors of $n$. I'm almost certain I've seen an evaluation of ...
- 1,990
3
votes
1
answer
758
views
Looking for a paper of Lagarias and Odlyzko
I have been studying about the Chebotarev Density Theorem and have been hunting for the following paper of Lagarias and Odlyzko for quite a while:
Effective versions of the Chebotarev density theorem, ...
- 739
25
votes
8
answers
3k
views
Relatively concise English expositions of the proofs of the various Weil conjectures
Where can I find relatively concise (i.e. not excessively wordy and waxing poetic about history and intuitions and such, doesn't spend an eternity carefully developing various parts of the theory of ...
user97565
1
vote
1
answer
153
views
Number of distinct near-squares primes dividing an odd perfect number
I'm curious about if the following question is in the literature or what work can be done about it.
Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
3
votes
1
answer
228
views
What fraction of the values of a quadratic polynomial can be prime?
I have an explicit, monic quadratic polynomial $P(x)$ and an integer $m$. Can I bound the number of prime values in $P(0), P(1), \ldots, P(m)$? A reference would be appreciated, if available. An ...
- 9,114
3
votes
1
answer
248
views
number of integers $n$ with $\|n \alpha \|$ small?
Let $\alpha \in \mathbb{R}$ and $N$ a positive integer. I am interested in the quantity
$$
D(\alpha, N) := \# \{ n \in [1, N]: \| n \alpha \| < 1/N \},
$$
$\| x \|$ denotes the distance to the ...
- 3,625
2
votes
0
answers
1k
views
Advanced texts on analytic number theory?
So a friend of mine is very interested in analytic number theory, and is looking for resources past the basic level.
He has studied analytic number theory from several books, among them are Hardy’s ...
1
vote
1
answer
220
views
Gaussian at $q=\pm1$, log-concave polynomials, Catalan numbers
Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ with $[0]_q!:=1$ and the Gaussian polynomials $\binom{n}k_q=\frac{[n]_q!}{[k]_q!\,\cdot\,[n-k]_q!}$. Adopt the convention that $\binom{n}k_q=0$ whenever $k&...
- 43.2k
11
votes
1
answer
699
views
Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?
Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion
$$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
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
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
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
12
votes
2
answers
1k
views
Short research articles
I am a masters student. I am interested in short articles which have counter examples and very few references. I want to write a short and interesting article.
For example; One of the best known ...
27
votes
3
answers
3k
views
Where's the best place for an algebraic geometer to learn some algebraic number theory?
There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- ...
- 64k
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
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
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
7
votes
3
answers
826
views
Analytic equivalents for primes in arithmetic progressions
By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$.
I would like ...
- 12.8k
17
votes
2
answers
1k
views
The GCD-matrix: generalizing a result of Smith?
Let $M$ be the $n\times n$ matrix, known as the GCD matrix, of entries $M_{ij}=\gcd(i,j)$. In the paper
H J S Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7:208-...
- 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) ...
14
votes
1
answer
749
views
reference for: no finite set of positive (integer) binary quadratic forms represents all primes
This recent question asks for a set of forms (binary quadratic) representing all primes.
Set of quadratic forms that represents all primes
When the question was asked on MSE last month
https://math....
- 25.7k
4
votes
1
answer
239
views
Yet, another numerical variant of the Vandermonde matrix
In my earlier (soft) MO post, an elementary response was given by Ofir Gorodetsky in regard to the determinant of the symbolic counterpart to the numerical matrix $\mathbf{M}_n=(i^j-j^i)_{i,j}^{1,n}$.
...
- 43.2k
30
votes
2
answers
4k
views
Motivation behind Analytic Number Theory
I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...
- 317
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$ ...
- 18.9k
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}})$$
based on the ...
- 43.2k
29
votes
4
answers
5k
views
What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?
The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
- 2,210
13
votes
3
answers
1k
views
linear independence of $\sin(k \pi / m)$
I have tried searching the literature for a result like the following, but have not found anything.
For a positive integer $m$, is it known that
$$\{ \sin (k \pi / m): 1 \leq k \leq m/2, (k,m)=1 \}$$
...
- 133
13
votes
1
answer
601
views
Congruences for "colored partitions" a la Ramanujan
Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by
$$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$
The numbers $p_t(n)$ can be regarded as enumerating ...
- 43.2k
9
votes
1
answer
638
views
Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$
Let $P,Q,R$ be the Fourier series of the Eisenstein series $E_2,E_4,E_6$, that is,
$$
P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n,
$$
$$
Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n,
$$
$$
R(q)=1-504\...
- 663
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
4
votes
1
answer
263
views
A refinment of Beck's conjecture
Let $\mathcal{O}(n)$ and $\mathcal{D}(n)$ denote the set of all integer partitions of $n$ into odd parts and distinct parts, respectively. Let $o(n)=\#\mathcal{O}(n)$ and $d(n)=\#\mathcal{D}(n)$. ...
- 43.2k
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
1
answer
413
views
Resources and outstanding conjectures about the Epstein zeta function
I am looking for a reference to the Epstein zeta function. For the Riemann zeta function, there is Titchmarsh's treatment. However, I do not know of any references regarding the Epstein zeta function ...
- 107
11
votes
1
answer
2k
views
Reference request: proof of Ramanujan's Cos/Cosh Identity
The Ramanujan Cos/Cosh Identity, as stated here, is
$$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+
\left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}=...
1
vote
0
answers
98
views
Reference request for a result in additive combinatorics
Let $p$ be a prime number and $[p-1]=\{1, 2, \ldots, p-1\}$.
The following proposition is proved: (but I cannot find out where)
Proposition: The non-empty subset sums of $[p-1]$ are equally ...
- 3,866
5
votes
1
answer
435
views
Limit on a certain double sum
While working with multi-zeta functions, I encountered the below (experimental) value for a certain evaluation (in a limit sense). Notice first this well-known fact in context
$$\sum_{n,m\geq1}\frac1{...
- 43.2k
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:
$$\sum_{0\le x\le p-1}\left(\frac{x}{p}\right)\left(\frac{x+1}{p}\right)\left(\frac{...
- 93
6
votes
4
answers
901
views
Mathematical induction vis-à-vis primes
One of the most used proof-techniques is mathematical induction, and one of the oldest subjects is the study of prime numbers. Thanks to Euclid, we can consider the primes as a infinite monotone ...
8
votes
2
answers
396
views
De Bruijn's sequence is odd iff $n=2^m-1$: Part I
Among the families of sequences studied by Nicolaas de Bruijn (Asymptotic Methods in Analysis, 1958), let's focus on the (modified)
$$\hat{S}(4,n)=\frac1{n+1}\sum_{k=0}^{2n}(-1)^{n+k}\binom{2n}k^4.$$
...
- 43.2k
4
votes
1
answer
233
views
About colossally abundant numbers - reference request
This post contains three related questions:
In the OEIS sequence 073751 ( https://oeis.org/A073751/ ) there is a short Mathematica program that is said to produce the prime factors of successive ...
- 1,018
9
votes
2
answers
709
views
Egyptian number theory
Might there be a good historical reference on Egyptian number theory ($ \sim 2000$ B.C.)? The following online reference by a professor at the UCLA indicates that they were aware of the Pythagorean ...
- 3,871
1
vote
1
answer
344
views
Is there a way to tie up even and "newly suggested odd" Riemann zeta values?
Define the sequence
$$a_s=(-1)^{\binom{s-1}2}\left(\frac{\pi}2\right)^s\frac1{2\cdot s!}\begin{cases} s\,E_{s-1}, \qquad \text{if $s$ is odd} \\ 2^{2s}B_s, \qquad \,\,\text{if $s$ is even};\end{cases}$...
- 43.2k
4
votes
1
answer
206
views
Relative density of primes in certain congruence classes
In "M. B. Nathanson - Elementary Methods in Number Theory" is shown (Theorem 7.14) that if $A$ is a set of positive integers such that $\sum_{a \in A} 1 / a$ converges then the set of ...
- 1,042
3
votes
1
answer
251
views
Congruence modulo 2 for q-series
This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their Dyson ranks.
I would like to ask:
QUESTION. Is this congruence true ...
- 43.2k
4
votes
1
answer
271
views
The highest power of $2$ dividing a polynomial evaluated at $x=3$
Let $\nu_2(a)$ be the $2$-adic valuation of an integer $x$, i.e. the largest power $t$ such that $2^t$ divides $x$.
Define the operator $D=x\frac{d}{dx}$ and the polynomial $\Phi_k(x)=\frac{x^{k+1}-1}{...
- 43.2k
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
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