All Questions
Tagged with reference-request nt.number-theory
204 questions
5
votes
1
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310
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Is there a statement in Presburger arithmetic about primes this simple heuristic fails for?
I came up with the following conjecture while thinking about ways to formulate some heuristics about primes:
Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary ...
- 3,319
5
votes
1
answer
613
views
generating $q$-Catalan numbers
An $n$-Dyck path (or a Catalan path) is a lattice path $P$, unit East and North steps, in an $n\times n$ square grid which stays (weakly) above the main diagonal. Let $\square_n$ denote all such paths....
- 43.2k
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
5
votes
2
answers
388
views
How much do these interval collections cover?
As usual any related references are appreciated.
Let $p \lt q$ be distinct primes, and for all such pairs, let $m=pq$ and let $\cal{C}$ be the collection $(m-p,m)$ of open intervals. Does (the union ...
- 13k
5
votes
0
answers
772
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The Grimm Machine(s): A Collatz Conjecture Rival?
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
Just as the Collatz ...
- 13k
5
votes
1
answer
235
views
Algebraic independence of $a^a$ and $b^b$ for algebraic irrationals $a,b$
Let $a,b$ be algebraic irrationals.
Are there conjectures or unconditional results about the algebraic
independence of $a^a$ and $b^b$?
Probably Schanuel's conjecture is related,
maybe only $\log{a},...
- 25.4k
5
votes
0
answers
425
views
Conjectured new primality test for Mersenne numbers
How to prove that this conjecture about a new primality test for Mersenne numbers is true ?
Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$
...
- 161
5
votes
1
answer
655
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A theorem of Stickelberger on the number of prime ideals in a decomposition
Suppose that $p$ is unramified in a number field $K$ of degree $n$. Apparently, Stickelberger proved that $\big( \frac{Disc(K)}{p}\big) = (-1)^{n - g}$, where $g$ is the number of prime ideal factors ...
- 7,347
5
votes
3
answers
300
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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
5
votes
1
answer
455
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Large gaps between P2s
Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two ...
- 9,114
5
votes
1
answer
227
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Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$
Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X \...
- 10.5k
5
votes
2
answers
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Request: Kato's article "Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions." Part II
The question (similar to MO.96531) is about the article by Professor Kazuya Kato in this book.
In this article, Professor Kato indicates the contents of the second part.
MathSciNet does not list it, ...
- 3,867
5
votes
4
answers
4k
views
Values of Dirichlet L-funcions at natural numbers
I want to know about reference of formulas for
$$
L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s}
$$
for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have ...
- 2,446
4
votes
1
answer
629
views
Is $\sum_{n\leq x}{z^{\Omega(n)}} = O(x^{\frac12 + \varepsilon})$ equivalent to the Riemann hypothesis for all roots of unity $z\neq1$?
$\Omega(n)$ is the number of prime divisors of $n$, counted with multiplicity.
For $z=-1$, $z^{\Omega(n)} = \lambda(n)$ is the Liouville function, and it's known that $\sum_{n\leq x}\lambda(n) = O(n^{\...
- 3,319
4
votes
2
answers
560
views
Asymptotics of special square-free numbers
What is the asymptotic number of square-free numbers less than $x$ with exactly $k$ prime divisors?
- 432
4
votes
2
answers
288
views
Best known bounds for $\left|\sum_{n<x}\mu(nk)\right|$ (Reference request)
What is the best known bound for the Mertens function along arithmetic progressions? More specifically, what is the best bound known for
$$\sum_{n<x}\mu(kn)$$
as $k,x\to\infty$. This paper of ...
- 2,902
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)
$$\hat{S}(2a,n)=\frac1{n+1}\sum_{k=0}^{2n}(-...
- 43.2k
4
votes
2
answers
1k
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Fricke groups and Fricke curves
The congruence subgroup $\Gamma_{0}(n) \subset PSL_{2}(\mathbb{Z})$ is normalized by the Fricke involution $F_n: z \mapsto -1/nz$ and so we may form the Fricke modular group $\Gamma_{0}^{+}(n)\langle \...
- 1,859
4
votes
0
answers
846
views
Reference request for a basic result on relative differents & discriminants
I am looking for a better reference for the results in this extremely short and elementary paper:
Tôyama, Hiraku,
`A note on the different of the composed field',
Kōdai Math. Sem. Rep. 7 (1955), 43–44....
- 1,499
4
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0
answers
313
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Four-distance problem
A Problem due to Steinhaus is the following: do there exists a point in the plane at rational distance from the corners of the unit square?
Please give me a reference if this problem is solved.
- 367
4
votes
1
answer
599
views
Reference request, zeta function is rational function via Riemann-Roch?
I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch ...
- 75
4
votes
1
answer
335
views
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ a square?
(I have asked a similar question in MSE four days ago, but did not receive any answers. I have therefore cross-posted it to this site, hoping to get some responses.)
An odd perfect number $N$ is ...
4
votes
1
answer
523
views
Siegel-Walfisz Theorem with smooth weights
Let
$$\psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\pmod q}\Lambda(n)$$
where $\Lambda$ denotes the von Mangoldt function and $\phi$ to be Euler's totient function.
Then the Siegel-Wafisz theorem states ...
- 3,625
4
votes
2
answers
470
views
On the natural density of almost perfect numbers
This question is pretty basic, so I apologize in advance if it is unsuitable for MO. If so, please do let me know and I will migrate it over to MSE.
Essentially, by work of Kanold, we know that the ...
4
votes
0
answers
332
views
Diophantine equations over cyclotomic fields
Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
- 11.3k
4
votes
0
answers
409
views
An unpublished note by Spencer Bloch and Kazuya Kato
I am looking for an unpublished note by Spencer Bloch and Kazuya Kato, p-divisible groups and Dieudonné crystals. This note is always cited as
Spencer Bloch and Kazuya Kato, p-divisible groups and ...
- 193
4
votes
1
answer
545
views
class number of biquadratic fields
Can any one provide some references which treat the relation between the class number of a biquadratic field and the class numbers of its sub-fields using the analytic class number formula ?
- 347
3
votes
2
answers
876
views
Asymptotics for the number of partitions of $n$ into odd prime parts
Hello!
I am interested in the asymptotic behavior of the function $p_o(n)$ defined as the number of partitions of $n$ into odd prime parts A099773 - http://oeis.org/A099773 .
I couldn't find any ...
- 3,463
3
votes
1
answer
562
views
Volume of PGL(2,F) \ PGL(2, A)
Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$?
This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...
- 11.2k
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
3
votes
4
answers
654
views
A generalization of Landau's function
For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$
the least common multiple of all ...
- 798
3
votes
1
answer
480
views
Reference for explicit formula for $\sum_n \Lambda(n) \chi(n)$ with smooth weights
Let $\Lambda$ be the von Mangoldt function and $\chi$ a primitive character mod $q$, then we have the explicit formula
$$
\sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X - \sum_{ |Im \ \rho| \leq ...
- 3,625
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
$$i(q) := \frac{\sigma(n^2)}{q^k}=\frac{...
3
votes
1
answer
447
views
A number array related to colored necklaces and the primes
I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...
- 10.5k
3
votes
1
answer
528
views
Karolyi's theorem for finite groups and its extensions
Suppose that $\mathbb A = (A, +)$ is a (possibly non-commutative) group, and denote by $p(\mathbb A)$ the minimum of $|S|$ as $S$ ranges in the set of non-trivial subgroups of $\mathbb A$, with the ...
- 10.5k
2
votes
2
answers
345
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Looking for a reference to a classical formula for the sum of the base-$b$ digits of an integer
Pick an integer $b \ge 2$, and for $n \in \mathbf N$ let $s_b(n)$ denote the sum of the base-$b$ digits of $n$. It is a nice exercise to prove that $$s_b(n) = (b-1) \sum_{i=1}^\infty \left\{\frac{n}{b^...
- 10.5k
2
votes
1
answer
1k
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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
2
votes
3
answers
515
views
Asymptotic number of zeros for Dirichlet series with functional equation
I think the usual proof for the asymptotic number of zeros of the Riemann zeta function
$$N(T) = \#\left\{\rho : \ \zeta(\rho)=0, \begin{array}{l}\scriptstyle Im(\rho)\ \in\ [0,T]\\ \scriptstyle Re(\...
- 3,403
2
votes
5
answers
611
views
A good introduction to the study of the Thue Equation
Hi,
I am interested in studying the Thue equation, where we are concerned with a binary form $F(x,y) = a_0 x^r + a_1 x^{r-1}y + \cdots + a_r y^r$ and solutions of the form
$$F(x,y) = h$$
for some ...
- 26.9k
2
votes
3
answers
912
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Reference on generators of subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
- 85
2
votes
0
answers
398
views
Counting factors: is this approach in the literature on multiperfect numbers?
Does the following approach (or something near it) exist in the number theory
literature?
I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$
and for this question. First, ...
- 2,158
2
votes
1
answer
322
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Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number
(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is ...
2
votes
1
answer
596
views
On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit
Let $b$ be a fixed integer $\ge 2$ and $A$ a proper subset of $\{0, \ldots, b-1\}$. Then define $X$ to be the set of all positive integers whose base-$b$ representation consists only of digits from $A$...
- 10.5k
2
votes
1
answer
91
views
Reference or a short argument that a certain subset generates the ring of p-typical symmetric functions under plethysm
Let p be a prime, and let $\Lambda_p$ be the subring of the ring of symmetric functions $\Lambda$ (over $\mathbb{Z}$) such that $$x \in \Lambda_p$$ iff there is an $i \in \mathbb{N}$ such that $p^ix \...
- 101
2
votes
2
answers
1k
views
Place stabilizers for the absolute Galois Group
Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
- 1,049
1
vote
1
answer
142
views
Algorithm for computing isogeny class of elliptic curve
Is there an algorithm for computing the entire isogeny class of a given elliptic curve $E/\mathbb{Q}$?
References/ideas are welcome. Thanks!
- 2,907
1
vote
1
answer
344
views
Products involving exponents of tribonacci numbers
The Fibonacci numbers $F_n$ can be given by
$$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$
Among many many properties of this sequence, consider the following two results:
(1) the coefficients of the ...
- 43.2k
1
vote
1
answer
204
views
Iterated exponential sums
Does anyone have any references for iterated exponential sums? That is, sums like
$$\sum_{1\leq n\leq X} e(f(n))\sum_{1\leq m\leq n} e(f(m)),$$
where $e(x)=e^{2\pi i x}$? I am looking for references ...
- 1,904
1
vote
1
answer
222
views
Sum over three squares
Let $x$ be a sufficiently large number. Is there an explicit or asymptotic formula for the following sum
$$\sum_{\substack{n\leq x\\ n=a^2+b^2+c^2}} 1.$$ Any reference would be helpful.
- 1,257
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