All Questions
Tagged with reference-request nt.number-theory
1,409 questions
-2
votes
1
answer
396
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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 ...
4
votes
0
answers
506
views
Collatz conjecture and a diophantine equation
Let $M \ge 2$. Inspired by the Collatz iteration / algorithm ($M=2$), I tried the following function:
$$C_M(n) = n/M, \text{ if } n \equiv 0 \mod M, \text{ otherwise } (M+1)n+\{(M-n) \mod M \}$$
We ...
- 3,074
23
votes
2
answers
2k
views
Dirichlet and the prime number theorem
I browsed Dirichlets Werke today and was kind of surprised by two remarks that he made on p. 354 (Über die Bestimmung ...) and p. 372 (Sur l'usage ...). In the second paper, he claims (my ...
- 32.6k
4
votes
1
answer
165
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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)
$$\hat{S}(2a,n)=\frac1{n+1}\sum_{k=0}^{2n}(-...
- 43.2k
15
votes
1
answer
943
views
BSD conjecture for rank 1 elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$
Thanks to the work of Gross-Zagier and ...
- 253
2
votes
1
answer
1k
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Sum of the digits in base $p+1$
Definition
Let $W$ be the function , defined as $W(a,b)=r$
given $a,b\in \mathbb{Z_+}$ and $a>1$
Take $m$ to be the integer s.t. $a^{m+1} \ge b > a^{m}$, i.e. $m = \...
- 599
3
votes
1
answer
220
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Reference request: relationship between discriminant and smoothness of a conic over arbitrary fields
I'm looking for an (ideally modern) reference of the relationship between the discriminant and smoothness of projective conics over arbitrary fields (including those of characteristic 2). Let $k$ be a ...
- 7,527
3
votes
1
answer
174
views
Sequences generated by sum & product of terms (with rotating indices): combinatorial?
Fix an integer $t\geq0$ and consider the sequence $T_{0,t}=1$ and for $n\geq1$, with $k*=n-1-k+t\mod n$,
$$T_{n,t}=\sum_{k=0}^{n-1}T_{k,t}T_{k*,t}.$$
EXAMPLES. Some initial terms:
(a) the case $t=0$: $...
- 43.2k
3
votes
0
answers
195
views
Congruence for the polynomials $(t+1)^n$
An interesting polynomial congruence is given by
$$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$$
where $A_n(t)$ are the Eulerian polynomials with ...
- 43.2k
5
votes
2
answers
1k
views
Request: Kato's article "Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions." Part II
The question (similar to MO.96531) is about the article by Professor Kazuya Kato in this book.
In this article, Professor Kato indicates the contents of the second part.
MathSciNet does not list it, ...
- 3,867
1
vote
1
answer
247
views
Gauss sum of imprimitive characters
On Wikipedia they state the following identity for the Gauss sum of an imprimitive character: suppose that $\chi: (\mathbb{Z}/m\mathbb{Z})^\times \rightarrow \mathbb{C}$ is a Dirichlet character with ...
6
votes
2
answers
453
views
Reduction to Lie algebra version of fundamental lemma?
Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration.
For the purposes of the trace formula, one actually needs the fundamental ...
- 2,516
3
votes
1
answer
405
views
Lower bound on Carmichael Function
What is the tightest lower bound currently known for the Carmichael function?
I imagine it must grow much more slowly than the Euler's totient function which according to here is bounded as
$$ \phi(...
- 2,689
13
votes
1
answer
2k
views
For a proof of the three-square theorem without using Dirichlet's theorem on primes in arithmetic progressions
The three-square theorem states that $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three squares if and only if it is not of the form $4^k(8m+7)$ ($k,m\in\mathbb N$). This was first proved by ...
- 15.6k
1
vote
2
answers
269
views
Average value of a fractional part of a function
Let $f(x): \mathbb{R} \to \mathbb{R}_{\geq 0}$ be a smooth function. I am interested in estimating
sums of the form
$$
\sum_{ A < n \leq B } \{ f(n)\}
$$
where $\{ c \}$ denotes the fractional part ...
- 3,625
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
14
votes
6
answers
10k
views
Frobenius number for three numbers
Given integers $a,b,c$ such that $\gcd(a,b,c) = 1$, it is well known that there exists only a finite set of numbers $n$ such that $n$ is not expressible as $ax+by+cz$ for non negative integers $x$,$y$,...
- 3,463
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
17
votes
4
answers
10k
views
Prime/undecomposable matrices
Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of ...
- 2,855
16
votes
1
answer
2k
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On J. T. Condict's Senior Thesis on Odd Perfect Numbers
I am trying to locate a copy of J. T. Condict's senior thesis on odd perfect numbers:
J. Condict, On an odd perfect number's largest prime divisor, Senior Thesis, Middlebury College (1978).
I am ...
24
votes
2
answers
1k
views
If $x_{n+1}= \frac{nx_{n}^2+1}{n+1}$ then $x_{n}=1$
I asked this question at MSE, but I think it's more appropriated to MO.
Let $x_{n}$ be a sequence, such that $x_{n+1}= \dfrac{nx_{n}^2+1}{n+1}$ and $x_n>0$ for all $n$.
There is a positive ...
- 3,153
13
votes
2
answers
586
views
How does the Bernstein-Zelevinsky construction of irreducibles from supercuspidals parallel the representations of the Weil-Deligne group?
In the Corvallis article Number Theoretic Background, here is what John Tate has to say on the local Langlands correspondence for a $p$-adic field $F$:
So, granting a correspondence between ...
- 6,180
8
votes
0
answers
240
views
Question on calculating character sums
I am wondering if there are any references that would help me with the following problem:
Let $p>2$ be a prime number, $n \in \mathbb{Z}^+$ odd such that $(n,p)=1$, $\chi$ an imaginary quadratic ...
- 707
20
votes
1
answer
1k
views
Curves over number fields with everywhere good reduction
My question is the following:$\newcommand{\Q}{\Bbb Q}
\newcommand{\Z}{\Bbb Z}$
What is known about number fields $K$ fulfilling the condition
$C_{g,K}$ "there is a smooth projective curve of ...
- 1,742
2
votes
1
answer
214
views
Looking for a paper by Landau and one by Watson
For the purposes of a project, I've been looking for the following two papers referred to in Serre's "Divisibilité de certaines fonctions arithmétiques":
Landau (E.), - Über die Eitenlung der ...
- 739
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
$$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ ...
- 1,776
5
votes
2
answers
388
views
How much do these interval collections cover?
As usual any related references are appreciated.
Let $p \lt q$ be distinct primes, and for all such pairs, let $m=pq$ and let $\cal{C}$ be the collection $(m-p,m)$ of open intervals. Does (the union ...
- 13k
3
votes
0
answers
97
views
Study of relative class number of 'non-abelian' CM field by using L-functions
I'm currently interested in finding good upper bounds for the relative class numbers of non-abelian CM-fields.
So I'm looking for some references to learn the techniques that can be useful.
So far, I ...
- 1,013
7
votes
0
answers
346
views
The space of $p$-adic norms
The 1963 paper by Goldman and Iwahori The space of $p$-adic norms deals with the space of norms on a finite dimensional vector space $E$ over a locally compact complete discrete valuation field $K$. I ...
- 433
4
votes
2
answers
676
views
Reference to a variant of Abel's summation formula
Edit. A stronger version of the formula is true (details follow).
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, $(\lambda_n)_{n \ge 1}$ a nondecreasing sequence of positive reals such that $...
- 10.5k
18
votes
6
answers
2k
views
Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic.
Let $K$ be a non-arch local field (I'm only interested in the char 0 case), let $\mathbb{G}$ be a connected reductive group over $K$ and let $G=\mathbb{G}(K)$. If $V$ is a smooth irreducible complex ...
- 41.4k
1
vote
0
answers
253
views
Who formulated the conjecture that the set of real parts of zeros of the Riemann zeta function is dense in $[0,1]$?
Does anyone know who formulated this conjecture related to Riemann's zeta function?
Conjecture. The set $$\{ x : \exists y \space \space \zeta (x+iy) = 0\}$$ is dense in $[0, 1]$.
In ...
10
votes
2
answers
4k
views
Reference book for Galois Representations
I am an undergrad. I have taken courses in algebraic number theory and have a basic idea about $p$-adic numbers. I have also read a little bit of infinite Galois theory. But I have no idea about ...
- 239
2
votes
2
answers
253
views
Approximation of a square with an irrational arithmetic progression
Let $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ be irrational. Does the arithmetic progression $(n\alpha )_{n\in\mathbb{N}}$ becomes arbitrarily close to squares?
More precisely, what can be said ...
- 3,574
8
votes
2
answers
1k
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How are such sets of natural numbers called?
I heard about this problem an year ago, but I just can't remember the name.
The problem goes like this: study the sets
$\{a_1,a_2,\dotsc,a_m\}\subseteq\mathbb{N}$ such that if $1\leq i<j\leq m$,...
- 103
21
votes
5
answers
5k
views
What arrangement of unit cubes minimizes surface area?
For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below).
Question A. How does one arrange $n$ unit cubes ...
- 7,831
0
votes
1
answer
192
views
English translation of Hasse's "Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage"
I want to read through Hasse's paper about cubic number fields: Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage, Mathematische Zeitschrift 31 (1930) pages 565-...
- 577
0
votes
1
answer
178
views
Size of parities in counting partitions into odd parts
Let $p_{odd}(n)$ be the number of partitions of $n$ into odd parts (see here). For instance, one has the generating function
$$\prod_{k\geq1}\frac1{1-q^{2k-1}}.$$
QUESTION. What is the size of this ...
- 43.2k
15
votes
4
answers
575
views
Are all partial consecutive harmonic subsums distinct?
Let $b \gt a \geq 0$ be integers, and as elsewhere let $H_n$ be $\sum^n_{i=1} 1/i$. A partial consecutive harmonic subsum is a number $H(a,b)$ of the form $H_b - H_a$ (with $ H_0=0$). If $c=a$ and $...
- 13k
3
votes
1
answer
195
views
English reference for the Brauer-Kuroda formula
I'm currently trying to understand the Brauer-Kuroda formula.
Although there are many recent papers on the formula but they seem to be purely algebraic.
They say that original analytic approach is ...
- 1,013
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
6
votes
1
answer
338
views
"Sub-logarithmic" zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem
For each $n\in\mathbb{N}$, let:
$\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$),
$\beta_n$ the largest real zero of $L(s,\chi_n)$,
$\delta_n := (1-\beta_n)\...
- 825
2
votes
2
answers
474
views
For what automorphic representations is Ramanujan-Petersson known?
I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that
If an automorphic representation on $GL(2)$ is ramified at a ...
- 609
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
8
votes
2
answers
564
views
Distribution of primitive roots, as p varies
For a prime number $p$, let $\Phi(p)$ be the subset of $\{ 1, 2, \ldots, p-1 \}$ consisting of primitive roots modulo $p$. (Thus $\# \Phi(p) = \phi(p-1)$, where $\phi$ denotes the totient.)
I am ...
- 13.3k
8
votes
0
answers
346
views
A generalization of Feit–Thompson conjecture, for square-free integers
I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...
2
votes
0
answers
186
views
Dyadic models in number theory and "spillover"
In a classic blog post, Tao discusses the appearance of "dyadic models" in various guises in various areas of math. The number-theoretic version of the idea is to study polynomials over a ...
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
29
votes
5
answers
5k
views
Partial sums of multiplicative functions
It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that $|\mu(1)+\mu(2)+\dots+\...
- 29k
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 ...