All Questions
Tagged with reference-request nt.number-theory
1,409 questions
3
votes
0
answers
164
views
Using the Hilbert symbol to find nice field extensions
Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
- 2,516
0
votes
0
answers
86
views
Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers
Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...
7
votes
1
answer
1k
views
Beilinson's height pairing vs. Néron–Tate
In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper Arithmetic intersection ...
- 5,637
1
vote
0
answers
69
views
Need help to check a quote from Hecke's "Lectures on Dirichlet Series, Modular Functions and Quadratic Forms"
I'd like to check the accuracy of a reference to Hecke's 1938 "Lectures on Dirichlet Series, Modular Functions and Quadratic Forms" implicit in eq. 4.8 on p. 50 of J. G. Leo's dissertation, which is ...
- 11
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
22
votes
2
answers
3k
views
$p$-adic Langlands correspondence
Basic question: Is it correct that the $p$-adic Langlands correspondence is known for $GL_2$ only over $Q_p$ but not other $p$-adic fields? If so, I would like to request some light to be shed on this ...
- 3,867
3
votes
1
answer
270
views
Eisenstein series for discrete subgroups of SL(2,C)?
I am looking for a reference for Eisenstein series for discrete subgroups of $SL(2,\mathbb C)$, in particular, finite index subgroups of $SL(2,\mathcal O_K)$ where $K$ is an imaginary quadratic field.
...
- 3,799
3
votes
0
answers
272
views
Reference request: Representing posets by integer divisibility
Does anyone know of an early published reference for the (very easy) fact that all finite posets can be represented as the poset of divisibility of a finite set of integers?
Page 1 of Birkhoff's ...
- 18.6k
18
votes
3
answers
745
views
Number of primitive $n$th roots with positive versus negative real parts
Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ ...
- 1,991
16
votes
1
answer
706
views
Connection between Bernoulli numbers and Riemann-Siegel theta function?
I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that
$$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n ...
- 1,903
5
votes
1
answer
1k
views
Concrete bounds on the discriminant of a number field
There exists a well known concrete bound on the discriminant of a number field by Minkowski.
Are there any concrete (completely explicit) improvements of this bound?
I know of a bound by Odlyzko, ...
- 11.3k
9
votes
1
answer
2k
views
Sums of two squares in (certain) integral domains
While giving the first of eight lectures on introductory model theory and its applications yesterday, I stated Hilbert's 17th problem (or rather, Artin's Theorem): if $f \in \mathbb{R}[t_1,\ldots,t_n]$...
- 65.4k
4
votes
2
answers
1k
views
Primes $p$ for which $2p-1$ is prime
It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$?
Seemingly it's also an open problem (see here and the linked ...
- 193
8
votes
1
answer
247
views
Origin of definitions of ramified Hecke operators
Consider a classical space $M_k(N)$ of elliptic modular forms of weight $k$ for $\Gamma_0(N)$. The definition of an unramified Hecke operator $T_{p^m}$ in terms of double cosets is the disjoint union ...
- 6,039
2
votes
2
answers
509
views
Question about Zeta Function of Singular Plane Curve
I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes).
I ...
- 53
7
votes
1
answer
487
views
Who proved the upper bound for the autocorrelation of higher order divisor functions?
Who first published a proof that
$$\sum_{n\leq x}d_{k}(n)d_k(n+h)=O(x(\log x)^{2k-2})$$
for fixed $k$ and $h$ please? I am struggling to find a reference. Thank you.
- 2,480
5
votes
3
answers
417
views
Exact bin packing the harmonic series: references?
Given $n$ and $B_n= \lceil H_n \rceil$, where the latter is the $n$th harmonic number $\sum^n_{i=1} 1/i$, for most $n$ it is easy to pack the first $n$ terms of the harmonic series into $B_n$ many ...
- 13k
0
votes
0
answers
133
views
What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?
It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...
6
votes
1
answer
288
views
Number of solutions for the inequality with square roots
Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality
$$|\sqrt{n_1}+\sqrt{n_2}-\sqrt{n_3}-\sqrt{n_4}|<\delta\sqrt{M},$$
where $...
- 7,193
7
votes
0
answers
379
views
Local properties of Galois representations attached to torsion classes
$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}} \DeclareMathOperator{\Sym}{Sym}$
Let $F$ be a number field, and let $\Gamma$ be ...
- 5,382
3
votes
1
answer
276
views
Almost-Primes in Short Intervals
Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number ...
- 14.7k
7
votes
2
answers
708
views
Eisenstein Series on Siegel Space
I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}_4$ is fine). Anything would be welcome, but in particular I'm interested in the ...
- 2,824
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
3
votes
2
answers
411
views
When are "normal" functions normal?
I expected that the fractional part of f(n), n being an integer, would be distributed uniformly over [0,1] (for positive functions - otherwise take [-1,1]) for any run-of-the-mill function, except ...
- 4,829
17
votes
1
answer
703
views
Combinatorics problem about sum of natural numbers
Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6)
Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each $...
- 422
1
vote
0
answers
274
views
On Primes in Arithmetic Progressions
I was wondering if the following approach is being attempted to prove the twin-prime conjecture.
Tao and Green proved in their paper (2006), that there are arbitrarily long arithmetic progressions ...
- 11
29
votes
1
answer
3k
views
The Riemann zeros and the heat equation
The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as
$$
\Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du,
$$
where $\Phi(u)$ is defined as
$$
2\sum_{...
- 11.1k
1
vote
0
answers
112
views
The $p$-adic valuation of powers of consecutive integers
Let $n > 0, K > 0$ integers and, for $i \in \{1,...,n\}$, let $k_i$ and $l_i$ be integers such that $k_i + l_i = K$. Assume that for some $i,j \in \{1,...,n\}$ we have $k_i \neq k_j$.
Claim: ...
- 365
5
votes
2
answers
862
views
Is every positive integer a sum of at most 4 distinct quarter-squares?
There appears to be no mention in OEIS: Quarter-squares, A002620. Can someone give a proof or reference?
Examples:
quarter-squares: ${0,1,2,4,6,9,12,16,20,25,30,36,...}$
2-term sums: ${2+1, 4+1, ...
- 3,443
6
votes
3
answers
811
views
Enumerating cosets of the modular group
Suppose we chose the generators $f(z) = z+1, g(z) = z-1, h(z) = -1/z$ for the modular group $\Gamma$ (i.e. the group of fractional linear tranformations $z \mapsto (az +b)/(cz+d)$ with $a,b,c,d \in \...
- 4,304
3
votes
1
answer
277
views
How do modular functions of level $N>1$ transform under the full modular group?
Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$
First question
What can we say in general about the factor $j(\...
- 2,375
10
votes
5
answers
2k
views
The Circle Method and the binary Goldbach Problem
I've been learning about the Circle Method (at the level of the book
"An Invitation to Modern Number Theory," by Miller and Takloo-Bighash). The arguments in the book show how the
Circle method can ...
- 1,077
8
votes
2
answers
2k
views
Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
- 11.1k
0
votes
2
answers
245
views
When the $o$-th division polynomial of an elliptic curve over finite vanishes only at $x$ coordinates?
Need this for probabilistic factoring algorithm.
Let $p$ be sufficiently large prime and $E$ the elliptic curve
$E /\mathbb{F}_p: y^2=x^3+ax+b$. Let $o=\#E(\mathbb{F}_p)$.
$\psi_n$ denote the $n$-...
- 25.4k
1
vote
2
answers
753
views
basis of the lattice generated by the integer points inside a subspace of R^L
Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{...
- 51
3
votes
1
answer
124
views
Provenance of a result on regular simplices with integer vertices
There are several MO questions related to the question of characterizing those integers $n$ for which there exists a regular $n$-simplex in $\mathbb{R}^n$ with integer vertices, e.g., coordinates of ...
- 50.8k
17
votes
2
answers
3k
views
Some unpublished notes of Hofstadter
I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because they'...
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 ...
10
votes
2
answers
962
views
Surveys of the items of Erdős' "toolbox"
Could you point out some survey papers and monographs that highlight the kernel of tricks, techniques, and tools that Paul Erdős employed the most in his research work (in particular in graph theory, ...
0
votes
1
answer
322
views
On the largest prime factor of $1+n^k$
For every positive integer $n>1$ , let $f(n)$ denote the largest prime factor of $n$. How fast does $f(1+n^k)$ grow with respect to $k$ ? Is it true that $f(1+n^k) > 2k, \forall n >2, \forall ...
user111492
7
votes
0
answers
291
views
What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of compact support)?
Let $G$ be a linear algebraic group over a number field $k$. Let $\mathbb A$ denote the adeles of $k$, $\mathbb A_f$ the finite adeles, and $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. Here are ...
- 6,180
8
votes
1
answer
3k
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Where to start reading into p-adic non-abelian Hodge theory?
I'm curious about Faltings' "A $p$-adic Simpson correspondence". Do you know more detailed, introductory, expositions, surveys, texts of seminars on that?
Edit: Annette Werner's survey &...
6
votes
2
answers
730
views
Definition of Hecke operators on orthogonal modular forms
In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5:
"Describe how the correspondence in this paper behaves under
the
action of Hecke operators."
Since ...
user94539
5
votes
1
answer
414
views
Primality test for $2p+1$
In 1750 Euler stated following theorem :
Let $p \equiv 3 \pmod 4$ be prime then $2p+1$ is prime iff $2p+1 \mid 2^p-1$ .
In 1775 Lagrange gave a proof of the theorem .
Recently I have formulated ...
- 2,661
14
votes
1
answer
755
views
Generating function of the Thue-Morse sequence
Let $T$ be the generating function of the Thue-Morse sequence; thus,
$T(x)=x+x^2+x^4+x^7+\dotsb$. It is known that $T$ satisfies the nice
congruence
$$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 \...
- 23k
18
votes
2
answers
3k
views
References for Artin motives
I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
- 7,442
4
votes
4
answers
3k
views
Where do Set Theory and Number Theory meet together?
As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...
- 2,381
10
votes
2
answers
870
views
Bound on gcd of two integers
Well this is a problem I was fiddling with. I came up with it but it probably is not original.
Suppose $a\in \mathbb{N}$ is not a perfect square. Then show that :
$$\text{gcd}(n,\lfloor{n\sqrt{a}}...
- 1,090
4
votes
0
answers
252
views
Height pairings of Heegner points of nontrivial conductor
I am studying Gross's and Zagier's proof of the BSD conjecture for elliptic curves of rank $\leq 1.$ Their calculation essentially boils down to the following ingredients:
(1.) Finding a suitable ...
- 1,805
5
votes
0
answers
2k
views
Jacobi's two-square theorem
Jacobi's theorem is: the number of ways of representing $N$ as a sum of two squares is $4(d_1(N)-d_3(N))$ where $d_i(N)$ is the number of divisors of $N$ that are of the form $4k+i$. I was wondering ...
- 221