All Questions
Tagged with reference-request nt.number-theory
1,408 questions
10
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Reference request: number theory of Z[1/p]
Can anyone suggest a good place to read up on the number theoretic properties of and techniques for $\mathbb{Z}[1/p]$, (that is, rational numbers with only powers of a prime $p$ in the denominator)?
...
- 2,235
22
votes
1
answer
2k
views
Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials
Let $H_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E_{...
- 4,898
15
votes
4
answers
2k
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Are any good strategies known for Erdos-Turan conjecture on additive bases of order two?
The following problem can become a bit of an obsession. I'm curious if there are any serious strategies for attacking it. The problem is a certain Erdos-Turan conjecture.
Let $ B \subseteq {\mathbb ...
- 7,057
2
votes
1
answer
386
views
Totient function inequality
Does any of you know if the inequality
$\displaystyle \frac{\phi(\sigma(n))}{n} < (\log \log \log n)^{-1/2}$
is true for all $n$ sufficiently large?
I remember reading something to that effect ...
- 8,842
8
votes
0
answers
875
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On Stark's conjecture for imaginary quadratic fields
In the famous paper "L-Functions at s = 1. IV. First Derivatives at s = 0" of Stark from 1980, it is shown that in the case of an imaginary quadratic field $K$ certain numbers of the form $$exp(-\frac{...
- 2,029
6
votes
0
answers
252
views
How big is the Fourier transform of the log of a polynomial over the p-adic numbers
Let $f(z_1,\dots,z_n)$ be a polynomial with $p$-adic coefficients, and let $g(z):=log\lvert f(z) \rvert$. If $\xi$ is a complex character of $\mathbb{Z}_p^n$ there exists a number $v=v(\xi)$ such that ...
- 527
23
votes
1
answer
4k
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Chapters 1--4 of the Artin-Tate notes on Class Field Theory
Emil Artin and John Tate held a seminar on class field theory at Princeton University in 1951--1952. Their notes were published in 1967 by Benjamin (New York), but the first four chapters covering (...
- 18.7k
8
votes
1
answer
2k
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Prof. Murty's B. Sc. Thesis
Can any of you guys help me to find out if there is a retrodigitized copy of M. Ram Murty's 1976 thesis available on the online database of Carleton University Library?
I really hope this question is ...
- 8,842
3
votes
6
answers
2k
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Teach a course in 1 month
I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?
15
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3
answers
1k
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Unit fraction, equally spaced denominators not integer
I've been looking at unit fractions, and found a paper by Erdős "Some properties of partial sums of the harmonic series" that proves a few things, and gives a reference for the following theorem:
$$\...
- 305
7
votes
1
answer
824
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Is the Landau-Ramanujan constant irrational?
Hi, here, in wikipedia, the Landau-Ramanujan constant appears under a list of suspected transcendentals. I could not find anywhere a statement or a proof of it's irrationality. So, my question is, is ...
3
votes
1
answer
2k
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Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)
The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao....
- 1,795
5
votes
0
answers
219
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Character tables of the p-core of the binary modular congruence group of p-power level
Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the
American Mathematical Society. 79 (1973), no. 4.), ...
- 2,396
2
votes
0
answers
687
views
Strong Bezout's Identity?
Let $\{ a_i \}_{i=1}^N $ be a set of elements of the ring of integers, $\mathbb{Z}_D$ and define $g = \text{gcd}(a_1, a_2,\ldots, a_N, D)$. Then Bezout's Identity states that there exists another set $...
- 133
20
votes
6
answers
4k
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Erik Westzynthius's cool upper bound argument: update?
Version 2 of this writeup is
available, and includes a newer and simple upper bound thanks to
MathOverflow 88777 as
well as indirect references to future writeups. Details of further work
...
- 13k
12
votes
3
answers
881
views
What does the computer suggest about the parity of p(n), for n in a fixed arithmetic progression?
Let p(n) be the number of partitions of n. A famous theorem of Euler allows one to compute
the parity of p(n) quickly for quite large n. In:
On the distribution of parity in the partition function, ...
- 5,422
11
votes
6
answers
2k
views
The Wiener-Ikehara approach to the PNT
Was providing an alternative proof of the PNT one of the main impulses that led to the discovery of the Tauberian theorem of Wiener and Ikehara or the other way around?
In any case, do you know who ...
- 8,842
2
votes
1
answer
775
views
Transcendence of $\log 2$
I am not number theorist, forgive me if this is a stupid question.
Recently I was curious about the ideas behind the transcendence of $\log 2$.
For the number $e$, It seems that the transcendence ...
- 2,044
-1
votes
1
answer
743
views
Taming this Conway-type sequence
(I started working on this problem after trying to get any "interesting" pattern out of the number that Gowers randomly wrote while answering:What is realistic mathematics?.)
The number was ...
- 2,855
3
votes
1
answer
2k
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What are Santilli's isonumbers?
A friend of mine asked me yesterday about Santilli's isonumbers. I told him that it was quackery. As I based my answer only on the general reputation of the guy and had no knowledge of the subject, I ...
- 12.3k
7
votes
2
answers
521
views
How large (small) can be the measure of a set where a polynomial takes small values ?
A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...
- 1,795
11
votes
3
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745
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Counting points on lattices
I expect that the following is a standard problem from analytic number theory, but I don't know where exactly to look for an answer.
Let f: ℤr→ H be a surjective homomorphism into a ...
- 527
29
votes
0
answers
3k
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What are the possible singular fibers of an elliptic fibration over a higher dimensional base?
An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one.
It is often required for the ...
- 3,022
9
votes
1
answer
2k
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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
6k
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Factorizing polynomials of several variables (in a different perespective)
I am looking for factorization of polynomials of several variables in the way outlined below.
Consider a second degree polynomial of two variables over the complex numbers.
"P(x,y) = Ax^2 + Bxy + Cy^...
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
21
votes
2
answers
1k
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Most squares in the first half-interval
It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one ...
- 14.7k
5
votes
1
answer
854
views
Rallis inner product formula for U(2,2) and U(3)
Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are:
"A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998.
"An ...
- 467
2
votes
2
answers
353
views
Reference request: given a divisor d of N, how quickly can I obtain the largest factor of N coprime to d?
This is quite likely to be a solved problem, perhaps even a standard exercise. However, being a non-[number theorist], I don't know where to look. A quick perusal of the basic starting references ...
- 1,444
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
15
votes
1
answer
1k
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If the tensor product of two representations are crystalline, are the original representations crystalline?
Let $K$ be a finite extension of the $p$-adic numbers. Suppose that $V$ and $W$ are two (finite dimensional, $p$-adic) continuous representations of $G_K$. Suppose that $V \otimes W$ is crystalline. ...
user631
8
votes
2
answers
852
views
Does anyone have access to a copy of Yury G. Teterin's 1984 (Russian) preprint "Representation of numbers by spinor genera"
Encouraged by
Does anyone have an electronic copy of Waldspurger's "Sur les coefficients de Fourier des formes modulaires de poids demi-entier"?
I realized I could ask for this rare item ...
- 25.7k
17
votes
5
answers
4k
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Fermat numbers and the infinitude of primes
Wonder whether any of you guys know why it is that the proof of the infinitude of primes that is based on the coprimality of any pair of (distinct) Fermat numbers is commonly attributed to Pólya.
In ...
- 8,842
6
votes
2
answers
976
views
References for modular polynomials
I am teaching a graduate "classical" course on modular forms. I try to achieve the most elementary level for presenting modular polynomials. Serge Lang's "Elliptic functions" cover the topic quite ...
- 13.5k
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
5
answers
2k
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The missing Euler Idoneal numbers
It is known that if GRH holds there does not exist additional Idoneal numbers. (see www.mast.queensu.ca/~kani/papers/idoneal.pdf this paper puts on the question of correctnes for Wikipedia and Wolfram ...
- 3,463
3
votes
1
answer
844
views
finite generation of the Mordell-Weil group over finitely generated fields
Does anyone know a reference for the proof of the finite generation of the Mordell-Weil group over finitely generated fields?
user19475
9
votes
2
answers
3k
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Transformation formulae for classical theta functions
I am looking for a reference for the transformation formulae
for the classical theta-functions
$$\theta_4(\tau)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}$$
and
$$\theta_2(\tau)=\sum_{n=-\infty}^\infty q^{...
- 20.8k
12
votes
3
answers
2k
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What is the etymology for the term conductor?
This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.
What motivated the use of the word "conductor" in the first place?
A friend ...
- 3,268
7
votes
1
answer
1k
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Can every finite graph be represented by an arithmetic sequence of natural numbers?
(This is a follow-up to my previous questions Natural models of graphs?.)
Erdös in The Representation of a Graph by Set Intersections (1966) states:
Theorem. Let $G$ be an arbitrary
graph. Then there ...
- 9,720
23
votes
2
answers
2k
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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
7
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4
answers
1k
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Reference for the expected number of prime factors of n larger than n^alpha is -log alpha
Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$.
It's my understanding that (...
- 14k
6
votes
2
answers
1k
views
Does anyone have an electronic copy of Waldspurger's "Sur les coefficients de Fourier des formes modulaires de poids demi-entier"?
Is there an electronic copy of Waldspurger's paper "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" floating around the internet somewhere? This appeared in J. Pures Math. ...
- 13.1k
11
votes
4
answers
4k
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Variants of Eisenstein irreducibility
In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...
- 32.6k
7
votes
1
answer
918
views
Integral expression for zeta(2)
By computing the sum of all Bernoulli numbers via Borel summation (I learned this technique from Varadarajan's excellent book Euler through time. A new look at old themes, 2006) I found that $$\sum ...
- 32.6k
11
votes
3
answers
461
views
citation for first statement of the Re(s) = 6 conjecture on zeros of Ramanujan L function
Hi, for the bibliography of a paper I'm writing I seek a citation for the first statement of the conjecture that the nontrivial zeros of $F(s) = \sum_n\tau(n)n^{-s}$ all lie on the line Re(s) = 6. (...
- 191
12
votes
2
answers
1k
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Dihedral extensions and the Ankeny–Artin–Chowla conjecture
Jensen and Yui (Polynomials with $D_p$ as Galois group
J. Number Theory 15, 347–375 (1982)) proved that if $p = 4n+1$
is a regular prime, then there is no normal extension of the
rationals with Galois ...
- 32.6k
5
votes
4
answers
2k
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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
17
votes
13
answers
6k
views
Probability in number theory
I am hearing that there are some great applications of probability theory (or more general measure theory) to number theory. Could anyone recommend some good book(s) on that (or other types of ...
11
votes
1
answer
875
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An arithmetic highest weight theory?
I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps ...
- 436