All Questions
Tagged with reference-request nt.number-theory
1,408 questions
2
votes
1
answer
626
views
Harmonic Analysis for Function Fields
Hi,
Where goes a characterictic $0$ person, in order to learn about the local harmonic analysis for local fields in characteristic $p$? Is there nice and conscise reference for the local fields in ...
- 11.2k
11
votes
1
answer
875
views
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
4
votes
0
answers
149
views
The number of representations of the positive integer $n$ as $a^{2}+b^{2}+p^{2}c^{2}$
Let $n$ be a positive integer and $p$ a prime number. I know that there are formulas by which one can compute the number of representations of $n$ as the sum of two or three squares etc.
I would to ...
- 81
1
vote
0
answers
272
views
Possible counterexample to the strong three exponentials conjecture
There is something wrong possibly either with me or with Wikipedia.
Wikipedia's article on the strong three exponentials conjecture
defines $L^\ast$ as the set of all complex numbers of the form
$$\...
- 25.4k
2
votes
1
answer
137
views
Quotients of number rings IZ[zeta_l]
Let $l=p^r$ a prime power and $\zeta$ a primitive l-th root of unity. It is classical result, that $(1-\zeta)^{\varphi(l)}=p\cdot\epsilon\in\mathbb{Z}[\zeta]$ for a unit $\epsilon$.
It should be a ...
- 155
7
votes
1
answer
1k
views
Shot in the dark: Is there an english translation of Deligne-Rapoport "Les schemas de modules..." anywhere?
Extensive googling (and searching here) has yielded nothing, unfortunately.
I knew a language genius once who offered to translate it for me as a favor, but I turned him down because it seemed like ...
- 73
1
vote
0
answers
207
views
Proofs for almost prime limits
A number $n$ with prime factorization $$n=\prod_{i=1}^rp_i^{a_i}$$
is a k-almost prime if it has a sum of exponents $$\sum_{i=1}^{r}a_i=k$$ i.e., when the prime factor (multiprimality) function $\...
- 1,903
1
vote
0
answers
192
views
Ideals with norm in arithmetic progression
Let K/Q be a number field extention. Is there an asymptotic formular for the numer of ideals $\sum\limits_{\substack{N(A)\leq x\\N(A)\equiv k(q)}}1$,where $(k,q)=1$ and $A$ runs over ideals in $O_K$. ...
- 21
4
votes
1
answer
448
views
Is that series-transformation known in the context of divergent summation?
Note: I asked this question in math.stackexchange but did not receive an answer
Background:
In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-...
- 5,342
1
vote
1
answer
262
views
Distribution of colors in the number of integer partitions of n
Given an integer $n$ the number of partitions of $n$ into two colors can be represented as
$$p_2(n)=\sum_{k=0}^n p(k)p(n-k)$$ where $p(k)$ counts the number of ordinary partitions of $k.$ What is the ...
- 1,306
2
votes
1
answer
219
views
RefReq: Algorithms for standard operations in Algebraic Number theory
Given an algebraic number field $F$ (I actually don't have an idea how to implement this data already, except for splitting fields of polynomials, but there is something in SAGE) is there free code ...
- 11.2k
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
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
2
votes
0
answers
224
views
Abel-Jacobi map isomorphism galois representations
Let $X/\mathbb{Q}$ be an irreducible smooth projective curve with a $\mathbb{Q}$-rational point $p$. Then there is a map $\phi: X \rightarrow \textrm{Pic}^{0}(X)$ with the property that $q \mapsto [q]-...
- 3,555
1
vote
0
answers
280
views
Would a proof of both (G)RH and Montgomery's pair correlation conjecture imply SOC?
Would a proof of both (G)RH and Montgomery's pair correlation conjecture imply SOC?
It seems, judging by the abstract of a 2002 paper of Ram Murty and a possibly Romanian co-author published on www....
- 6,984
2
votes
1
answer
776
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
0
votes
0
answers
234
views
On the irrationality measure of generalized Stoneham numbers
Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and $\gcd(a,...
- 10.5k
1
vote
0
answers
74
views
Equivalent of Lauricella $F_D$ on an elliptic curve?
Lauricella's hypergeometric function $F_D$ is related to (weighted) configurations of points on $\mathbb{P}^1$. I am looking for generalizations to weighted point configurations on an elliptic curve. ...
- 111
5
votes
1
answer
374
views
Where can I read about exponential sums corresponding to Jones Polynomial?
I remember reading that a number theoretic analogue of Witten's path integral formula for the Jones polynomial:
$$\text{Jones}_K(e^{2\pi i/(k+2)})=\int_{\text{$SU(2)$ connections on $\mathbb S^3$}/\...
- 18.7k
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
2
votes
0
answers
236
views
Is there a generalization of Granville-Langevin conjecture for number fields?
According to Wikipedia and other sources the Granville-Langevin conjecture
states:
If $f$ is a square-free binary form of degree $n > 2$, then for every real $\beta > 2$ there is a constant $...
- 25.4k
4
votes
0
answers
242
views
Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$
As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion $E_{\ell,2}(q)=E_{2}(q)-\...
- 1,537
29
votes
0
answers
3k
views
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
7
votes
1
answer
824
views
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
views
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
4
votes
1
answer
288
views
Reference request for an identity for tangent numbers
The tangent numbers $(T_{2n+1})=(1,2,16,272,7936,...)$ (cf. OEIS: A000182) satisfy many recurrences. I would be interested to find references for the following which I think must be very old:
$T_3 -...
- 5,622
4
votes
1
answer
338
views
Reference for Rank Distribution Conjecture.
I am currently writing my master's thesis and I was wondering if the rank distribution conjecture was ever formally written down. Recall that it says that:
Half of all elliptic curves have rank $0$, ...
- 1,458
0
votes
0
answers
215
views
Invariance of the (Liouville-Roth) irrationality measure under rational Möbius transformations
For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the (...
- 10.5k
2
votes
0
answers
293
views
Zeta function of abelian variety with CM
In his book, "Introduction to the arithmetic theory ..." (Ch 7, section 8) Shimura constructs the Zeta function of an abelian variety with CM and expresses it as a product of L-functions.
Since I ...
- 21
2
votes
1
answer
387
views
Zero-cycles on an arithmetic surface
Could anyone give a reference for the following statement, which I believe is true.
"Let X be a regular scheme, flat over $Spec( \mathbb{Z}) $, with fiber dimension $1$. Then the Chow group $CH^2(X)$ ...
- 5,637
4
votes
0
answers
291
views
Translation of an article by Wolfgang Schmidt on normality for real numbers in different bases.
I would greatly appreciate a pointer to a translation from German into English of the article by Wolfgang Schmidt, Über die Normalität von Zahlen zu verschiedenen Basen, from Acta Arithmetica VII, ...
- 1,780
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
2
votes
0
answers
152
views
Reference Request: Properties of the Integer Factorization Polytope
The complexity of Integer Factorization is to my knowledge still an open problem, whereas deciding, whether a given integer is a prime number is known to be in $P$ and a proof is available online here:...
- 13.2k
5
votes
0
answers
288
views
Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?
Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...
- 2,396
1
vote
1
answer
453
views
Are large numbers the sum of two or more large primes? [Hoping for reasonable constants]
Is it true that for all $n>N$ that n is the sum of two or more distinct primes that are either large or (for parity reasons) 2?
I feel like I've seen a result allowing this with $p\gg n^e$ for ...
- 9,114
-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
427
views
Work exploring application of probability to metric number theory problems
I am interested in studying the application of probabilistic tools to study metric number theoretic problems, specifically the Duffin-Schaeffer conjecture (http://www.math.osu.edu/files/duffin-...
- 26.9k
5
votes
1
answer
856
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
5
votes
0
answers
219
views
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
1
vote
0
answers
452
views
Reference Request - Jakob Weisblat's "The Search for the Odd Perfect Number" [closed]
Hi All!
I am currently trying to locate an online copy of Jakob Weisblat's paper titled "The Search for the Odd Perfect Number". I could only get hold of the abstract:
"A perfect number is a number ...
5
votes
0
answers
504
views
A conjecture on the relative size of Goldbach pairs?
On leafing through some papers of John Nash (available online on his webpage) I found this intriguing little observation:
Noticing that with larger even numbers it seemed to become
possible to ...
- 497
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
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
2
votes
0
answers
172
views
Bost's slopes method for Lindemann theorem
Hello,
I am looking for a proof of Lindemann theorem (transcendence of $e^a$ for $a$ algebraic) that uses Bost's slopes method. Does anyone know where I could find that?
- 3,998
9
votes
0
answers
605
views
Hilbert symbol and Weil index, beyond the quadratic case?
Let $F$ be a local nonarchimedean field. Let $n$ be a positive integer for which the group $\mu_n(F)$ of $n^{th}$ roots of unity in $F$ has order $n$. Let $\epsilon: \mu_n(F) \rightarrow C^\times$ ...
- 13.3k
2
votes
0
answers
228
views
Progress towards Selberg's conjecture A
Hello,
Selberg's conjecture A states that whenever $F$ is an element of the Selberg class, there exists a non negative integer $n_F$ depending only on $F$ such that $\sum_{p\leq x}\dfrac{\vert a_{p}(...
- 6,984
15
votes
0
answers
476
views
Any references on zeta-function like sums of inverse determinants over lattices of matrices?
I'm sorry for the title, it was little difficult to phrase..
Let us consider a matrix lattice $L\subset M_n(\mathbb{C})$. By this I mean a discrete additive group in $M_n(\mathbb{C})$. Let us ...
- 201
3
votes
1
answer
318
views
Minor Arc Estimates for an Exponential Sum for a Quadratic Polynomial Over the Primes
Let $f$ be a quadratic polynomial with leading coefficient $\alpha$, and suppose $\alpha$ is in a "minor arc" in the sense that $\alpha$ is not within $\frac{(\log N)^A}{q N^2}$ of any rational number ...
- 33
9
votes
0
answers
462
views
$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$
Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
- 3,076
8
votes
0
answers
877
views
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