All Questions
Tagged with reference-request nt.number-theory
1,408 questions
2
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0
answers
217
views
A high dimensional generation of Dirichlet approximation theorem, linear case and nonlinear case
I am working with something on Diophantine approximation, and I found a high dimensional generation of Dirichlet approximation theorem which may be true; I will be very happy if this is true. The ...
- 697
4
votes
2
answers
1k
views
Reference request for Kato's paper: A generalization of local class field theory by using K -groups
I would like to ask for the paper of Kato: A generalization of local class field theory by using K -groups I, J. Fac. Sci. Univ. Tokyo Sec. IA 26 No.2, 1979, 303–376. I could not find it. Could anyone ...
- 91
19
votes
2
answers
2k
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Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)?
In Wikipedia's page for Bertrand's postulate, it is said that its (2n,3n) version was proved by El Bachraoui in 2006. Seems likely that it was first proved way before than that! Can anyone point to ...
- 2,992
3
votes
0
answers
540
views
Questions about the exceptional zeros of Dirichlet $L$-functions
I have couple questions regarding the exceptional zeros of Dirichlet $L$-functions. We have the following result:
There is a constant $c_1 > 0$ such that $L(\sigma, \chi) \not = 0$ whenever
$$
\...
- 3,625
5
votes
1
answer
243
views
How to find a set of integers that satisfy certain linear conditions
Suppose I have a sequence of non-negative integers $J=\{j_1,j_2,\ldots,j_n\}$
and want to find (if possible) a set of integers $I=\{0=i_1<i_2< \cdots < i_m\}$
such that $j_t$ counts the ...
- 51
12
votes
3
answers
1k
views
Chow Groups of varieties over number fields
I believe that there is a conjecture that for any smooth projective variety $X$ over a number field $K$, its Chow groups $CH^i(X)$ (or at least $CH^i(X)\otimes_{\mathbf Z} \mathbf Q$) are finitely ...
- 2,923
12
votes
2
answers
1k
views
Has there been further work on Bender-Brody-Müller approach to RH?
Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I ...
- 6,984
0
votes
0
answers
98
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Eigenvalues of a sequence of matrices involving the divisor function
Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
- 10.4k
2
votes
0
answers
519
views
Good place to learn about arithmetic schemes?
Where is a good place to learn about arithmetic schemes? There is discussion in Eisenbud-Harris's book The Geometry of Schemes (and also Mumford's red book) and I hear that there is discussion in Liu'...
- 21
3
votes
0
answers
162
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Hecke eigensystem in cohomology vs. compactly supported cohomology
What follows is a question that's probably well-known to experts, but I haven't been able to find a reference.
Let $\mathrm G$ be a connected, semisimple $\mathbb Q$-group. Let $K \subset \mathrm G(\...
- 391
7
votes
3
answers
551
views
Minkowski's theorem for non-0-symmetric sets
Let $\Lambda \subseteq \mathbb{R}^n$ be a full-rank lattice, i.e. $\Lambda = A \mathbb{Z}^n$ for some $A \in \mathrm{GL}_n (\mathbb{R})$, and let $C \subseteq \mathbb{R}^n$ be a $0$-symmetric convex ...
user92170
13
votes
1
answer
3k
views
A good reference to the general Chinese Remainder Theorem
I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following
General Chinese Remainder Theorem. For integer numbers $a_1,\dots,a_n$ and positive ...
- 42k
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
0
votes
1
answer
107
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Variation on the definition of the uniform distribution mod 1 [closed]
A sequence $x_{n}$ is said to be uniformly distributed mod 1 if $\forall a,b$ with $0\leq a<b<1$,
$$\lim_{n\rightarrow \infty}\frac{1}{n}|\lbrace j=1,...,n :\lbrace x_{j}\rbrace\in [a,b]\rbrace|...
7
votes
1
answer
858
views
Teichmuller groupoids in Grothendieck's esquisse d'un programme
Grothendieck in his Esquisse d'un programme mentioned without any precise definition and construction that the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the whole "tower" ...
- 21.8k
5
votes
0
answers
145
views
Character sums over a sumset
Suppose that $p$ is a prime, $A$ is a subset of $\mathbb F_p$, and $P$ is a polynomial over $\mathbb F_p$ of degree $d$. Using Weil's bound, it is not difficult to show that
$$ \left| \sum_{a,b\in A}...
- 23k
3
votes
1
answer
481
views
Reference for explicit formula for $\sum_n \Lambda(n) \chi(n)$ with smooth weights
Let $\Lambda$ be the von Mangoldt function and $\chi$ a primitive character mod $q$, then we have the explicit formula
$$
\sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X - \sum_{ |Im \ \rho| \leq ...
- 3,625
0
votes
1
answer
210
views
Are there Vaughn's identity type decompositions for other arithmetic functions?
Vaughn's identity is a useful way to decompose the von Mangoldt function $\Lambda(n)$ into Type I and Type II components, and this is used in many problems involving prime numbers. I was wondering if ...
- 3,625
6
votes
1
answer
419
views
independence of $\ell$ of characteristic polynomial of Frobenius on $\ell$-adic Tate module of Abelian varieties over number fields
I am looking for a reference for the independence of $\ell$ of the characteristic polynomial of the Frobenius $\mathrm{det}(1-|\kappa(v)|^{-s}\mathrm{Frob}_v \mid (V_\ell A)^{I_v})$ acting on the $\...
user19475
8
votes
0
answers
161
views
Looking for an elliptic curve E st ${\large Ш}(\mathbb Q,E)$ cont. an element of order $p^2$ and certain other properties
I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied:
(1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ ...
- 1,805
7
votes
1
answer
675
views
Short divisor sum
Let $d(n)$ denote the number of positive divisors of the positive integer $n$.
Pick some positive $X,h \in \mathbb{R}$ and consider the sum
$$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$
In view of ...
- 11.3k
0
votes
0
answers
234
views
whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture?
I vaguely recall that whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture, could any one give the reference? or ...
- 3,094
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
$$i(q) := \frac{\sigma(n^2)}{q^k}=\frac{...
1
vote
1
answer
136
views
How resolution of singularity is linked to continued fracton? [closed]
I vaguely recall that resolution of singularity may be linked to continued fracton, possibly it is cusp that links to CF. Could any one give concrete reference and give example? Thanks.
- 3,094
2
votes
2
answers
257
views
Reference request for function by which to compute coefficients of continued fraction of algebaic number
The simple continued fraction is in the form
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance. Obviously,the coefficients $x_i$can be computed by computable function $x_i=f(i),...
- 3,094
7
votes
1
answer
1k
views
Szpiro's conjecture for function fields and Mochizuki's approach to the number field case
Where can I find more details on the proof of Szpiro's conjecture for function fields, as mentioned in Minhyong Kim's answer to this MO question?
I am looking at this in the context of Mochizuki's ...
- 3,309
1
vote
1
answer
81
views
Reference request for multiple free sequences
Erdos usually named a sequence of integers no one of which is divisible by any other as an $M$- sequence (M stands for "multiple-free") or primitive sequence.
For example it is easy to see that $\...
- 3,866
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
4
votes
1
answer
150
views
About Averages of Incomplete Additive Character Sums
Let $p$ be a prime. Let $\omega$ be a $ p $-th root of unity. We know that $ \chi_\alpha (x) = \omega^{\alpha\cdot x} $ are the additive characters of $ \mathbb{Z}_p $.
I have a question about ...
- 43
24
votes
2
answers
1k
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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
12
votes
1
answer
307
views
Partition of [3n] into summoids
Let $ [n] $ be the set $ \{1,2,\ldots n\}$.
A summoid is a subset $ A \subset [n] $ of the form $ \{a,b,a+b\} $ (you can choose a better name, if it doesn't exist already).
Now, I developed by ...
24
votes
0
answers
1k
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Exotic 4-spheres and the Tate-Shafarevich Group
The title is a talk given by Sir M. Atiyah in a conference with the following abstract:
I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
- 1,629
1
vote
0
answers
148
views
Estimating the sum of Dirichlet character $\sum_{0 \leq x < q} \chi(F(x))$ where $F(x)$ is a polynomial
Let $q \in \mathbb{N}$ and $\chi$ a Dirichlet character mod $q$. Let $F(x)$ be a polynomial with integer coefficients. I was wondering if a bound for the following sum was available or not:
$$
\sum_{0 ...
- 3,625
2
votes
0
answers
325
views
Reference/PDF request for paper by Sathe and related note
I am looking for a PDF version of the following articles:
Sathe, L. G. - On a problem of Hardy on the distribution of
integers having a given number of prime factors, (I. - IV.) J. Indian Math. Soc. ...
- 593
2
votes
1
answer
212
views
When will the value of automorphic function $f(x)$ satisify an algebraic equation?
When will the value of automorphic function $f(x)$ satisfy an algebraic equation? Or what is the value of $x$ such that the value of automorphic function $f(x)$ is algebraic?
If the question is too ...
- 3,094
-4
votes
2
answers
272
views
Has nontrivial solution in positive integers of a diophantine equation: $x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$ [closed]
Has nontrivial solution in positive integers of a diophantine equation as follows ?
$$x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$$
Where trivial solutions are $x_i=y_j$.
Can you send me any ...
- 477
11
votes
0
answers
324
views
Why is the CM-type preserved after base changing from char 0 to char p?
There is a transition in the theory of complex multiplication which seems to be glossed over in all expositions I can find. I would like to explicitly find a theorem that allows me to do this.
...
- 3,539
5
votes
1
answer
278
views
Are there primitive quartic CM fields whose norms of units give all totally positive units of the real quadratic subfield?
Let $K$ be a primitive (i.e. not biquadratic) quartic CM-field. That is we have $[K:\mathbb{Q}]=4$ and let $K_0=\mathbb{Q}(\sqrt{d})$ be the totally real quadratic subfield, here $d> 1$, that is we ...
- 1,025
2
votes
0
answers
188
views
Solution of the Diophantine equation $x^4+y^4+z^4=2t^4$ are well-known? [closed]
Are solutions of the Diophantine equation $x^4+y^4+z^4=2t^4$ well-known?
I give a solution:
$x=m^2-n^2, y=m^2-2mn, z=n^2-2mn, t=m^2+n^2-mn$
- 477
1
vote
2
answers
334
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(Reference) A Shimura operator acting on Hermitian modular forms
In his book Arithmeticity in the theory of automorphic forms (http://bookstore.ams.org/surv-82-s) Shimura introduces at page 146 an operator $\Delta_p^q$ which should act on nearly holomorphic modular ...
- 253
5
votes
2
answers
274
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The Heegner hypothesis for a mean value result of Murty-Murty/Bump-Friedberg-Hoffstein
The classical mean value result of Murty and Murty (1991) and Bump, Friedberg, and Hoffstein (1990) on derivatives of modular form L-functions $L(s,f)$ proves (roughly speaking) the existence of ...
- 3,799
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
6
votes
4
answers
900
views
Mathematical induction vis-à-vis primes
One of the most used proof-techniques is mathematical induction, and one of the oldest subjects is the study of prime numbers. Thanks to Euclid, we can consider the primes as a infinite monotone ...
11
votes
2
answers
369
views
Harmonic congruence
There are a number of interesting congruences for harmonic sums, not the least of which is Wolstenholme's theorem: $H_{p-1}:=\sum_{j=1}^{p-1}\frac1j\equiv 0\mod p^2$.
It appears that $\sum_{j=1}^{p-1}...
- 309
2
votes
2
answers
338
views
Weak form of Brocard's conjecture
I ask this out of curiosity, motivated by a question asked by one of my students.
The Brocard's conjecture claims that there exist at least four prime numbers between $p_{i}^2$ and $p_{i+1}^2$, where ...
- 66.3k
4
votes
1
answer
524
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
5
votes
0
answers
327
views
Is there a concrete way to show the existence of canonical model for non-modular Shimura curves?
I am trying to read Carayol's article on the construction of Galois representations associated to Hilbert modular forms (http://archive.numdam.org/article/CM_1986__59_2_151_0.pdf). The main geometric ...
- 2,842
6
votes
1
answer
144
views
Proportion of pairs of integer polynomials with a bounded value of the resultant
Let $f(x), g(x) \in \mathbb Z[x]$ be polynomials of positive degrees $r$ and $s$ respectively, and let $\operatorname{Res}(f,g)$ denote their resultant. Further, let $H(f)$ denote the naive height of ...
- 1,625
1
vote
1
answer
443
views
Distribution of Mobius function
Let $\mu(n)$ be the Mobius function.
Let us define $\mu^+(n)$ to be $\mu(n)$ if $\mu(n)>0$ and $0$ otherwise. Is there a known asymptotic formula for
$$
\sum_{n \leq N} \mu^+(n),
$$
and ...
- 3,625
3
votes
1
answer
330
views
Turan Inequalities
A real entire function
$$\psi(x)=\sum_{k=0}^{\infty} \gamma_k\frac{x^k}{k!}$$
is said to be in the Laguerre-Polya class, denoted $\psi(x) \in \mathcal{LP}$, if it can be represented in the form
\...
- 73