Prime numbers, diophantine equations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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2
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0answers
87 views

characteristic ideal of the Iwasawa module

Let $H$ be a complex biquadratic Galois extension of $\mathbb{Q}$ such that the galois group of $H$ is isomorphic to the Klein Group. Let $H_{\infty}$ be an anticyclotomic $\mathbb{Z}_p$-extension of $...
3
votes
0answers
144 views

Can someone explain this appearance of the Fibonacci series in the formula of the Fibonacci series? [closed]

I have found the Fibonacci series as a function. The function is as follows :- $$F(x) = 1 - 0×f_1(x) + 1×f_2(x) - 1×f_3(x) + 2×f_4(x) - 3×f_5(x) + 5×f_6(x) - 8×f_7(x) + 13×f_8(x) - 21×f_9(x) + 34×f_{...
2
votes
2answers
270 views

Asymptotic estimate of the probability of $(n, P(\sqrt{x})) \leq x$?

Let $P(x)$ be the product of all primes less or equal to $x$. The probability of $(n, P(\sqrt{x})) \leq x$ for an arbitrary $n$ is then given exactly by $$ \prod_{p\mid P(\sqrt{x})}{\left(1-\frac{1}{p}...
4
votes
1answer
226 views

How to construct an abelian variety with CM by a given CM field?

Let $F$ be a totally real number field, and let $K$ be a quadratic extension of $F$ which cannot be embedded into $\mathbb{R}$. Then $K$ is a so called CM field. For instance, take $F = \mathbb{Q}(\...
1
vote
0answers
109 views

Probability that two integers selected from a fixed interval are relatively prime [closed]

I found the answer to a very similar question already asked here on mathoverflow: what is the probability that two natural numbers are relatively prime? The answer given in the link below was $\frac{6}...
10
votes
1answer
239 views

Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?

Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...
1
vote
0answers
148 views

Optimal Diophantine approximation of $\pi$

If the 'optimal' Diophantine approximation of $\pi$ is given by the maximum value of $M=-\log_q(\min_{\forall p \in \mathbb{N}} |\frac{p}{q}-\pi|)$ for $q \geq 2$, what is the maximum value of $M$?
1
vote
1answer
203 views

Approximation of sets

Is the following true? For every $\varepsilon>0$ there is a finite subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that $$|p_1(W)\cap p_2(W)\cap \{p_1(x)+p_2(x):x\in W\}\cap \{...
2
votes
1answer
156 views

Restriction to the diagonal of Hilbert eigenforms

Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?
12
votes
3answers
606 views

2-torsion in class groups of cubic fields

I was wondering if there are good bounds for the $p$-parts of the class group of a number field $F$ in terms of its discriminant $D_F$. More precisely, the bound for the order of the full class group ...
2
votes
0answers
31 views

Embedding of quadratic field orthogonal to a rational element in quaternion algebra

Let $a,b \in \mathbb{Q}$ and $\mathbb{K}$ be a number field. Consider the quaternion algebra $\left(\dfrac{a,b}{\mathbb{Q}}\right)$ and its $\mathbb{K}$-points $\left(\dfrac{a,b}{\mathbb{K}}\right)$. ...
1
vote
1answer
222 views

new proof of Halasz inequality

In Mean values of multiplicative functions over function fields the mention a proof of Halasz inequality in one of their future pre-prints. In fact here is an a proof from 1999. I would like to see a ...
1
vote
0answers
89 views

partial sum of a power series is almost a power [closed]

Using three or more terms of a partial sum within a power series, one gets $$8+16+32+64=11^2 - 1$$ $$4+8+16+32+64=5^3 - 1$$ $$2+4+8+16+32+64=5^3 + 1$$ $$9+27+36+243=19^2 - 1$$ $$64+128+256+512=31^2 - ...
2
votes
0answers
76 views

Relation between the sign of the Stieltjes constants and some zero-free region of $\zeta$

One may recall that the Stieltjes constants $\gamma_{k}$ appear as the scaled coefficients in the regular part of the Laurent series expansion of the Riemann zeta function about $s = 1$: $$ \begin{...
10
votes
0answers
792 views

Recent progress on the verification of Mochizukis proof of the abc conjecture? [closed]

Apparently in preparation for the upcoming workshop on "Interuniversal Teichmüller Theory" in Kyoto in two weeks, which is intended to bring more light into Mochizukis proposed proof of the $abc$ ...
0
votes
2answers
83 views

Dependence on parameters of solvability of a non-linear Diophantine system

For which $(k,t)\in\mathbb Z^2\times\mathbb Z$ does there exist $(v,s)\in\mathbb Z^2\times\mathbb Z$ so that $|v|^2=s^2\neq0$ and $v\cdot k+st=0$? I do not care what the solutions $(v,s)$ are but only ...
2
votes
1answer
315 views

An elliptic curve for Ramanujan-type cubic identities?

Given the roots $x_i$ of the depressed cubic, $$x^3+px+q=0$$ with rational coefficients. It can be shown that, in general, one can find rational $u,v$ such that, $$(u-x_1)^{1/3}+ (u-x_2)^{1/3}+ (u-...
0
votes
0answers
75 views

Prime Ideal Theorem for Real Quadratic Number Rings over Hyperbolic sectors?

As a follow-up to my previous question about variants of Prime Ideal Theorems about Imaginary quadratic number rings found here, I am now asking about the existence of such a companion result for Real ...
2
votes
1answer
172 views

Trace of roots of unity has valuation more than 1 in uramified field

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...
18
votes
1answer
636 views

On the history of the Artin Reciprocity Law

At the beginning of Milne's notes on class field theory, he has a quote by Emil Artin (as recalled by Mattuck in Recountings: Conversations with MIT mathematicians): I will tell you a story about ...
4
votes
0answers
96 views

Is there a natural way to generalize the so-called Bhargava-Shankar height for counting binary forms?

Let $F(x,y) = a_4 x^4 + a_3 x^3 y + a_2 x^2 y^2 + a_1 xy^3 + a_0 y^4$ be a binary quartic form with real coefficients. It is well-known the action of $\operatorname{GL}_2^{(\pm)} (\mathbb{R})$ ...
1
vote
0answers
136 views

Spivakovski-Popescu-Neron desingularisation

For $A \colon= {\Bbb F}_p[[X_1,...,X_d]]$, by generalising Popescu-Neron's method, Spivakovski proved that $A$ is written by smooth sub-algebras. That is, $A \cong \underset{\lambda \in \Lambda}{\...
0
votes
0answers
184 views

Number Theory and p-Power-Partitioned Numbers

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, we're going to define its digits-partition as the next set $D_{N} = \bigcup_{j=1}^{n}\bigcup_{k=1}^{p(a_{j})}\{(P_{k},j)\}$, where each pair $(...
7
votes
0answers
198 views

A variant of the equidistribution of primes in an imaginary quadratic number ring

It is known that the arguments of prime elements of $\mathbb{Z}[i]$ are equidistributed in $(0,2π)$ (by Theorem 5.36 of Iwaniec and Kowalski, or one of Kubilius' papers cited below). This theorem ...
0
votes
1answer
137 views

Minimal number of different values in the sequence $(\mu(d_i)\varphi(d_i))_{i=\overline{{1,\tau(m)}}}$

Let $m=p_1\ldots p_k$ be the prime factorization of some positive integer $m$ and $k\geq 2$. Let $d_1,\ldots,d_{\tau(m)}$ be all divisors of $m$, where $\tau(m)$ counts the number of divisors of $m$....
0
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0answers
147 views

How do these primes jump?

I have several questions regarding the analysis, behaviour, and expression of a simple sieving algorithm which uses associative arrays. The pseudocode below assumes integer addition, string ...
1
vote
0answers
154 views

The number of fixed points of an automorphism of $\mathbb{Z}_m\times\mathbb{Z}_n$

Let $m$ and $n$ be two positive integers such that the groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have no common direct factor. Then an automorphism $f$ of $\mathbb{Z}_m\times\mathbb{Z}_n$ is of type $$\...
8
votes
0answers
284 views

Semisimplicity of Frobenius on *integral* Tate module

Let $K$ be a number field and $A/K$ an Abelian variety; let $l$ be a (rational) prime. Do there exist infinitely many primes $\mathfrak{p}$ of $K$ such that the Frobenius at $\mathfrak{p}$ acts ...
12
votes
1answer
315 views

Rational curves on the Fermat quartic surface

Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many ...
5
votes
2answers
223 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 ...
2
votes
0answers
134 views

When two non-equivalent binary forms primitively represent the same infinite subset of the integers

Let $F(x,y)$ be an irreducible binary form with integer coefficients, degree $d \geq 3$ and content 1. We say that an integer $n$ is primitively represented by $F$ if there exist coprime integers $x$ ...
2
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0answers
73 views

If $N = q^k n^2$ is an odd perfect number, if $n < q^{k+1}$, does it follow that $k > 1$?

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. According to Dickson (as pointed out recently by Beasley), Descartes conjectured $k=1$ in a letter to Mersenne in 1638, with Frenicle'...
2
votes
1answer
107 views

Higher dimensional analogs of logarithmic density

For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by \begin{align*} \underline{d}(A)=\liminf_{N\to\infty} \frac{|A\cap [1,N]|}{N},\\ \bar{d}(A)=\limsup_{N\to\...
2
votes
1answer
172 views

Katz $p$-adic L function and ordinary condition

Let $H$ be a CM field and $F$ be the maximal totally real subfield of $H$. Can we construct a Katz $p$-adic L-functions of Hecke characters without the ordinary condition (i.e every prime of $F$ above ...
0
votes
0answers
127 views

Localisation of the formal power series ring

Let $A \colon= K[[X_1,...,X_d]]$ be a formal power series ring of $d$-variables over a field $K$. Let ${\frak a}$ be a height $r$ prime of $A$ given by ${\frak a} \colon= (f_1,...,f_r)$, where $f_1 ...
0
votes
0answers
55 views

Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...
0
votes
1answer
89 views

Application of the EGZ theorem

Given $r$ numbers $a_1,a_2,...,a_r$ and $n=qP$ where $P$ is the product of these $r$ numbers. $q$ is a natural number such that $q \geq 2$. Also given is a matrix $A$ of the following form: $$A=\...
1
vote
2answers
632 views

Statements going against the grain of Riemann Hypothesis (R.H.) [closed]

Let $M(N) := \sum_{n=1}^N \mu(n)$ It is known that bounding $M(N)$ by $N^{1/2+\epsilon}$ implies R.H. A bound of $M(N)$ by $K\sqrt N$ for say $ K\ge2 $ is also sufficient. $K = 1$ is excluded as ...
7
votes
2answers
374 views

Power sums of p-th roots of unity

The following question was asked by a colleague of mine. For any prime $p$ consider $$ M_p:=\min_{z_1,\dots,z_p}\max_{j,k}\left|z_1^k+\dots+z_j^k\right|,$$ where $z_1,\dots,z_p$ are the complex $p$-th ...
5
votes
1answer
136 views

Stabilizers of pairs of ternary quadratic forms

Let $A,B$ be two ternary quadratic forms with real coefficients, given by symmetric matrices $$\displaystyle 2A = \begin{pmatrix} 2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23}...
0
votes
0answers
375 views

Number Theory and d-Self-Contained Numbers

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...
3
votes
3answers
433 views

sum of four squares with some coefficients

Is there an ordered 4-tuple of rational numbers $(a,b,c,d)$ such that $(b,d)\ne(0,0)$ and $2a^2+3b^2+30c^2+45d^2=2$? The former (deleted) question was just about cases $(a,b,c,d)\ne(1,0,0,0)$ but it ...
2
votes
0answers
108 views

On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers [closed]

(Note: This question has been cross-posted to MSE.) Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$. A number $M$ is called almost perfect if $\sigma(M) = 2M -...
13
votes
3answers
625 views

Collatz-like properties of finite fields

I was wondering what an equivalent of the Collatz conjecture might be for finite fields. In a Collatz sequence a number is moved down within a set $\{2^k n : k \in \mathbb{Z}^* \}$ for some odd $n$ or ...
2
votes
1answer
82 views

An inequality on partitions into distinct bounded parts

Let $P(n,m)$ denote the set of all positive integer partitions of $n$ into parts that are pairwise distinct and bounded by $m$. Let $p(n,m) = |P(n,m)|$. After some numerical experiments it appears $...
0
votes
0answers
62 views

Normalizing factor in Reciprocity of traces of Frobenius with solutions of equations mod p

One kind of Reciprocity tells us that we can count solutions to polynomial equations over finite fields and relate them to traces of Galois Representations with some normalization factor. For ...
1
vote
1answer
75 views

Parametric solutions to a system of equations

Let $s,t$ be two independent real parameters, and let $a_2(s,t), a_1(s,t), a_0(s,t)$ be linear forms in $s, t$ with real coefficients. Put $a_4 = s, a_3 = t$ and consider the quadratic form $$\...
1
vote
0answers
95 views

Relative Leopoldt defect

Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$. Is there a bound of the Leopoldt defect of $M$ ...
1
vote
0answers
77 views

The Linnik problem for dimension $2$

For $N$ an integer, let $$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$ For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...
2
votes
0answers
203 views

On the size of residue class

Let $n \in \mathbb{N}$ be a odd number. Let $S \subseteq \{1,3,5,7,...,n-2,n\}$ and $|S|$ is even number. Let $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...