# Tagged Questions

**2**

votes

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

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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**

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**0**answers

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

**1**answer

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**

votes

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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

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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**

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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

**1**answer

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

**2**answers

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

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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 $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...