# Tagged Questions

**10**

votes

**1**answer

594 views

### The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity

What is the origin of the Ramanujan's approximate identity $$\pi^4\approx 2143/22,\;\;\tag 1$$ which is valid with $10^{-9}$ relative accuracy? For comparison, the relative accuracy of the well known ...

**7**

votes

**3**answers

434 views

### Which groups are Galois over some p-adic field?

Suppose I have some finite $p$-group $G$, or a little extension of it.
How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...

**29**

votes

**2**answers

3k views

### Have there been any updates on Mochizuki's proposed proof of the abc conjecture?

In April 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known ...

**29**

votes

**3**answers

878 views

### Unexpected applications of transcendental number theory?

In the last pages of "Equations Différentielles à points singuliers réguliers", Deligne provides a proof, attributed to Brieskorn, of the so-called local monodromy theorem (on the quasi-unipotence of ...

**1**

vote

**0**answers

64 views

### Complex Structure Moduli of Elliptic Fibrations

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.
I know how it is done for the generic Weierstrass ...

**3**

votes

**1**answer

127 views

### Counting lattice points inside an ellipsoid subject to congruence conditions

Let $A,B,C$ be positive integers, and let $Z$ be a positive parameter. Let $M$ be a positive integer, and consider the set of points
$$\displaystyle \{(x,y,z) \in \mathbb{Z}^3 : Ax^2 + By^2 + Cz^2 ...

**2**

votes

**0**answers

203 views

### Identity with Ramanujan's generalized continued fraction

Let $F(x,q)=\sum_{n\geq 0}x^n\dfrac{q^{n^2}}{(q)_n}$, where $(q)_n=(1-q)(1-q^2)\dots(1-q^n)$. Then:
$$H(x,q)=\frac{F(-xq,q)}{F(-x,q)}=\dfrac{1}{1-\dfrac{qx}{1-\dfrac{q^2x}{1-\dots}}}$$ is the ...

**2**

votes

**2**answers

168 views

### A lower bound involving the divisor function and primorial numbers

It is known that
$\lim$ $\sup \dfrac{\sigma(N_k)}{e^{\gamma}N_k \log\log N_k}=\frac{6}{\pi^2}$,
where $\gamma$ is the Euler-Mascheroni constant and $N_k$ is the $k-th$ primorial number.
But is it ...

**1**

vote

**0**answers

152 views

### Number of minimal primes for UFD

Let $R$ be a UFD which is NOT noetherian. It is well-known that $R$
is a Krull ring. Let $I$ be an ideal of $R$ such that the height of $I$
is $d$ which is finite.
Question. Is the number of minimal ...

**5**

votes

**0**answers

154 views

### Fermat's Little Theorem in function fields

There is a well-known generalisation of Fermat's Little Theorem as for any $a\in\mathbb{Z}$ and $n\in\mathbb{N}$ we have $$\sum_{d\mid n} \mu(n/d) a^d\equiv 0\pmod n.$$ where $\mu$ is the Möbius ...

**5**

votes

**2**answers

265 views

### Galois representations along eigenvarieties

This question is about the status of the following.
Meta-hypothesis. Let $X$ be an irreducible component of an eigenvariety. Then there exist: (a) a pseudo-representation/character $\psi$ along $X$, ...

**17**

votes

**1**answer

342 views

### Convergence of zeta functions for schemes of finite type over the integers

In his lecture "Zeta functions and $L$-functions", Serre presents a very elegant proof of the convergence of the zeta function
$ \zeta (X,s) = \prod_{x \in |X|} (1- N(x)^{-s})^{-1}$ in the half plane ...

**4**

votes

**3**answers

339 views

### Integral quaternary forms and theta functions

The following question arises when I attempt to understand the modular parameterization of the elliptic curve $$E:y^2-y=x^3-x$$
In Mazur-Swinnerton-Dyer and Zagier's construction, a theta function ...

**4**

votes

**1**answer

165 views

### Computing coefficients for the slash operator of a modular form

Suppose $f$ is a classical modular form of weight $r$ for a (congruence) group $\Gamma$. Let $\gamma$ be any matrix in $\operatorname{SL}_2(\mathbb{Z})$. Then the slash operator $|_\gamma$ is usually ...

**2**

votes

**0**answers

58 views

### Normaliser of image of induced representation

I would like to compute the normaliser of the image of an induced representation. So it would be a representation from $GL_{2}(\mathbb{Z}_{p})$ to $GL_{n}(\bar{\mathbb{Z}}_{p})$. Is this maybe some ...

**16**

votes

**1**answer

705 views

### A converse of the abc conjecture?

Let ${\rm rad}(n)$ denote the radical of a positive
integer $n$, i.e. the product of its distinct prime divisors.
Given positive integers $a$ and $b$, the triple $(a,b,a+b)$ is
called an abc triple if ...

**0**

votes

**1**answer

100 views

### What is the relative size of the radical of an ABC-triple relative to the number of primes up to its largest element?

Write $\bf N$ for the set of natural numbers, and $P$ for the set of primes. For $x$ in $\bf N$ let $p(x)$ be the product of the primes dividing $x$ (that is, the "radical" of $x$). Also write $\#(x)$ ...

**8**

votes

**0**answers

133 views

### Positivity of coefficients of Ehrhart polynomial of n-Tetrahedron

A set of positive integers $d_1, \dots, d_n$ describe two n-dimensional closed lattice tetrahedron:
$$ T = \left\{ (x_1, \dots, x_n) \in \mathbb{R}^n: \sum_{i=1}^n \frac{x_i}{d_i} \leq 1 \textrm{ and ...

**1**

vote

**1**answer

92 views

### Does positive relative density imply asymptotic additive basis behaviour?

First definitions: let $A, B \ \subset \mathbb{Z_{>0}}$ and $1\in A, 1\in B$. We define the relative density of $A$ with respect to $B$ to be $$rel(A, B) = \inf_n \frac{|A \cap [1,n]|}{| B \cap ...

**0**

votes

**1**answer

170 views

### Is it consistent with Cramer's conjecture to conjecture that $x/\pi_{2}(x)>2B_{2}/6\times\log^{2}x$?

Brun's constant $B_{2}$ is defined as $B_{2}=1/3+1/5+1/5+1/7+1/11+1/13+...$ where the sum is taken on $p$ such that $p$ is an element of a couple of twin primes. The number of twin primes below is ...

**2**

votes

**0**answers

54 views

### When is Coppersmith method polynomial? (Factorization related)

From pari's implementation of Coppersmith method
zncoppersmith(P, N, X, {B=N}): finds all integers $x$ with $|x| \le X$ such that
$\gcd(N, P(x)) \ge B$. $X$ should be smaller than
...

**5**

votes

**1**answer

305 views

### Unique quadratic subextension of a ray class field

Let $K_q$ denote the unique quadratic subextension of the ray class field over $\mathbb{Q}$ of conductor $q\times\infty$. Then $K_q$ should be $\mathbb{Q}(\sqrt{q})$ if $q$ if 1 mod 4 and ...

**1**

vote

**0**answers

130 views

### A weighted ergodic average

According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ ...

**1**

vote

**1**answer

121 views

### Antiholomorphic cusp forms of negative weight

Let $k\geq 2$ be an even integer and let $\Gamma=\Gamma_0(N)$. Let $f\in S_k(\Gamma)$. To $f$, one may associate an antiholomorphic cusp form of weight $k$ and level $\Gamma$ by defining ...

**0**

votes

**0**answers

56 views

### the shifted convolution sums and the sub convexity problem for l functions

in the paper of gergely harcos, an additive problem in the fourier coefficients of cusp forms, a bound for the shifted convolution sums for hecke eigenvalues was explicited and i thought that his ...

**3**

votes

**1**answer

260 views

### Simplest PA theorem whose proof requires encoding of sequences even though the statement itself doesn't

What is the simplest number-theoretic theorem whose proof requires exponentiation or finite sequences/sets (so any proof in Peano Arithmetic would need to use encodings of such things using e.g. ...

**9**

votes

**1**answer

280 views

### Largeness and arithmetic progression properties of generic reals

Consider the following properties for a subset $A$ of $\mathbb{N}$:
(1) $A$ is large: $\sum_{n \in A}$$ 1\over n$$=\infty,$
(2) $A^\infty=\limsup \frac{|A \cap \{ 1, \dots, n\}|}{n} >0$,
(3) ...

**5**

votes

**1**answer

186 views

### Is there a primitive recursive real number which cannot have a primitive recursive expansion?

Using the definitions given below, my question can be restated as
Does there exist a primitive recursive (PR) real $\{s_n\}$ such that for every scale $r \geq 2$ and every PR sequence $a_n$ with ...

**1**

vote

**0**answers

61 views

### Counting power-free values inside in bounded regions in $\mathbb{R}^3$

I apologize if this question is simple and the answer is obvious, but I was not able to find a reference that could be used immediately.
Let $k_1, k_2, k_3$ be three integers which are each at least ...

**5**

votes

**1**answer

157 views

### Levelt-Turrittin Theorem over p-adics (or the monodromy theorem)

Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying
$$
...

**6**

votes

**1**answer

124 views

### An explicit description of neighborhoods of the rank 2 boundary in the Satake Compactification of $\mathbf{A}_2$

My Motivation: I'm having a hard time following the description of the topology in the Satake Compactification of locally symmetric spaces. The group theory is something I'm finding a bit tricky to ...

**1**

vote

**0**answers

87 views

### On exponential sum weighted with von-Mangoldt function

Suppose we have $\alpha \in \mathbb{R}$ such that $|\alpha - a/q| < 1/q^2$,
where $(a,q)=1$. Then we know that the exponential sum
$$
S(\alpha) = \sum_{1 \leq n \leq X} \Lambda(n) e(n \alpha)
$$
...

**12**

votes

**1**answer

323 views

### Is Faltings' $p$-adic Eichler-Shimura isomorphism the $p$-adic comparison isomorphism?

This is a question about Faltings' $p$-adic Eichler-Shimura isomorphism from his 1987 article "Hodge-Tate structures and Modular Forms".
Let $N\ge5$, $k\ge2$ be integers. Denote by $X(N)$ the proper ...

**4**

votes

**0**answers

165 views

### The density of quartic polynomials whose Galois group is a subgroup of $D_4$

Let $D_4$ denote the (isomorphism class of) dihedral group of 8 elements. For a given binary quartic form $f$ with integer coefficients, define $\text{Gal}(f)$ to be the Galois group of the Galois ...

**3**

votes

**1**answer

125 views

### Question absolutely cuspidal representation

Let F be a local field, and $\pi$ an absolutely cuspidal representation of $GL_2(F)$. Then the Kirillov model of the representation is given by the space of locally constant functions with compact ...

**4**

votes

**0**answers

395 views

### What is the best way to learn about Modular Forms?

I am a senior Mathematics Major, and I am interesting in learning about Modular Forms. I have a layman's general sense of what they are but I was wondering if there is a lecture(I am willing to pay) ...

**15**

votes

**1**answer

787 views

### Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?

Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners ...

**7**

votes

**1**answer

249 views

### Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...

**1**

vote

**0**answers

111 views

### Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes? [closed]

This question follows from the information provided below.
Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...

**22**

votes

**2**answers

777 views

### Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element
$$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$
...

**0**

votes

**0**answers

66 views

### Algebraic operations with memory hardness properties

In cryptography, there are password hash functions like scrypt and argon2 for which the fastest known algorithms employ large ...

**6**

votes

**3**answers

323 views

### Asymptotics for the number of abelian groups of order at most $x.$

The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for ...

**3**

votes

**1**answer

466 views

### Use of infinitude of primes in the Green-Tao theorem [closed]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...

**3**

votes

**1**answer

194 views

### A decreasing sequence involving the divisor function?

Define $N_k \geq 6$ to be the $k-th$ primorial number and let $\sigma(n)$ be the divisor function.
It seems that $u_k = \dfrac{\sigma(N_k)}{N_k \log\log N_k}$ is a decreasing function ?
By ...

**1**

vote

**0**answers

34 views

### Standard term for parametrisation where heights of parameters and values are correlated

Suppose an algebraic variety over Q, or subvariety of one, has a parametrization, also over Q.
Clearly an infinite number of birationally equivalent parametrisations can be obtained from this. But ...

**4**

votes

**1**answer

107 views

### Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...

**2**

votes

**0**answers

90 views

### When is $(1^2+1)(2^2+1)\dots (n^2+1)$ a perfect square? [duplicate]

Find all such $n$. Natural guess is that $n=3$ is the only solution.
It is natural to try something like Bertrand's postulate for Gaussian integers with imaginary part 1, but what is known?

**4**

votes

**1**answer

245 views

### Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is
For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m.
I don't see why ...

**2**

votes

**0**answers

146 views

### Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, ...

**6**

votes

**1**answer

220 views

### Up to $2000$, $A145722(n-1) \equiv \sigma(4n-3) \pmod{5}$

A145722 is
Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.
Using the pari program and offset 0, up to ...