**12**

votes

**1**answer

461 views

### A combinatorial identity involving harmonic numbers

The harmonic numbers are given by $$H_n=\sum_{k=1}^n\frac{1}{k}.$$
Numerical calculation suggests
$$
\sum_{k=1}^{n}(-1)^k{n\choose k}{n+k\choose k}\sum_{i=1}^{k}\frac{1}{n+i}=(-1)^nH_n.
$$
I can not ...

**2**

votes

**1**answer

137 views

### Want more details about the image of a Maass form in the AIM press release concerning LMFDB

Actually I came upon this through MO a couple of days ago: in here
(http://aimath.org/aimnews/lmfdb/) there is a mesmerizing image
The caption reads
A Maass form, one of the 20 different types ...

**1**

vote

**1**answer

105 views

### Linear functions

Let $(f_1, f_2, \ldots, f_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative ...

**1**

vote

**2**answers

189 views

### Implicit constant in Tenenbaum's result

In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5):
Let $g$ be a positive multiplicative function and ...

**15**

votes

**1**answer

332 views

### Combinatorics problem about sum of natural numbers

Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6)
Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each ...

**1**

vote

**0**answers

103 views

### Ramanujan sum type

I try to show
$$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...

**2**

votes

**1**answer

89 views

### On attempting a proof for $r > 1$, if $M = {2^r}{b^2}$ is an even almost perfect number which is not a power of two

(Preamble: I first thought that this question might be more appropriate for MSE. However, I posted it here nonetheless in the hope that someone with that brilliant idea can help with answering my ...

**6**

votes

**0**answers

117 views

### Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?

There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$.
We usually call it $\mathbb{C}$, but by this we impose a ...

**1**

vote

**1**answer

248 views

### Is this function $\mathbb{Q}$ periodic?

Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ?
$$F(x)= \sum\limits_{q =1}^{\infty} \; \sum\limits_{n =1}^{\infty} \; \sum\...

**9**

votes

**2**answers

305 views

### Neron models and ramification

I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me:
let $E$ be ...

**1**

vote

**1**answer

334 views

### Integer Polynomial solutions to functional equation

Recently I came across a functional equation which always has a polynomial with integer coefficients solution.
Let
$$
L_n(x)=(2 x+1)^2f(x+1)-4x(x+n+1)f(x)-((2 n+1)!!)^2\prod_{i=1}^n(x+i).
$$
Problem:
...

**5**

votes

**2**answers

840 views

### The number of prime factors of a natural number

It seems to be true that OEIS sequences A001222 and A257091 are closely related. First one is the number of prime divisors of n counted with multiplicity. The second one is the logarithms to the base ...

**3**

votes

**1**answer

185 views

### Connected-étale sequence for ordinary CM elliptic curves

Let $E/k$ be an elliptic curve over algebraically closed field of characteristic $p$ with CM, for simplicity, by the maximal order of a quadratic imaginary field $K/\mathbb{Q}$.
Suppose that $p$ is ...

**3**

votes

**0**answers

60 views

### Can we express the degree 10 and degree 15 Galois resolvents of sextic binary forms in terms of its basic invariants?

Let $V_6$ denote the 7 dimensional $\mathbb{C}$-vector space of binary sextic forms. For $T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \operatorname{GL}_2(\mathbb{C})$, $T$ ...

**1**

vote

**2**answers

283 views

### Is the singular integral that come up in circle method independnet of the representatin of the equations?

Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial.
For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$
we have the ...

**1**

vote

**0**answers

115 views

### Class field theory, Ideles class

Let $H$ be a totally complex Galois extension of $\mathbb{Q}$ and $g:G_H \rightarrow \bar{\mathbb{Q}}_p$ be a continuous morphism. By class field thoery we have $\mathrm{Hom}(G_H, \bar{\mathbb{Q}}_p)\...

**10**

votes

**1**answer

235 views

### On $\eta(6z)\eta(18z)$ and the splitting / modularity of $x^3 - 2$

Consider one of the simplest non-abelian examples of modularity. Let $$\eta(6z)\eta(18z) = q\prod_{n=1}^\infty (1 - q^{6n})(1 - q^{18n}) = q - q^7 - q^{13} -q^{19} + q^{25} + 2q^{31} - q^{37} + 2q^{43}...

**1**

vote

**1**answer

158 views

### A number theory problem related to graph spectra?

Let $p$ be a prime number. Is there any $n$ such that $2^n-1=p^2$?
In general is there any n such that $2^n-1=p^k$ for some integer $k$? If there isn't, Is there any proof for that?

**5**

votes

**1**answer

126 views

### Showing the positivity of a singular integral that came up in circle method

Let $F(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous form. Let
$$
I(\alpha) = \int_{[0,1]^n} e^{2 \pi i F(\mathbf{x}) \alpha} dx_1...dx_n.
$$
Then the singular integral is ...

**7**

votes

**1**answer

162 views

### Do arbitrary $K$-twin primes exist?

Polignac's conjecture states that for any positive even integer $K$, there exist infinitely many pairs of primes such that their difference is $K$.
I am interested the status in a much weaker form of ...

**4**

votes

**3**answers

689 views

### Hypersurface missing just one point

Let $\mathbb F_q$ be a finite field and $n$ an integer.
What is the minimal degree $d = d(q,n)$ of a polynomial $f \in \mathbb F_q[X_1,\dots,X_n]$ such that the set $Z(f)$ of zeros of $f$ in the ...

**7**

votes

**2**answers

187 views

### Obstruction to rationality of del Pezzo surfaces of degree 4

Let $X$ be a del Pezzo surface over a number field $k$. (A del Pezzo surface over $k$ is a smooth, projective, geometrically connected surface whose anti-canonical class $K_X$ is ample.) Let $d := K_X^...

**16**

votes

**1**answer

1k views

### Distinctness of products of Fibonacci numbers

Suppose that $S$ and $T$ are sets of Fibonacci numbers greater than $1$. Let $S^*$ be the product of numbers in $S$, and likewise for $T^*$. If $S^*=T^*$, must $S=T$?

**5**

votes

**2**answers

380 views

### BSD and generalisation of Gross-Zagier formula

The classical Gross-Zagier formula and the modularity theorem leads to a proof of half-BSD (i.e. an inequality and not equality) for elliptic curves of analytic rank 0.
The Gross-Zagier formula gives ...

**0**

votes

**0**answers

106 views

### Simple congruence problem

In a paper the author makes a statement which, I think, can only be true if the following holds for all $a,b,m\in\mathbb N$:
If $ab\equiv 1\mod(m^2\ )$, then $a\equiv b\mod (m)$.
Can somebody tell ...

**3**

votes

**1**answer

196 views

### Non-vanishing of L-function of modular form

There is a theorem by Langlands and Shalika (link) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be ...

**2**

votes

**3**answers

343 views

### Primes in arithmetic progressions in number fields

My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of ...

**4**

votes

**0**answers

119 views

### Special fibre of Shimura curves over totally real number fields

The works of Katz-Mazur and Deligne-Rapoport provide a description of an integral model of classical modular curves like X(N), X_0(N) and various variants of X_1(N). In particular, if p is a prime ...

**9**

votes

**2**answers

544 views

### Number of polynomials whose Galois group is a subgroup of the alternating group

Let $f = x^n + a_{n-1}x^n + \cdots + a_0$ be a monic polynomial of degree $n \geq 2$ with integer coefficients. By $\text{Gal}(f)$ we mean the Galois group over $\mathbb{Q}$ of the Galois closure of $...

**10**

votes

**0**answers

705 views

### Is it worth the efforts to read books/papers written in Weil's algebraic geometry language

There is much important work written in Weil's language of algebraic geometry rather than schemes (besides Weil himself, I can think of Shimura, Neron immediately).
My question is: is it worth the ...

**2**

votes

**0**answers

76 views

### Iwasawa lambda Invariant for CM type fields

In the following I will follow the notations as in Chapter $11$, section $3$ of the book 'Cohomology of number fields' by Neukirch and others. Let $k_{\infty}$ be a $\mathbb{Z}_p$-extension of a ...

**2**

votes

**1**answer

112 views

### On the construction of a certain sequence of integers

Let $\alpha \in \mathbb{R}$ be a fixed (positive) number. For each $k \in \mathbb{N}$ we choose $\varepsilon_k >0$ with the property that $\lim_k \varepsilon_k =0$.
If $\alpha \in \mathbb{R} \...

**4**

votes

**2**answers

713 views

### Divergent Series as a topic of research

About a year ago, while studying real analysis, I got very much interested in divergent series. I discussed possible research topics related to divergent series with my teachers but couldn't find any. ...

**10**

votes

**1**answer

456 views

### What is the motivation behind Ramanujan's conjecture?

One motivation I have seen given for Ramanujan's conjecture for the estimate
$$ |a_p|< C p^{k - \frac{1}{2}} $$
for the Fourier coefficients of a cusp form of weight $2k$ is that it allows one to ...

**2**

votes

**0**answers

115 views

### $f(x)$-th largest number of prime factors

Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...

**1**

vote

**0**answers

135 views

### On the cardinality of the set of right-truncatable primes

We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...

**7**

votes

**1**answer

227 views

### Integral representation of adjoint L-factor for GL(2)

My question is about a local computation in the paper of Gelbart and Jacquet, "A relation between automorphic representations of GL2 and GL3", from 1978.
Let $\sigma$ be an irreducible smooth complex ...

**5**

votes

**0**answers

105 views

### Choice of digits for extensions of $\mathbb{Q}_p$

I am interested in writing (in base $p$) elements of the maximal unramified extension $\mathbb{Q}_p^{\mathrm{unr}}$ of $\mathbb{Q}_p$, or (its completion) the field $\mathrm{W}(\mathbb{F}_p^{\mathrm{...

**0**

votes

**1**answer

193 views

### Approximation of real numbers [closed]

Is there any function $f(x)$, such that for all real $\alpha$ and rational $p/q$
$$\left|\alpha-\frac{p}{q}\right|>\frac{1}{f(q)}.$$
or at least
$$\left|\alpha-\frac{p}{q}\right|\leq\frac{1}{f(q)}$$...

**1**

vote

**0**answers

79 views

### Why is the kernel of an algebraic Hecke character open in the ideles?

I've been reading about algebraic Hecke characters, and how one obtains one dimensional $p$-adic representations from them. I have a question about why the kernel of a map defined on the ideles is ...

**23**

votes

**0**answers

264 views

### Does every ring of integers sit inside a monogenic ring of integers?

Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has $\mathcal{...

**3**

votes

**1**answer

89 views

### Distribution of decomposition types of primes in non-Galois extensions of number fields

Let $L/K$ be an extension of number fields. If $p$ is a prime of $K$ that is unramified in $L/K$ and $(f_1, \dots, f_r)$ is a partition of $n = [L:K]$, say that $p$ has "decomposition type" $(f_1, \...

**5**

votes

**2**answers

612 views

### Langlands program vs Shimura-Taniyama-Weil conjecture

Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...

**1**

vote

**0**answers

147 views

### What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$? [closed]

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...

**0**

votes

**0**answers

149 views

### $z^n-t^m=x^3+y^3$ and Vojta's more general abc conjecture

In A more general abc conjecture, p. 7 Paul Vojta conjectures:
If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$
$$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C \prod_{...

**5**

votes

**1**answer

242 views

### Average size of $p$-part of the Tate-Shafarevich group for elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$. For a given prime $p$, the $p$-Selmer group $\operatorname{Sel}_p(E)$ of $E$ and the $p$-part of the Tate-Shafarevich $Ш_E[p]$ group ...

**2**

votes

**1**answer

160 views

### The exponential Diophantine equation $a^n-b^m=x^3+y^3$ for arbitrary large $n,m$

Let $a,b$ be coprime integers, neither a perfect power, $n,m$ naturals and $x,y$ integers.
Consider the exponential Diophantine equation
$$ a^n-b^m=x^3+y^3 \qquad (*) $$
Nontrivial solution ...

**3**

votes

**1**answer

146 views

### 4-th order diophantine equation

I met in many places the equation
$(a^4-b^4)(c^4-d^4)=\square$
It is well known that this was investigated by Euler.
But I was unable to find the general solution of this equation. Could you please ...

**1**

vote

**1**answer

118 views

### Is it possible to have an even superperfect number and an odd superperfect number whose product is an almost perfect number?

A number $n \in \mathbb{N}$ is said to be superperfect if
$$\sigma(\sigma(n)) = 2n.$$
A number $m \in \mathbb{N}$ is said to be almost perfect if $$\sigma(m) = 2m - 1.$$
Here is my question:
Is ...

**1**

vote

**1**answer

95 views

### Formula for negative polylogarithms

Theorem. We have that $\displaystyle \underbrace{x\left(\dfrac{d}{dx}\left(\cdots x \left(\dfrac{d}{dx} \left( \dfrac{x}{1-x}\right)\right)\cdots\right)\right)}_{\text{$x \frac{d}{dx}$ $m$ times}}=\...