# Tagged Questions

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

175 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 ...
29 views

### On modulus of powers

Given $n\in\Bbb N$ is there an $a_n\in\Bbb N$ such that for every $a>a_n$ there are $b,c\in\Bbb N$ such that $b^i\bmod a,c^i\bmod a\in(\sqrt a,\sqrt a\log a)$ for every $i\in\{1,\dots,n\}$? If so ...
72 views

### The non-abelian Gauss-Manin connection; non-abelian M_dR; a Grothendieck lemma for cyrstals

I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme ...
171 views

### Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...
273 views

### Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational numbers $\ \mathbb Q_{_{>0}},\$, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...
170 views

### Uniformizer for splitting field of p^{1/p^n} over p-adics

Does anyone know an explicit uniformizer for $\mathbf{Q}_p(\zeta_{p^n}, p^{\frac{1}{p^n}}) / \mathbf{Q}_p$? I was reading the question "adding an n-th root to Q_p" where dke mentions this question but ...
225 views

### Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...
53 views

### Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...
325 views

### How big are the prime factors of $2^kp - 1$?

I have already asked this question here. No answers despite the bounty (which has now ended) Let $p$ be a prime number, $p > 3$. Does there always exist $k \in \mathbb N_{\ge 1}$ such that the ...
486 views

### If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark: One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...
34 views

### If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n^2$ solitary?

(Note: A similar question was asked in MSE two months ago.) Let $\sigma(x)$ be the sum of the divisors of the natural number $x$, and denote the abundancy index $\sigma(x)/x$ by $I(x)$. Here is my ...
65 views

### writing an integer as particular summation [on hold]

I think my question is an elementary question. Thanks for any help or comment. Is there any formula for the number of writting a natural number $n$ in a summation as follows, $n=a_1+\dots+a_k$, ...
107 views

### I am looking for a general solution for when $n$ and a rational function $f (n)$ are both integers [on hold]

I am looking for a general solution for when $n$ and a rational function $f \left({n}\right)$ are both integers. One example is below. This seams simple, how to prove that the only integer solutions ...
50 views

### Pairing for non-uniformizable Anderson T-motives

Let $M$ be an Anderson T-motive (the simplest case, i.e. abelian in the meaning of [G] Goss, Basic structures of function field arithmetic, Def. 5.4.12, over $A^1$, having $N=0$), and let $H_1(E)$ ...
165 views

### Weight 12 cusp forms for $\Gamma_0(p)$

Let $S_k$ be the space of weight 12 cusp forms of $\Gamma_0(p)$, ($p$ prime), then Sage tells that $\dim S_k^{\text{new}}=\dim S_k-2$. Thus the old forms spans a 2-dimensional subspace. One of the ...
347 views

### Asymptotics of product of Euler's totient function (A001088)?

Conjecture: \begin{align} \lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...} \end{align} The numerical result from 100000 terms is: My questions ...
310 views
+50

### Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm. In the ...
66 views

### Degree of a rational Function [closed]

This might sound a very trivial question but I found different answers on the web. Assume on has a rational function f(x)/g(x) where f(x) and g(x) are polynomials. What is the degree of the rational ...
152 views

### Where I can find a Carlitz's paper? [closed]

I am looking for a PDF of the paper: Carlitz, Lewis, Mills, Straus - Polynomials over finite fields with minimal value sets. Only I can find about it is this data: Mathematika / Volume 8 / Issue ...
125 views

### Rational valued functions on the Cantor set with $\int_{C} f^{3}d\mu=1$

Let $C$ be the Cantor set as a compact Abelian topological group, isomorphic to countable product of $\mathbb{Z}/2\mathbb{Z}$. Its normalized Haar measure is denoted by $\mu$. Is there a ...
110 views

### Are prime gaps of even index essentially larger than those of odd index?

Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let's introduce the following summatory functions: $$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$ $$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$. Let's ...
128 views

### Computing intersection number of two arithmetic line bundles

I have some questions in Arithmetic Arakelov geometry Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and ...
126 views

### Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and ...
233 views

### If $N = qn^2$ is an odd perfect number, is it possible to have $q + 1 = \sigma(n)$?

The title says it all. Question If $N = qn^2$ is an odd perfect number with Euler prime $q$, is it possible to have $q + 1 = \sigma(n)$? Heuristic From the Descartes spoof, with quasi-Euler ...
55 views

151 views

### On even almost perfect numbers other than the powers of two, as compared to odd perfect numbers given in Eulerian form

This question has been cross-posted from MSE. Let $\sigma(x)$ be the sum of the divisors of $x$. We say that $X$ is almost perfect if $\sigma(X) = 2X - 1$. Antalan and Tagle (in a 2004 preprint ...
72 views

### Simple Diophantine equations for Cartan matrices of Kac-Moody algebras

Let us consider the matrix $A$ defined as follows: $A_{i i} =2$, $A_{i j} = - \frac{2 d_i}{d - 1 - d_{j}}, \qquad i \neq j$; $i, j = 1, \dots, n$. Here $d_1,...,d_n$ are natural numbers, $n > 1$ ...
99 views

### Local character expansion for discrete series representations of $GL_n(F)$

I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field. First, some notation: let $G$ be a ...
81 views

### A connection between basic hypergeometric series and number theory

I am studying functions given by the power series: $$f(z)=1+\sum_{n=1}^{\infty}\frac{z^n}{(1-q)(1-q^2)\cdots(1-q^{n})}.$$ The parameter $q$ is usually assumed to be such that $|q|<1$. Then it is ...
118 views

### Does the Tate pairing agree with the Brauer-Manin pairing

Let $X$ be a proper, smooth, geometrically integral variety over a field $k$. Let $A$ be (the identity component of) its Picard variety and let $B$ be (the identity component) of its Albanese variety. ...