Tagged Questions

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

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Rational points and finite étale covers

This question is broadly about non-trivial examples where a map $f:Y\to X$ of smooth projective $k$-varieties ($k$ not algebraically closed) is such that existence of $k$-points (or in fact ...
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How does this small change in the Pollard Rho method affect its complexity?

In finding the smaller factor $p$ of an input number $n$, the Pollard Rho method takes time bounded by a function in $O(\sqrt{p})$. (Did I get that right?) Now let's say I tweak the method just a ...
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Parallel algorithm for modular multiplication of polynomials over Z/nZ

Is there a parallel algorithm for doing modular multiplication of polynomials over Z/nZ? n is a very large number (for hundreds and thousands of bits). Normally, the method used is binary ...
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Examples when one can use the the symmetric power $L$-functions to study topics related to the number theory

"The symmetric power $L$-functions are a powerful tool for studying algebraic or geometric objects through analytic methods." I read this sentence in the introduction of a Master thesis. I want to ...
218 views

Average digit sum in different bases

Given two natural numbers $n\geq 1$ and $b\geq 2$, denote by $S_b(n)$ the sum of the digit of $n$ in its representation in base $b$. Clearly $S_b(n)$ varies from 1 (when $n$ is a power of $b$) to ...
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Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$. I'll start with a somewhat vague question and make my question more specific further down: How do ...
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Smooth complete intersections and sharpness of the Chevalley-Warning theorem

Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d_1,\ldots,d_r)$. If we're working over a finite field $\mathbb{F}_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the ...
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Langlands-Shahidi method carried out in the simplest case?

I would like to read the simplest examples of Langlands-Shahidi method carried out to prove the functional equation of $L$-function. Could the constant term of GL(2)-Eisenstein series to prove any ...
139 views

Young-Fibonacci lattice and purely periodic continued fractions

The Fibonacci lattice $\mathcal{F}$ is the poset of all finite words consisting of 1's and 2's where a word $v$ covers a word $u$ if $v$ is obtained from $v$ by either (a) inserting a 1 in $u$ prior ...
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Question on some coverings of the euclidean space

Edit : no answer, no comment ... let's try with a chocolate bar. Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form ...
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The density of square-free integers represented by a cubic polynomial

Suppose that $f(x)$ is an irreducible cubic polynomial with integral coefficients. Suppose further that for all primes $p$, there exists an integer $n_p$ for which $p^2 \nmid f(n_p)$. Then it is a ...
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Avoiding multiples of $p$

Let $p$ be a prime number and $P=\{1,2,...,p-1\}$ In how many ways we can sum all the elements of $P$ in such a way that we will reach a multiple of $p$ only when we sum the last summand? For ...
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Improvement on $\phi\sigma$ bound [migrated]

We have: $$\dfrac{6}{\pi^2}\lt\dfrac{\phi(n)\sigma(n)}{n^2}\le1$$ with equality iff $n=1$. Wikipedia Are there any known improvements on these bounds?
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Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction. Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...
137 views

Kloosterman sum

Does anybody know the non-trivial bound for this sum? $S(m,n,c,q)=\sum_{a,b\in \mathbb{Z}/cq\mathbb{Z}, \;ad\equiv 1\text{ mod }c} e^{2\pi i(am+nd)/qc},$ where $m,n\in\mathbb{Z},\;q,c\in\mathbb{N}$. ...
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Factorial Sums over Compositions or Unlabeled Permutations"

Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$ In a divergent sum, the sequence $$a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i!$$ frequently shows up and one ...
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Formal generic fibre and Fermat's Last Theorem

Set $A_{n} \colon= {\Bbb F}_p[[S_1,...,S_n]]$ and $A_{n,d} \colon= {\Bbb F}_p[[S_1,...,S_n]][[X_1,...,X_d]]$ be a $d$-variables formal power series ring over $A_n$. We denote by $K$ the fractional ...
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Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?

http://mathworld.wolfram.com/ChoquetTheory.html Is the claim in the link true? Here's the reference given there: https://www.renyi.hu/~p_erdos/1934-01.pdf Erdős proved that there exist at least ...
Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the American Mathematical Society. 79 (1973), no. 4.), ...