Tagged Questions

6
votes
6answers
513 views

Sequences equidistributed modulo 1

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results. H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidist …
0
votes
0answers
64 views

Zeroes of a homogeneous function

I am interested in the zero-set of a homogeneous function $f(x_1, \cdots, x_n)$, where $f$ is not necessarily a polynomial. In particular, I would like to know if there are any gen …
0
votes
0answers
108 views

What is most current greatest lower bound on gaps between P2 almost primes

What is the current best result on the greatest lower bound on gaps between P2 almost primes where P2 represents a prime or the product of two semi-primes?
0
votes
0answers
76 views

Bounding number of solutions to an equation:

I have an equation that I think should not have too many solutions, but I don't see a way to argue this. Given $a, b, c, N \in \mathbb{N}$, how many positive integer solutions $x, …
2
votes
1answer
72 views

A question about the second Chebyshev function $\psi(x) = \sum_{m=1}^{\infty}\vartheta(\sqrt[m]{x})$

Using a simple java application, I have noticed that for $x > 25$: $$\psi\left(\frac{x}{5}\right) \ge \psi\left(\frac{x}{3}\right) - \psi\left(\frac{x}{4}\right)$$ where: $$\psi …
3
votes
1answer
157 views

short character sums averaged on the character

Let $a$ be an integer, $p$ a prime (much) greater than $a$, and $\chi$ a Dirichlet character. There is an abundant literature on the sums $$S(\chi,a)=\sum_{i=1}^a \chi(i),$$ called …
3
votes
1answer
197 views

Least non primitive root

There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more gener …
4
votes
3answers
229 views

A divergent series related to the number of divisors of of p-1

Let $d(n)$ denote the number of divisors of $n$. Is it known that the series $$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$ diverges? This would follow immediately from the Sophie Ge …
1
vote
0answers
49 views

estimate for i-th smooth number, gap between consecutive smooth numbers

Does anyone know of the best estimates for $n_i$ and $n_{i+1}-n_i$ where $n_i$ is the $i-$th $y-$smooth number? The best I could find was Tijdemann's estimate for the gap in terms …
13
votes
3answers
728 views

Good uses of Siegel zeros?

The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "expectional zeros" …
1
vote
0answers
130 views

Convergent series, asymptotics and truncation

In regard to the characteristics of certain "explicit formulae" arising in number theory, I am pondering the connection between the rate of convergence of series and the asymptotic …
1
vote
1answer
166 views

Bounds for the largest divisor of n less than n^0.5

Let $d(n)$ denote the largest divisor of $n$ less than $\sqrt{n}$. Are there good lower bounds for $d$ that hold for almost all natural numbers? More precisely, is there a functi …
1
vote
1answer
85 views

What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$

I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$. In the paper, he uses the following bounds for the second …
4
votes
1answer
240 views

What analytic tools can provide a lower bound for this Diophantine equation?

The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken. I felt …
6
votes
3answers
263 views

Jacobi sums on tori

The Jacobi sum of $n$ multiplicative character $\chi_1,\dots,\chi_n$ on a finite field $\mathbb F_q$ is defined as $$J(\chi_1,\dots,\chi_n) = \sum_{x_1,\dots,x_n \in \mathbb F_q, …

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