All Questions
8 questions
6
votes
2
answers
685
views
Number of divisors which are at most $n$
I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
- 541
5
votes
0
answers
200
views
A modern reference for the Piltz divisor problem
apparently, the Dirichlet hyperbola method is no longer up-to-date, and instead Voronoi's identity is used in order to establish good bounds on the Dirichlet divisor problem.
The same applies to the ...
- 559
3
votes
1
answer
410
views
Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes
I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ...
3
votes
1
answer
309
views
How to estimate the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
Let $\tau(n)$ be the number of positive divisors of $n\in \mathbb{N}$.
Is it possible to get some good estimate for the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
I know that the sum is $\mathcal O(x^2)$...
- 3,876
5
votes
0
answers
179
views
The total number of divisors of those integers with the most divisors
I'm interested in summing $\tau(m)$, the number of positive divisors of $m$, not over all integers in an interval but rather over only the integers with the most divisors. More specifically:
Given a ...
- 12.8k
7
votes
1
answer
675
views
Short divisor sum
Let $d(n)$ denote the number of positive divisors of the positive integer $n$.
Pick some positive $X,h \in \mathbb{R}$ and consider the sum
$$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$
In view of ...
- 11.3k
5
votes
1
answer
232
views
Improvement of a bound on divisor distributions from "Divisors" (Hall and Tenenbaum)?
In the classic text referred to in the title of this question, the bound
$$
H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x})
$$
is given, where $\delta=1-\frac{...
- 10.4k
4
votes
0
answers
415
views
Maximal order of Hooley's Delta function?
There is a large literature on Hooley's
$$
\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1
$$
giving its normal and average order. What is known of its maximal order?
Clearly $\Delta(n)\le d(n)$ ...
- 9,114