All Questions
23 questions with no upvoted or accepted answers
8
votes
0
answers
271
views
Restricted divisor summatory function
I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where
$$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$
and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
- 361
8
votes
0
answers
643
views
Divisor problem: find the fallacy!
The following approach to the divisor problem (that is, the problem of estimating $D(x) = \sum_{n\leq x} d(n)$, where $d(n)$ is the number of divisors of $n$; more precisely, we are meant to bound the ...
- 20.2k
6
votes
0
answers
201
views
Smooth integers with lower bound on $\omega(n)$
Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
- 1,826
5
votes
0
answers
233
views
What is known about the mode of the number of divisors $\le x$?
Let $d(x)$ be the divisor function. Let $M(x)$ ($x$ a positive integer) be the most frequent value of $d(y)$ for $1 \le y \le x$. Is an asymptotic known for $M(x)$, and failing that, can $M(x)$ at ...
- 251
4
votes
0
answers
86
views
On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
4
votes
0
answers
413
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
3
votes
0
answers
76
views
Divisor of given order in short intervals
Is the following Open question or Conjecture already known, or eventually settled ?
Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
- 433
3
votes
0
answers
179
views
The binary additive divisor problem in arithmetic progressions
I find quite a few results about the binary additive divisor problem, that is evaluating
\[ \sum _{n\leq x}d(n)d(n+h)\]
for certain ranges of $h$.
Are there any known results about the same count ...
- 1,381
3
votes
0
answers
121
views
On consecutive superabundant numbers
Define $\sigma(n)=\sum_{d\mid n} d$. A number $n>1$ is said to be superabundant (SA) if it is an integer and $\frac{\sigma(n)}{n}>\frac{\sigma(s)}{s}$ for every positive integer $s<n$. Let $n$...
- 31
3
votes
0
answers
280
views
Magnitude and distribution of largest prime factor?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors.
What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number?
What is ...
- 13.9k
2
votes
0
answers
156
views
Exponential sum of $k$-fold divisor function
Can anyone point me to a reference for the main term when approximating the exponential sum of the 3-fold divisor function? Specifically I want the main term in $$\sum _{n\leq x}d_3(n)e\left (an/q\...
- 1,381
2
votes
0
answers
75
views
Least number of factors $\sigma(p^e)$ of representation of $\sigma(N)$ to get the least multiple of $\operatorname{rad}(N)$, for odd perfect numbers
I've cross-posted this from the post of Mathematics Stack Exchange that I've asked (Apr, 2nd 2020) with title On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\...
2
votes
0
answers
192
views
The multiplicative constant in the estimate for $S_a(x)=\sum_{n\leq x} d(n)^a$
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S_a(x)=\sum_{n\leq x} d(n)^a.
$$
For $a=1,$ the following is well known
$$
S_1(x)=\sum_{n\leq x} d(n)...
- 10.4k
2
votes
0
answers
112
views
Queries on distribution of prime divisors by magnitude?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$.
What is the probability distribution of ...
- 13.9k
1
vote
0
answers
305
views
About inequalities that involve the sum of divisors, the Euler's totient and the aliquot part $\sigma(n)-n$
In this post, for integers $n\geq 1$, I denote the sum of divisors $\sum_{1\leq d\mid n}d$ as $\sigma(n)$ and the Euler's totient function as $\varphi(n)$. It's easy to check* that if we assume that $...
1
vote
0
answers
106
views
Lower bound on a Truncated Divisor Sum
Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum_{k\mid n} 1$ of the positive integer $n$.
I am interested in estimating, the following sum
$$
A(a,x)=\sum_{n\leq x} \min[ d(n), M]^a
$$
...
- 10.4k
1
vote
0
answers
222
views
Attempt of exploit the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$ in the context of even perfect numbers, and a related conjecture
It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...
1
vote
0
answers
101
views
Size of a set defined by divisor function
After some computations, I guessed the following conjecture.
How can I prove or disprove it? thanks!
Let
$$
A(k)=\#\left\{\left(t,\frac{k+t+a}{4t-1}\right):1\leq t\leq k,\ 1\leq a\leq k+t,\ a\mid(k+...
- 841
1
vote
0
answers
141
views
On certain number theoretic sextuples?
Given small parameters $0<\epsilon<\epsilon'$ is there an $n_\epsilon>0$ such that at every $n>n_\epsilon$ if we are given a prime $n^2<p<2n^2$ then can we always find integers $a,b,...
- 13.9k
1
vote
0
answers
90
views
An arithmetic function involving arbitrary (but fixed) number of divisors
I need at least basic information about generating functions of the following class of arithmetic functions, grouped by levels $k$.
Fix some $k$ and some family $\varepsilon_*=(\varepsilon_\sigma)_{\...
- 18.4k
0
votes
0
answers
152
views
On the $\mathsf{LCM}$ of a set of integers defined by moduli of powers
For integers $a,b,t$ define $$\mathcal R_t(a,b)=\{q\in\mathbb Z\cap[1,\min(a^t,b^t)]: a^t\equiv b^t\bmod q\}$$ and $\mathsf{LCM}(\mathcal R_t(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal ...
- 1,826
0
votes
0
answers
89
views
Partial sums involving Gregory coefficients that cannot be an integer
For integers $n\geq 1$ let $G_n$ be the Gregory coefficients or reciprocal logarithmic numbers, see the Wikipedia [Gregory coefficients.]
(https://en.wikipedia.org/wiki/Gregory_coefficients)
$${z\...
0
votes
0
answers
47
views
Approximation of $\sum_{\substack{n\geq 1\\n\text{ is abundant}}}\frac{\sigma(n)}{n^3}$, where $\sigma(n)$ denotes the sum of divisors function
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors function then, from the theory of Dirichlet series, it is well-known the value of $$\sum_{n=1}^\infty\frac{\sigma(n)}{n^3},$$
in terms of ...