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2 votes
1 answer
162 views

Closed form expression for this zeta-like series involving GCD and LCM

I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$: $$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...
Alexandre's user avatar
  • 634
2 votes
1 answer
202 views

Exponential sums involving smooth truncated divisor functions

Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as $...
user152169's user avatar
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 ...
TheBestMagician's user avatar
7 votes
2 answers
1k views

Convolution sum of divisor functions

Let $\sigma_0(n)$ be the divisor counting function $$\sigma_0(n) = \sum_{d \vert n} 1.$$ I'm interested in the convolution sum $$ S(n) := \sum_{k=1}^{n-1} \sigma_0(k) \sigma_0(n-k)$$ I ran some quick ...
Adithya Chakravarthy's user avatar
9 votes
1 answer
558 views

Is the divisor counting function equidistributed mod $p$?

Let $\sigma_0(n)$ be the divisor counting function: $$\sigma_0(n) = \sum_{d \vert n} 1.$$ I ran some numerical experiments that showed when $p$ is prime, the function $\sigma_0(n)$ is equidistributed ...
Adithya Chakravarthy's user avatar
28 votes
3 answers
3k views

Expressing the Riemann Zeta function in terms of GCD and LCM

Is the following claim true: Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased, $$ \frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...
Nilotpal Kanti Sinha's user avatar
2 votes
1 answer
167 views

Estermann's argument for the binary additive divisor problem

In the paper https://eudml.org/doc/149759 an estimate for the binary additive divisor problem is given with a power saving. I don't get the main bit of the argument - I'm obviously missing something. ...
tomos's user avatar
  • 1,381
2 votes
0 answers
158 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\...
tomos's user avatar
  • 1,381
20 votes
2 answers
2k views

Is every prime the largest prime factor in some prime gap?

Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...
Nilotpal Kanti Sinha's user avatar
36 votes
2 answers
7k views

Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
Nilotpal Kanti Sinha's user avatar
5 votes
1 answer
187 views

Small covering of divisors

Let $D_n$ be the set of divisors of $n$. Does there always exists a $B\subseteq D_n$ such that $D_n = \{\gcd(ab,n) \mid a\leq \sqrt{n}, b\in B\}$ and $\sum_{b\in B} \frac{n}{b}=O(n)$?
Chao Xu's user avatar
  • 613
4 votes
0 answers
87 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 ...
user142929's user avatar
4 votes
1 answer
271 views

Around the equation $\sigma\left(\square\right)=\text{prime}$: counterexamples or a proof for some of these conjectures

For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible ...
user142929's user avatar
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 $[x,x+\log^2(x)]$ ...
G. Melfi's user avatar
  • 433
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 $n$...
qsq's user avatar
  • 31
11 votes
3 answers
605 views

Number of matrices with bounded products of rows and columns

Fix an integer $d \geq 2$ and for every real number $x$ let $M_d(x)$ be number of $d \times d$ matrices $(a_{ij})$ satisfying: every $a_{ij}$ is a positive integer, the product of every row does not ...
Kate's user avatar
  • 213
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 $...
user142929's user avatar
15 votes
3 answers
3k views

On Robin's criterion for RH [closed]

\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation} In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984)....
Roupam Ghosh's user avatar
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 ...
tomos's user avatar
  • 1,381
5 votes
2 answers
403 views

Estimating $\sum_{n\leq x: n \in A} d(n)^a$ from below for large sets $A\subset \{1,2,\ldots,x\}.$

I apologise for the long-windedness of this question. 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 $a=1,$ ...
kodlu's user avatar
  • 10.4k
3 votes
1 answer
437 views

Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?

Robin's inequality $$\sigma_1(n)<e^\gamma n\log\log n$$ at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...
Turbo's user avatar
  • 13.9k
5 votes
3 answers
716 views

Good books on the divisor sum function $\sigma(n)$?

I would like gain detailed knowledge about properties of the divisor sum function $\sigma(n)$, special equation that have been studied (e.g. $\sigma(n) = 2n$ perfect numbers, ...) and progress that ...
Fabian Schn.'s user avatar
7 votes
1 answer
370 views

If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide $n-1$

If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? The conjecture has ...
Nilotpal Kanti Sinha's user avatar
11 votes
2 answers
1k views

Does the Prime Number Theorem have anything to do with Erdos-Kac law or vice versa?

The prime number theorem says on average we can find $\frac n{\log n}$ primes of magnitude $n$. Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ primes. Somehow the fact $e^{\...
user avatar
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 ...
Cloudscape's user avatar
3 votes
1 answer
411 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 ...
user142929's user avatar
1 vote
1 answer
245 views

Moments of number of interval restricted divisors

I have previously asked the question A truncated divisor function sum where the sum $$ S_f(x)=\sum_{n\leq x} \min\{f(x),d(n)\}\quad (1) $$ was of interest, and it was answered satisfactorily. Here, I ...
kodlu's user avatar
  • 10.4k
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 $$ A(a,x)=\sum_{n\leq x} \min[ d(n), M]^a $$ ...
kodlu's user avatar
  • 10.4k
8 votes
0 answers
272 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 $n$ ...
Nick Belane's user avatar
0 votes
1 answer
260 views

Generalized Erdős multiplication table problem

Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$. What is the cardinality of the range? At $k =2$ ...
VS.'s user avatar
  • 1,836
6 votes
1 answer
258 views

How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?

I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that $$ \prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...
Johnny T.'s user avatar
  • 3,625
2 votes
1 answer
280 views

On a problem that equates $\frac{\text{prime}-1}{\operatorname{rad}(\text{prime}-1)}$ with the sequence of primorials

We denote for integers $m>1$ the product of the distinct prime numbers dividing $m$ as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p,$$ with the definition $\operatorname{rad}(...
user142929's user avatar
9 votes
1 answer
1k views

Sum of divisors below threshold

Let $\sigma(n)$ denote the sum of divisors of $n$, that is, $$ \sigma(n) = \sum_{d | n} d. $$ It is known that $\sigma$ can have values as large as order $n \log \log n$. However, obviously the sum is ...
Kurisuto Asutora's user avatar
2 votes
1 answer
198 views

Bounds for two arithmetic functions, when one assumes that $n$ are odd perfect numbers

For an integer $n>1$ in this post we denote the Dedekind psi function as $\psi(n)=n\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$ and the product of distinct primes dividing ...
user142929's user avatar
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 ...
H A Helfgott's user avatar
  • 20.2k
5 votes
1 answer
960 views

There at least 4 divisors of $n-1$ which do not divide $\phi(n)$ if $n$ is a composite of the form $6k+1$

If $n$ is composite then $\phi(n) < n-1$ (Euler's totient function) hence there must be one or more divisors of $n-1$ which do not divide $\phi(n)$. For lack of a better terminology, let us call ...
Nilotpal Kanti Sinha's user avatar
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 $\mathcal O(x^2)$...
Konstantinos Gaitanas's user avatar
4 votes
1 answer
646 views

Piltz Divisor Problem

Let $\tau_k(n)$ count the number of ways of representing $n$ as the product of $k$ natural numbers. It is known that: $$D_k(x) = \sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O(x ^{1 - \frac{1}{k-1}}(\...
user366818's user avatar
0 votes
1 answer
146 views

On $\mathsf{LCM}$ of a set of integers

For integers $a,b$ define $$\mathcal R(a,b)=\{q\in\mathbb Z\cap[1,\min(a,b)]: a\equiv b\bmod q\}$$ and $\mathsf{LCM}(\mathcal R(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal R(a,b)$. How ...
VS.'s user avatar
  • 1,836
7 votes
1 answer
231 views

The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are ...
user142929's user avatar
1 vote
1 answer
259 views

Sum of divisors of Stirling numbers of the second kind

In this post we denote the Stirling number of the second kind as ${n\brace k}$, I add as reference the article Stirling numbers of the second kind from the encyclopedia Wikipedia. And we denote the ...
user142929's user avatar
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 ...
VS.'s user avatar
  • 1,836
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)...
kodlu's user avatar
  • 10.4k
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 ...
VS.'s user avatar
  • 1,836
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 ...
user142929's user avatar
14 votes
3 answers
1k views

On the number of consecutive divisors of an integer

Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq ...
Dr. Pi's user avatar
  • 3,062
10 votes
4 answers
4k views

Sum of the sum-of-divisors function

I was looking at the abstract of a paper 1 which claims that [2] and [3] prove $$ \sum_{n\le x}\sigma(n)-\frac{\pi^2}{12}x^2=\Omega(x\log\log x). $$ But I cannot find the above—or indeed, ...
Charles's user avatar
  • 9,114
2 votes
1 answer
284 views

A truncated divisor sum

I am interested in an upper bound for $$\sum_{\substack{d\mid N\\ d>A}}\frac{1}{d^3},$$ in particular, I can show that above is $$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^...
user164144's user avatar
2 votes
0 answers
76 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 $\...
user142929's user avatar
3 votes
2 answers
795 views

Estimate about primes

Can anyone give an estimate (upper bound or lower bound) for the number of divisors $d\mid P_r$ such that $\frac{\sqrt{P_r}}{2}< d < \sqrt{P_r}$, where $P_r$ is the product of the $r$ smallest ...
Farzad Aryan's user avatar