Results 21 to 30 of about 41,624 (236)

Distribution mod p of Euler’s Totient and the Sum of Proper Divisors [PDF]

The Michigan mathematical journal, 2021
. We consider the distribution in residue classes modulo primes p of Euler’s totient function φ(n) and the sum-of-proper-divisors function s(n) := σ(n)−n.
Noah Lebowitz-Lockard, P. Pollack, A. Roy   +2 more
semanticscholar   +1 more source

New Proof That the Sum of the Reciprocals of Primes Diverges

Mathematics, 2020
In this paper, we give a new proof of the divergence of the sum of the reciprocals of primes using the number of distinct prime divisors of positive integer n, and the placement of lattice points on a hyperbola given by n=pr with prime number p.
Vicente Jara-Vera, Carmen Sánchez-Ávila   +1 more
doaj   +1 more source

On the sum of Divisors Function

Journal of Number Theory, 1995
A classical identity of Ramanujan states that, for \(\sigma_ z (n)= \sum_{d\mid n} d^ z\) and \(\text{Re } s>1+ \max (0, \text{Re } a\), \(\text{Re } b,\text{Re} (a+b))\) we have \[ \sum_{n=1}^ \infty \sigma_ a (n) \sigma_ b (n) n^{-s}= {{\zeta (s) \zeta (s-a) \zeta (s-b) \zeta (s-a-b)} \over {\zeta (2s-a -b)}}.
U. Balakrishnan
semanticscholar   +2 more sources

A formula for the number of solutions of a restricted linear congruence [PDF]

Mathematica Bohemica, 2021
Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\pmod{n^s}$ for $b\in\mathbb Z$, $n,s\in\mathbb N$. Let $(a,b)_s$ denote the generalized gcd of $a$ and $b$ which is the largest $l^s$ with $l\in\mathbb N$ dividing $a$ and $b ...
K. Vishnu Namboothiri
doaj   +1 more source

Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function [PDF]

Journal of Number Theory, 2019
Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $\phi(p-1) \geq \phi(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes.
S. Garcia, F. Luca, Kye Shi, G. Udell
semanticscholar   +1 more source

There are no multiply-perfect Fibonacci numbers [PDF]

, 2011
Here, we show that no Fibonacci number (larger than 1) divides the sum of its ...
Alain Togbé, Bugeaud Y., Florian Luca, Kevin A. Broughan, Luca F., Marcos J. González, Robbins N., Rosser J. B., Ryan H. Lewis, V. Janitzio Mejía Huguet   +9 more
core   +2 more sources

Divisors of Bernoulli Sums [PDF]

Results in Mathematics, 2007
We study the asymptotic behavior of the sums of divisors when the integers are modelled with the Bernoulli random walk; We prealably study the correlation properties of the corresponding system.
openaire   +3 more sources

On unitary Zumkeller numbers [PDF]

Notes on Number Theory and Discrete Mathematics
It is well known that if n is a Zumkeller number, then the positive divisors of n can be partitioned into two disjoint subsets of equal sum. Similarly for unitary Zumkeller number n, the unitary divisors of n can be partitioned into two disjoint subsets ...
Bhabesh Das
doaj   +1 more source

Etebari Perceptions: Assessing the Definitions and Offering Alternative Definitions [PDF]

حکمت و فلسفه, 2022
Allameh Tabatabai has used various interpretations to define credit perceptions, but none of them are suitable. This article, after examining the proposed definitions and measuring them, based on the characteristics of a desirable definition, attempts to
Hosein Kalbasi ashtari, mahdi kooshki
doaj   +1 more source

The algebraic independence of the sum of divisors functions

Journal of Number Theory, 2010
Daniel Lustig
semanticscholar   +3 more sources