ID: 1405.4784

An Explicit Formula For The Divisor Function

May 16, 2014

View on ArXiv
N. A. Carella
Mathematics
General Mathematics

The details for the construction of an explicit formula for the divisors function d(n) = #{d | n} are formalized in this article. This formula facilitates a unified approach to the investigation of the error terms of the divisor problem and circle problem.

Similar papers 1

Analytical Representations of Divisors of Integers

February 25, 2017

90% Match
Krzysztof Maślanka
General Mathematics

Certain analytical expressions which "feel" the divisors of natural numbers are investigated. We show that these expressions encode to some extent the well-known algorithm of the sieve of Eratosthenes. Most part of the text is written in pedagogical style, however some formulas are new.

Find SimilarView on arXiv

A Successive Approximation Algorithm for Computing the Divisor Summatory Function

June 15, 2012

87% Match
Richard Sladkey
Number Theory

An algorithm is presented to compute isolated values of the divisor summatory function in O(n^(1/3)) time and O (log n) space. The algorithm is elementary and uses a geometric approach of successive approximation combined with coordinate transformation.

Find SimilarView on arXiv

Rapidly Convergent Series of the Divisors Functions

July 28, 2014

87% Match
N. A. Carella
General Mathematics

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results of some special values of the sums of divisors functions.

Find SimilarView on arXiv

On the Number of Divisors of $n^2 -1$

July 30, 2015

86% Match
Adrian Dudek
Number Theory

We prove an asymptotic formula for the sum $\sum_{n \leq N} d(n^2 - 1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our proof, we also furnish an asymptotic formula for the sum $\sum_{d \leq N} g(d)$, where $g(d)$ denotes the number of solutions $x$ in $\mathbb{Z}_d$ to the equation $x^2 \equiv 1 \mod d$.

Find SimilarView on arXiv

The sum of the unitary divisor function

December 17, 2013

86% Match
Tim Trudgian
Number Theory

This article establishes a new upper bound on the function $\sigma^{*}(n)$, the sum of all coprime divisors of $n$. The article concludes with two questions concerning this function.

Find SimilarView on arXiv

Counting primitive subsets and other statistics of the divisor graph of $\{1,2, \ldots n\}$

August 15, 2018

86% Match
Nathan McNew
Number Theory
Combinatorics

Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = \alpha^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this constant $\alpha$. We also show that the method developed can be applied to many similar problems that can be stated in terms of the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the di...

Find SimilarView on arXiv

An Exact Formula for the Prime Counting Function

May 24, 2019

86% Match
Jose Risomar Sousa
General Mathematics

This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $\pi(x)$, and the prime-power counting function, $J(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given $F_a(s)$, we know $a(n)$, which implies the Riemann hypothesis, and enabled the creation of a formula for $\pi(x)$ in the f...

Find SimilarView on arXiv

Divisor and Totient Functions Estimates

June 23, 2008

86% Match
N. A. Carella
Number Theory
General Mathematics

New unconditional estimates of the divisor and totient functions are contributed to the literature. These results are consistent with the Riemann hypothesis and seem to solve the Nicolas inequality for all sufficiently large integers.

Find SimilarView on arXiv

A Divisor Function Inequality

December 9, 2009

85% Match
N. A. Carella
Number Theory

This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely elementary.

Find SimilarView on arXiv

On sums of the small divisors of a natural number

October 25, 2019

85% Match
Douglas E. Iannucci
Number Theory

We consider the positive divisors of a natural number that do not exceed its square root, to which we refer as the {\it small divisors\/} of the natural number. We determine the asymptotic behavior of the arithmetic function that adds the small divisors of a natural number, and we consider its Dirichlet generating series.

Find SimilarView on arXiv