Analytical Representations of Divisors of Integers

A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n<=x that are in S, for specific choices of S. The asymptotical precision of their respective approaches are being compared and related to Euler-Kronecker constants, a generalization of Euler's constant gamma=0.57721566.... This paper claims little originality, its...

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.

We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the connections among the nodes are made based on the divisibility pattern among the numbers. We derive analytical expressions for the centrality measures of this network in terms of the floor function and the divisor functions. We validate thes...

Prime factorization is an outstanding problem in arithmetic, with important consequences in a variety of fields, most notably cryptography. Here we employ the intriguing analogy between prime factorization and optical interferometry in order to obtain, for the first time, analytic expressions for closely related functions, including the number of distinct prime factors.

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.

In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result is deeply related with the divisor function. In the course of our study we get an independent result about it. Concretely we are able to deduce a new upper bound for the divisor function valid for any integer and fully explicit.

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindro...

Let $p_{\textrm{dsd}} (n)$ be the number of partitions of $n$ into distinct squarefree divisors of $n$. In this note, we find a lower bound for $p_{\textrm{dsd}} (n)$, as well as a sequence of $n$ for which $p_{\textrm{dsd}} (n)$ is unusually large.

We present an improved version of the analytic method for calculating $\pi(x)$, the number of prime numbers not exceeding $x$. We implemented this method in cooperation with J. Franke, T. Kleinjung and A. Jost and calculated the value $\pi(10^{25})$.

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...