Analytical Representations of Divisors of Integers
Krzysztof Ma\'slanka

TL;DR
This paper explores analytical expressions that represent divisors of natural numbers, revealing connections to the sieve of Eratosthenes, with a focus on pedagogical presentation and some novel formulas.
Contribution
It introduces new formulas and insights into how analytical expressions can encode divisor information and relate to classical algorithms like the sieve of Eratosthenes.
Findings
Expressions encode sieve of Eratosthenes
Some formulas are new
Pedagogical presentation
Abstract
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.
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Taxonomy
TopicsMatrix Theory and Algorithms · Advanced Optimization Algorithms Research · Polynomial and algebraic computation
Analytical Representations
of Divisors of Integers
Krzysztof Maślanka
e-mail: [email protected]
Institute for the History of Science
Polish Academy of Sciences
Abstract
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.
MSC: Primary 11A51; Secondary 26A06
1 Notation and Conventions
Throughout this paper we shall adopt the following notation and conventions: is a given natural number and is a possible divisor of . If actually divides then . Let denotes any real analytic function defined in the neighborhood of the origin by a power series
[TABLE]
with all (). It will be shown that is also the exponent of in the expansion (1) around zero and labels half-lines or rays of divisors (see below).
2 Motivation
The theory of divisors of integers is the cornerstone of elementary number theory. It is convenient to introduce the characteristic function for divisors:
Definition. For any
[TABLE]
Another pretty obvious (and rather useless in numerical calculations) representation of (2) is:
[TABLE]
where denotes the Euler gamma function and gives the remainder on division of by . In fact (3) is more general than (2) since it may be calculated also for non-integer or even complex values of and but this leads to some interpretation difficulties which we shall not discuss here.
Consider the following expression for some natural numbers and :
[TABLE]
We will prove the following
Theorem. Apart from a trivial normalization factor, defined in formula (2) is equal to defined in (4).
Proof. Expanding the exponential function in (4) in power series and performing term-by-term differentiation we get:
[TABLE]
Recall the general formulas for the -th derivative of with respect to
[TABLE]
[TABLE]
where the second formula stems from properties of the gamma function and is suitable for integer negative (see e.g. [4]). Note that the order of derivative does not have to be integer but for integer both (6) and (7) reduce to the well-known elementary differentiation rule. Using (6) we get:
[TABLE]
By simple inspection of (8) we see why this expression ”feels” the divisors of the integer . Indeed, when taking the limit the only non-zero term in the series appears when for some integer , and this occurs if and only if divides . All terms with disappear in the limit whereas those with , although singular in , vanish since the binomial coefficient term is zero. Therefore, in the summation (8) at most only one term can survive in the limit process.
The above reasoning might appear far too excessive. However, it guarantees that among divisors none have been omitted. It should also be stressed that it may be used as a starting point for various generalizations since need not to be integer.
It is easy to guess the normalizing factor:
[TABLE]
Using the same reasoning we can derive similar expression for :
[TABLE]
3 Simple example
In a natural way coefficients may be regarded as a square matrix of arbitrarily large dimension where the running integer labels rows and the potential divisor labels columns. The entries of this matrix are either one or zero depending on whether divides or not. This matrix is always triangular, since of course no divisor can exceed a given number, and its determinant (for any dimension) is 1.
[TABLE]
(Matrix (11) is closely related to the Redheffer matrix, see e.g. [5], [6].) Introducing
[TABLE]
we see that just counts the number of all divisors of a given including both unity and itself.
It is known (see e.g. [1]) that the inverse of matrix (11) is:
[TABLE]
where denotes the Möbius function:
[TABLE]
[TABLE]
Note that the numbers in (15) when summed in rows give zero except for the first row which stems from the following identity:
[TABLE]
Matrices (11) and (15) are visualized in Figure 1.
Somewhat similar but purely qualitative results have been published in [3].
Figure 1. Graphic distribution of divisors (11) for as a square matrix (left panel). Each blue square denotes . In the inverse matrix (15) blue square denotes and red square denotes (right panel).
4 General case
The particular choice of the exponential function in (4) is not crucial to our reasoning. Indeed, instead of this function we can take any regular function provided that it has all non-zero coefficients in its power series expansion
[TABLE]
Thus in general we have (up to appropriate normalizing factor)
[TABLE]
For example, taking
[TABLE]
we get:
[TABLE]
or simply
[TABLE]
Taking
[TABLE]
we get:
[TABLE]
The general explicit formula for using arbitrary function satisfying (17) is:
[TABLE]
where denotes the -th derivative of with respect to taken at . (If in (24) is non-integer then the value of fractional derivative is unimportant since in this case the sum vanishes.)
The table below contains normalizing factors for , for several different choices of function , obtained using (24).
[TABLE]
( is the Lambert -function and in the last row denotes the -th Fibonacci number.)
5 Interpretation
Let us now explain in more details how it all works. The thing is that all formulas for presented so far encode, at least to some extent, the ancient algorithm known as the sieve of Eratosthenes.
Indeed, consider as the function and let us temporarily restrict ourselves to the linear case:. According to the general formula (18) we have
[TABLE]
and this produces a single line of ones on the diagonal in the divisor matrix (11) – cf. Figure 2 below. This is equivalent to the trivial statement that all integers are divisible both by one and by themselves. Let us further consider more precise approximation . We get from (18) another sequence of ones on the line . This is equivalent to selecting all even integers and adding to the divisor matrix (11) their divisors . Taking into account higher powers of we select all numbers which are multiplies of and this adds to the matrix further lines of divisors: , respectively.
Proceeding in the same way we finally arrive at the full expansion of :
[TABLE]
which produces the entire sequence of lines labelled by parameter . In this way we have selected and visualized all divisors for all integers. It is clear that there are certain well-defined numbers (marked in bold in Figure 2) which have exactly two divisors: unity and themselves, i.e. prime numbers: At the same time we see the importance of condition in (17) since even a single coefficient would cause a skipping of certain divisors. In view of this the characteristic function for divisors may also be written in a very natural form as a sum over Kronecker deltas:
[TABLE]
Note that combining (12), (18) and (26) gives:
[TABLE]
Hence
[TABLE]
which is consistent with the theory of Lambert series (see e.g. [2]) which is the generating function for the sequence where is the total number of divisors for a given integer .
Figure 2. Distribution of divisors of integers computed from . This figure illustrates how various terms in the sum (27) contribute to the whole pattern of divisors. Each term corresponds to a ray of divisors. Rows are labelled by consecutive integers and columns are labelled by potential divisors . Each colored disc means that given actually divides , otherwise there is small black circle. To better visualize the whole pattern discs are in 3 different colors and lines connecting them are drawn. Of course, above the diagonal () there can’t be any divisors.
6 Concluding remarks
A few elementary comments at the end of this note. As we have seen, all divisors of integers lie on rays passing through the origin of the coordinate system on the () plane and are labelled by an integer parameter
[TABLE]
We have also seen that this simple condition has a natural interpretation since may be identified with the exponent in in the expansion (17). The key thing is that these rays must pass through certain points of an integer lattice and only then a potential divisor can be an actual divisor. For large these rays typically get closer and closer to one another. Therefore we qualitatively see why it is so difficult to factorize large integers.
Moreover, numerical experiments suggest that all divisors lie on countable families of parabolas passing through the origin (see Figures 3, 4 and 5 below). These parabolas are ”quantized” in the sense that each family is characterized by two discrete parameters and and inside any family parabolas are labelled by another integer parameter :
[TABLE]
Careful simulations using Mathematica revealed that parameter assumes equidistant values with integer constant step:
[TABLE]
starting from where denotes greatest common divisor, i.e. , ,
Figure 3. Various families of parabolas (31) for and . Step (32) described in the main text is also indicated.
Figure 4. Family of parabolas (31) for and (red) (orange) (green),and (cyan) for . For clarity of the plot parameter assumes only consecutive values. Prime numbers among s are indicated by vertical lines.
Figure 5. Family of parabolas (31) for and (red) (yellow) and (green) around . For clarity of the plot parameter assumes only consecutive values. Prime numbers among s are indicated by vertical lines.
As far as I am aware the unexpected parabolas in the distribution of divisors have been independently noticed by Jeffrey Ventrella (see his popular book [7], page 33) but with no quantitative considerations.
Finally, it should be stressed that, unfortunately, expressions presented in this note do not tell us much about distribution of primes. They are even not very suitable for numerical calculations for large therefore may be treated merely as a curiosity. Nevertheless, we have shown some unexpected relationship between number theory and calculus.
Acknowledgments. The author would like to thank Prof. Jeffrey Lagarias for his encouragement and several suggestions and to Prof. Andrzej Schinzel for several remarks.
The results presented in this paper were inspired by experimenting with Wolfram Mathematica. Also all calculations were checked using this powerful software.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] On-Line Encyclopedia of Integer Sequences , sequence number A 054525.
- 2[2] Tom M. Apostol, Modular Forms and Dirichlet Series in Analysis , 1976, Springer-Verlag.
- 3[3] David N. Cox, Visualizing the Sieve of Eratosthenes , Notices of the AMS, vol. 55, nr. 5, May 2008, p. 579-582.
- 4[4] Kennet S. Miller, Bertram Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993, John Wiley and Sons.
- 5[5] Raymond M. Redheffer, Eine explizit lösbare Optimierungsaufgabe , Internat. Schriftenreihe Numer. Math. vol. 36, 1977.
- 6[6] Michael Trott, The Mathematica Guide Book for Programming , 2004, New York: Springer-Verlag.
- 7[7] Jeffrey J. Ventrella, Divisor Drips and Square Root Waves – Prime Numbers are the Holes in Complex Composite Number Patterns , 2010, Eyebrain Books (book web site: www.divisorplot.com).
