The Schnirelmann Density of the Set of Deficient Numbers

THE SCHNIRELMANN DENSITY

OF THE SET OF DEFICIENT NUMBERS

A Thesis

Presented to the

Faculty of

California State Polytechnic University, Pomona

In Partial Fulﬁllment

Of the Requirements for the Degree

Master of Science

Mathematics

Peter Gerralld Banda

2015 SIGNATURE PAGE

THESIS: THE SCHNIRELMANN DENSITY OF THE SET OF DEFICIENT NUMBERS

AUTHOR: Peter Gerralld Banda

DATE SUBMITTED: Summer 2015

Mathematics and Statistics Department

Dr. Mitsuo Kobayashi Thesis Committee Chair Mathematics & Statistics

Dr. Amber Rosin Mathematics & Statistics

Dr. John Rock Mathematics & Statistics

ii ACKNOWLEDGMENTS

I would like to take a moment to express my sincerest appreciation of my ﬁanc´ee’s never ceasing support without which I would not be here. Over the years our rela tionship has proven invaluable and I am sure that there will be many more fruitful years to come. I would like to thank my friends/co-workers/peers who shared the long nights, tears and triumphs that brought my mathematical understanding to what it is today. Without this, graduate school would have been lonely and I prob ably would have not pushed on. I would like to thank my teachers and mentors.

Their commitment to teaching and passion for mathematics provided me with the incentive to work and succeed in this educational endeavour. Last but not least, I would like to thank my advisor and mentor, Dr. Mitsuo Kobayashi. Thanks to his patience and guidance, I made it this far with my sanity mostly intact. With his expertise in mathematics and programming, he was able to correct the direction of my efforts no matter how far they strayed. His never wavering confidence in me drove me through many a pot of coffee and finally paid off with this thesis.

iii ABSTRACT

The goal of this discussion is to determine the Schnirelmann density of the deﬁcient

numbers, which is not currently known. The Schnirelmann density for deﬁcient

numbers is defined as σpDq “ infrDpnq{ns, where Dpnq denotes the cardinality of the set of deficient numbers not exceeding n, and inf is the infimum operator. This density is of particular interest because of the useful properties that it possesses.

The Schnirelmann density of the deficient numbers can be found by finding a local minimum density in an initial interval of the natural numbers and then lower- bounding the density over the rest of the set of natural numbers. The lower bound for the density of deficient numbers is equivalent to an upper bound for the density of the complement set of the deficient numbers, the non-deficient numbers. To achieve our goal we incorporate Felix Behrend’s procedure for bounding the density of the non-deficient numbers with some improvements to reduce the effects of the approximations he made. With this information about the two intervals, it is expected to arrive at the Schnirelmann density of deficient numbers.

iv Contents

Signature Page ii

Acknowledgements iii

Abstract iv

List of Tables vii

List of Figures viii

1 Exposition 1

1.1 Where did it all start? ...... 1

1.2 Density in Mathematics ...... 3

1.3 The Schnirelmann Density ...... 5

1.4 More from the Number Theorist’s Handbook ...... 6

2 Developing The Bound 11

2.1 The Candidate Number ...... 11

2.2 Bounding the Contributors ...... 12

2.3 Calculating the Density of Non-Deﬁcients ...... 17

2.4 Revisiting the Bound ...... 19

v 3 Results and Interpretations 22

3.1 Initial Bounds: Behrend’s Result ...... 22

3.2 An Illustration of the Bounding Method ...... 24

3.3 The Reﬁning Process ...... 25

4 The Schnirelmann Density of Deﬁcient Numbers 31

4.1 Improvements on Bound and Execution Time ...... 32

4.2 Second-Order Terms and Further Research ...... 35

Bibiliography 38

Appendices 39

A Deﬁcient Data 39

B Programs 41

B.1 Program 1 ...... 41

B.2 Program 2 ...... 44

vi List of Tables

3.1 Conditions for improvement ...... 26

3.2 Revised conditions for improvement ...... 28

3.3 Final conditions for improvement ...... 28

4.1 τ Speciﬁc Densities of Non-deﬁcient Numbers ...... 31

4.2 Improved τ Speciﬁc Densities of Non-deﬁcient Numbers ...... 34

4.3 Further Improved τ Speciﬁc Densities of Non-deﬁcient Numbers . . 34

A.1 Density of Deﬁcient Numbers up to n ...... 39

A.2 Density of Deﬁcient Numbers up to n continued ...... 40

vii List of Figures

2.1 Density of Deﬁcient Numbers ...... 11

3.1 Partition Scheme ...... 24

3.2 Cardinality Partition Scheme up to n ...... 25

viii Chapter 1

Exposition

1.1 Where did it all start?

Euclid (300 B.C.) made mention of diﬀerent types of numbers in his thirteen-

book treatise called The Elements. This collection of mathematics is considered by

many to be one of the most inﬂuential works in the history of mathematics. In the

Elements, Euclid realized the principles of what is now called Euclidean geometry from a small set of axioms. This collection contains works on perspective, conic sections, spherical geometry, number theory and rigor. Euclid talks about a speciﬁc type of number called perfect or complete. Nicomachus (100 C.E.) continued this discussion and categorized the natural numbers into three types.[1] These numbers have the property that the sum of their proper divisors either is greater than the number itself, is the number itself, or is less than number itself.

The perfect numbers are the numbers where the sum of their proper divisors is the number itself. To understand mathematically and symbolically what a perfect number is, consider the following deﬁnition and developed notation.

1 (Note that N denotes the number theorist’s set of natural numbers, t1, 2, 3,... u.)

Notation. If n P N, let σpnq denote the sum of all the divisors of n, σpnq “ d. d|n ÿ Deﬁnition. If n P N and σpnq “ 2n, then n is perfect.

Euclid’s Elements contains the first recorded mathematical result regarding per fect numbers. Proposition 36 of Book IX of the Elements states: For some k ą 1 with 2k ´1 prime, 2k´1p2k ´1q is a perfect number.[8] Using the k values t2, 3, 5, 7u, we find that the first four perfect numbers are 6, 28, 496 and 8,128. Notice that

σp6q “ 1 ` 2 ` 3 ` 6 “ 12 “ 2p6q. A similar calculation verifies the membership of 28, 496 and 8,128 in the set of perfect numbers which will be notated P. It should be noted that the cardinality of this set is neither known to be finite nor infinite.

For future use, it will be beneﬁcial to have some general notation for sets of natural numbers and their corresponding cardinalities.

Notation. Let S Ď N with S “ |S| the cardinality of S. We will use the notation Spnq if we desire the subset S X r1, ns and Spnq for the cardinality of Spnq.

As one might observe, for the set of perfect numbers that Euclid’s formula pro

vides, there is a sizeable subset of the natural numbers that is unaccounted for.

Making use of the divisor function, σpnq, it is observed that the set of natural

numbers, N, can be partitioned into three sets of numbers, each possessing a dis tinct divisor sum characteristic. The characteristics are σpnq ą 2n, σpnq “ 2n and σpnq ă 2n. We shall call these subsets the set of abundant numbers, per fect numbers and deﬁcient numbers, respectively. We will use the notation Λ for

2 the abundant numbers and D for the deﬁcient. Note that Λ Y P Y D “ N and

Λpnq Y Ppnq Y Dpnq “ Npnq. To assert that D is a non-empty set that is inﬁnite, consider the fact that for

any prime number p, σppq “ p ` 1 ă 2p and hence D contains the inﬁnite subset

P, the set of prime numbers. To assert that Λ is an inﬁnite set, consider that for any positive multiple mn of

n P P where m ą 1, σpmnq ą 2n. (We know that σpnq “ d “ 2n. Regardless d|n ÿ of what factors n and m have in common, we have that σpmnq ě m ¨ d ` 1 ě d|n mp2nq ` 1 ą 2mn.) ÿ

1.2 Density in Mathematics

In mathematics, the density of a subset of numbers contained in a larger set is de

ﬁned to be the ratio of the cardinality of the subset to the cardinality of the larger

set if this ratio exists. It should be clear that the density of any subset of a set will have a value in the interval r0, 1s. In the case that both the subset and parent set are ﬁnite, the calculation is quite elementary. For example, one could ask what

is the density of numbers that are even in the set t1, 2, 3, 4, 5, 6, 7, 8, 9, 10u? The elements of the set that are even are t2, 4, 6, 8, 10u and hence the answer to the question is exactly determined by the calculation:

|t2, 4, 6, 8, 10u| 5 ““ 0.5. |t1, 2, 3, 4, 5, 6, 7, 8, 9, 10u| 10 In the case where the two sets are not ﬁnite, the ratio of cardinalities is in indeterminate form and determining the density is not elementary, so we need a

3 new deﬁnition for the density of a subset and as it pertains to this paper, a subset

of natural numbers. To aid in the calculation process, let’s develop some working notation to handle potentially inﬁnite sets.

Notation. Let S Ď N. Denote the natural density of S by dpSq, where dpSq “ Spnq lim , provided the limit exists. nÑ8 n

With this notation, we can determine the natural density of any subset, S, of natural Spnq Spnq numbers, provided dpSq “ dpSq where dpSq “ lim and dpSq “ lim . This n n is also known as the asymptotic density of a set and dpSq “ dpSq “ dpSq. Consider

Opnq 1 2 1 3 1 4 the set of odd numbers, O. We have that “ 1, , , , , , ,... . n 2 3 2 5 2 7 "*"8 * Notice that the function

1 : n is even fpnq “ 2 $ 1 1 ’ ` : n is odd & 2 2n Opnq %’ 1 accurately calculates for all n P . Notice that limfpnq “ limfpnq “ and n N 2 1 hence dpOq “ . A similar calculation asserts that the natural density of the even 2 1 numbers, dpEq, is . 2 This result for the even numbers could also have been calculated the following way. Notice that E YO “ N and E XO “H. Now dpNq “ dpE YOq “ dpEq`dpOq´ 1 1 dpE X Oq “ dpEq ` dpOq and lastly we have dpEq “ dp q ´ dpOq “ 1 ´ “ . This N 2 2 calculation makes use of the fact that the odd numbers and even numbers partition the natural numbers and hence their natural density sum is the natural density of the natural numbers.

4 1.3 The Schnirelmann Density

Opnq 1 2 1 3 1 4 Recall the observation “ 1, , , , , , , . . . . The inﬁmum of n 2 3 2 5 2 7 " *8 " * this sequence of natural densities is deﬁned to be the Schnirelmann density. The

Schnirelmann density of a set of numbers is another way to measure how “dense” the set is.

Deﬁnition. The Schnirelmann density of a set of natural numbers, S, is deﬁned Spnq as δpSq “ inf . n n

Spnq The Schnirelmann density is well-deﬁned even if the limit of as n Ñ 8 n 1 fails to exist. Notice how the Schnirelmann density of the odd numbers O is . It 2 is not always the case that the Schnirelmann density of a set is the same as the

natural density. To demonstrate this consider the sequence of values representing

Epnq 1 1 1 2 the density of even numbers up to n, “ 0, , , , ,... . The inﬁmum n 2 3 2 5 "*"8 * of this set is 0 and hence the Schnirelmann density is 0 which is not the natural

density of even numbers.

The Schnirelmann density has proved to be a resource for solving important

problems, such as Waring’s problem. The Schnirelmann density is of particular

interest because of the properties that it possesses. For instance, Lagrange’s four

square theorem could be restated and shown in the following way given Mann’s

Theorem.

5 Theorem 1.1. (Mann’s Theorem) : δpA ‘ Bq ě mint1, δpAq ` δpBqu

where A ‘ B is the sum set a ` b a P A Y t0u, b P B Y t0u . ´ ! ˇ )¯ ˇ

Theorem 1.2. (Lagrange’s Theorem)

Every natural number can be expressed as the sum of at most four square natural

numbers.

Letting S2 k2 8 , Mann’s theorem takes Lagrange’s theorem and makes “ k“1 the statement (

2 2 2 2 2 2 2 2 δ S ‘ S ‘ S ‘ S “ 1 iﬀ S ‘ S ‘ S ‘ S “ N. ` ˘ 1.4 More from the Number Theorist’s Handbook

To better follow the calculations in this paper and understand the results, we should

familiarize ourselves with a few standard number theoretic functions used, as well

as Landau notation and a few well-known results in number theory.

Deﬁnition. (Euler’s Totient Function) : φpvq is an arithmetic function that counts the totatives of v ( i.e., the positive integers less than v that are relatively prime to 1 v). This function has the closed form φpvq “ v 1 ´ where p runs over the p p|v ˆ ˙ distinct prime divisors of v. This function is multiplicativeź (i.e. if m and n are

relatively prime to each other then φpmnq “ φpmqφpnq).

Notation. We will write pm, nq to denote the greatest common divisor between m

and n. Note that when m and n are co-prime, pm, nq “ 1.

6 Deﬁnition. (Principal Character Modulo k): χkpvq is an indicator function that

is deﬁned to be χkpvq “ 1 for pv, kq “ 1 and χkpvq “ 0 for pv, kq ą 1.

This function is a periodic function with period k. Further it is a completely multi

plicative function in both v and k. (I.e., this function has the following properties:

χkpm ¨ nq “ χkpmq ¨ χkpnq and χk¨lpmq “ χkpmq ¨ χlpmq for all k, m, n, l P N.)

Notation. (Landau notation) If f and g are two functions deﬁned on a subset of

the real numbers, then we write fpxq “ Opgpxqq if and only if there exists a pos itive real number M and a real number x0 such that |fpxq| ď M|gpxq| for all x ě x0.

Deﬁnition. (Dirichlet product): Given two arithmetic functions f and g, f ˚ g will

denote a third arithmetic function deﬁned by pf ˚ gqpnq “ fpaqgpbq where the ab“n sum runs over pairs of positive integers pa, bq such that ab ÿ“ n. Alternatively we

could write fpdqg pn/dq to represent this product. d|n ÿ

Notation. We will write tau to denote the greatest integer less than or equal to a.

Notation. We will write tau to denote the fractional part of a i.e., tau “ a ´ tau.

7 Results in number theory:

1. χm piq “ k ¨ φpmq ` φpr, mq, i“1 ÿ r n where φpr, mq “ χ piq, r ” npmod mq, 0 ď r ă m and k “ . m m i“1 ÿ Y] k t d u 2. gpvqfpdq “ fpdq gpidq, where k P R and id “ v. vďk d|v dďk i“1 ÿ ÿ ÿ ÿ 8 χ pdq π2 1 3. k “ ¨ 1 ´ , d2 6 p2 d“1 p|k ÿ ź ˆ ˙ 8 1 π2 1 ´1 which follows from the result ““ 1 ´ . d2 6 p2 d“1 pP ÿ źP ˆ ˙ σpnq d 1 4. ““ , where the sum is over divisors d of n. n n d d|n d|n ÿ ÿ n χ pdq 5. k “ Oplog nq. d d“1 ÿ log n 1 log n 6. O ` O “ O . n n n ˆ ˙ ˆ ˙ ˆ ˙

f ¨ g f ¨ g f ¨ g 7. pmnq “ pmq ¨ pnq h h h ˆ ˙ ˆ ˙ ˆ ˙ provided f, g and h are multiplicative arithmetic functions and pm, nq “ 1.

8. pf ˚ gqpmnq “ pf ˚ gqpmqpf ˚ gqpnq

provided f and g are multiplicative arithmetic functions, hpnq ‰ 0 for all

n P N. k k ei ei 9. f p i “ f pp i q ˜i ¸ i providedź f is a źmultiplicative arithmetic function, where p1, . . . , pk are distinct

primes with ei P N for all i “ 1, . . . , k.

8 8 fpnq fppq fpp2q 10. “ 1 ` ` ` ¨ ¨ ¨ ns ps p2s n“1 p ÿ ź ˆ ˙ provided f is multiplicative and Repsq ą αapfq where αapfq is the abscissa of absolute convergence.

Results 3-10 are common results covered in most Number Theory books, however

I crafted result 1 and 2 for use in this paper. For this reason, I will provide their proofs below.

Theorem 1.3. (A Useful Theorem) n

Let φpn, mq “ χm piq. Then φpn, mq “ k ¨ φpmq ` φpr, mq, i“1 ÿ n where r ” npmod mq, 0 ď r ă m and k “ . m Y] Proof. Suppose n “ lm where l P N. lm l

Then φpn, mq “ φplm, mq “ χm piq “ φpmq “ l ¨ φpmq. i“1 i“1 ÿ ÿn n t m um n

Let n ě m. Now φpn, mq “ χm piq “ χm piq ` χm piq i“1 i“1 i“ n m`1 ÿ ÿ t mÿ u n´t n um n m n “ ¨ φpmq ` χ j ` m m m m j“1 Y] r ÿ ´ Y ] ¯ n “ ¨ φpmq ` χ pjq p1q m m j“1 Y]n ÿ “ ¨ φpmq ` φpr, mq, m where in (1) we use that Y] n n χ j ` m “ 1 iff j ` m, m “ 1 iff pj, mq “ 1 iff χ pjq “ 1. m m m m n Lastly,´ for Y the ] case ¯ that n´ ă m Y, we ] have¯ that k ““ 0 and r “ n. m Y]

9 Lemma 1.4. (Sum Re-indexing)

k t d u gpvqfpdq “ fpdq gpidq, where k P R and id “ v. vďk d|v dďk i“1 ÿ ÿ ÿ ÿ

Proof. Consider the double sum gpvqfpdq where k P R and the functions g vďk d|v ÿ ÿ and f are arithmetic functions deﬁned on N. Note that the outer sum runs over v P r1, ks with v P N and the inner sum runs over all d P N such that d|v. Suppose 1 k id “ v with i, d P N. Now it must be the case that id P r1, ks or i P , . But d d k „ j k i P N, so equivalently i P 1, . Now since id “ v and v ď k, we have d ď d i k „Z ^j and with 1 ď i ď , we have that d ď k. Now if summed over i then over d, we d Z ^ have the following:

k k t d u t d u gpvqfpdq “ gpvqfpdq “ gpidqfpdq “ fpdq gpidq ¨ ˛ vďk d|v vďk id“v dďk i“1 dďk i“1 ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ˝ ‚ .

10 Chapter 2

Developing The Bound

2.1 The Candidate Number

Dr. Mitsuo Kobayashi conjectured that it would be possible to ﬁnd the Schnirelmann density of deﬁcient numbers by observing that there is a local minimum value for the density up to n in an initial interval and developing a lower bound over the rest of the natural numbers using the work of Behrend. Consider the following graph

of the density of deﬁcient numbers versus n, which depicts this idea.

Figure 2.1: Density of Deﬁcient Numbers

11 Inspection of the graph and its data, provided in the Table A.1 and A.2 of the

Appendix, suggests that the density of deficient numbers achieves a local minimum at the number 114 of 0.7456140 ... and the tail end of the chart seems to converge on a value. In Dr. Mitsuo Kobayashi’s dissertation, he calculates this value to four decimal places. In particular, he calculates the natural density of non-deficient numbers Λ Y P, the complement of the deficient numbers, to be 0.2476 ... and this gives us the value 0.7523... for the natural density of deficient numbers.[7] For the

remainder of this thesis we will denote the set of non-deﬁcient numbers, Λ Y P, by

A and the set of all non-deﬁcient numbers up to n, pΛ Y Pqpnq, by Apnq.

2.2 Bounding the Contributors

As it pertains to this paper, determining a lower bound on the density of deﬁcient

numbers requires the use of an upper bound on the density of non-deﬁcient num

bers. I accomplish this by making use of Felix Behrend’s work that asserts that the quantity Apnq of non-deﬁcient numbers below n is less than the quantity Dpnq of deﬁcient numbers below n.[2] Behrend proved his assertion by considering the sets

of natural numbers that are divisible by 6, divisible by 2 and not 3, and divisible

by 3 but not 2. To do this he examined the following sums:

1 σpvq 1 σp2vq 1 σp3vq ¨ , ¨ , ¨ . n v n 2v n 3v vďn 2vďn 3vďn pv,ÿ6q“1 pv,ÿ3q“1 pv,ÿ2q“1

One might ask why Behrend decided to investigate these sums. To answer this, σpkq consider the expression . This ratio will produce a value that will be indicative k

12 of whether the number k is deﬁcient or non-deﬁcient in the following fashion:

σpkq σpkq ě 2 if k P A and ă 2 if k P D. k k 1 σpkq Now let Ω be a finite set of natural numbers. The sum, , is the average |Ω| k kPΩ indicator value for all k P Ω and hence is an way to estimateÿ how many deficient or non-deficient members are in the set Ω.

As it pertains to Behrend’s sums, the indexing makes sure to only consider the natural numbers that belong to each partition he called for. The following calculation I extracted and generalized from Behrend’s proof that the quantity of non-deﬁcient numbers is greater than that of the deﬁcient numbers.[2] Consider the following calculation where τ is a square-free product of prime numbers and τ “ α¨γ. 1 σpαvq This calculation produces an estimate, from above, of the sum . n αv αvďn pv,γÿq“1

1 σpαvq 1 1 “ χ pvq n αv n γ d αvďn αvďn d|αv pv,γÿq“1 ÿ ÿ

1 i ¨ χi pdq “ ¨ χγ pvq αn ¨ ¨ d ˛˛ αvďn d|v i|α ÿ ÿ ÿ ˝ ˝ ‚‚ 1 i ¨ χi pdq “ ¨ χγ pvq αn n ¨ ¨ d ˛˛ vď α d|v i|α ÿ ÿ ÿ n ˝˝ t αd u ‚‚ 1 i ¨ χi pdq “ ¨ χγ pidq Result 1. αn ¨d ˛ dď n i|α i“1 ÿα ÿ ÿ ˝ ‚ t n u 1 i ¨ χ pdq αd “ ¨ i¨γ χ piq αn d γ n ¨ ˛ i 1 dď α i|α “ ÿ ÿ ÿ ˝ ‚ n 1 i ¨ χ pdq “ ¨ i¨γ αd φpγq ` φpr, γq Result 2. αn ¨ d ˛ γ dď n i|α ˜[X \_ ¸ ÿα ÿ ˝ ‚

1 n n i ¨ χ pdq “ ¨ φpγqαd ´ αd ` φpr, γq i¨γ αn n ˜ ˜ γ# γ +¸ ¸ ¨ d ˛ dď α X \ X \ i|α ÿ ÿ n n ˝ ‚ 1 n i ¨ χ pdq “ ¨ φpγq ´ αd ` αd ` φpr, γq i¨γ αn τd γ γ ¨ d ˛ dď n ˜ ˜ ˜ ( #X\+¸¸ ¸ i|α ÿα ÿ ˝ ‚ 1 φpγqn i ¨ χi¨γ pdq “ ¨ ¨ 2 αn τ n ¨ d ˛ dď α i|α ÿ ÿ n ˝ n ‚ φpγq i ¨ χ pdq ´ ¨ αd ` αd i¨γ αn γ γ ¨ d ˛ dď n ˜ ( #X \+¸ i|α ÿα ÿ ˝ ‚ 1 i ¨ χ pdq ` ¨ φpr, γq i¨γ αn ¨ d ˛ dď n i|α ÿα ÿ ˝ ‚ 1 φpγq i ¨ χ pdq φpγq i ¨ χ pdq ď ¨ i¨γ ` ¨ i¨γ α τ ¨ d2 ˛ αn ¨ d ˛ dď n i|α dď n i|α ÿ α ÿ ÿα ÿ ˝ 8 ‚ ˝ ‚ 1 φpγq χ pdq φpγq i ¨ χ pdq ď ¨ i ¨ i¨γ ` ¨ i¨γ α τ ¨ d2 ˛ αn ¨ d ˛ i|α d“1 dď n i|α ÿ ÿ ÿα ÿ ˝ ‚ ˝ ‚ φpγq 1 π2 φpγq i ¨ χ pdq ď ¨ i ¨ 1 ´ ` ¨ i¨γ τα ¨ p2 ˛ 6 αn ¨ d ˛ i|α p|γï dď n i|α ÿ ź ˆ ˙ ÿα ÿ ˝ ‚ ˝ ‚ φpγq 1 π2 log n “ ¨ i ¨ 1 ´ ` O . p1q τα ¨ p 2 ˛ 6 n i|α p|γï ÿ ź ˆ ˙ ˆ ˙ ˝ ‚ γ Now, let Aαpnq denote the quantity of non-deficient numbers that are multi

γ γ ples of α relatively prime to γ up to n. Further, let Dαpnq and Nα pnq denote the quantity of deﬁcient numbers and natural numbers, respectively, with the same

α and γ requirements up to n. produces an expression that estimates the sum 1 σpαvq from below in terms of Aγ pnq. n αv α αvďn pv,γÿq“1

14 1 σpαvq 1 σpαvq “ χ pvq n αv n γ αv αvďn αvďn pv,γÿq“1 ÿ 2 ¨ Aγ pnq ` σpαq ¨ Dγ pnq ě α α α n σpαq Aγ pnq σpαq Aγ pnq ` Dγ pnq “ 2 ´ ¨α ` ¨ α α α n α n ˆ ˙ σpαq Aγ pnq σpαq N γ pnq “ 2 ´ ¨α ` ¨ α α n α n ˆ ˙ τ t n u γ τ n σpαq Aαpnq σpαq “ 2 ´ ¨ ` ¨ ¨ χγ piq ` χγ piq˛ α n αn ˆ ˙ i“1 i“τtn u`1 ˚iÿPαN ÿτ ‹ ˚ iPαN ‹ ˝ ‚ γt n u γ τ n σpαq Aαpnq σpαq “ 2 ´ ¨ ` ¨ ¨ χγ piq ` χγ piq˛ α n αn ˆ ˙ i“1 i“τ n `1 ˚ ÿ ÿtτ u ‹ ˚ iPαN ‹ ˝ ‚ γ n 2α ´ σpαq Aαpnq σpαq n “ ¨ ` ¨ ¨φpγq ` χγ piq˛ α n αn τ ˆ ˙ i“τ n `1 ˚ Y] ÿtτ u ‹ ˚ iPαN ‹ ˝ n´τ n ‚ γ t τ u 2α ´ σpαq Aαpnq σpαq n n n “ ¨ ` ¨ φpγq ¨ ´ ` χγ i ` τ α n αn ¨ τ τ τ ˛ i“1 ˆ ˙ ´ ! )¯ iPα ´ Y ]¯ ˚ ÿN ‹ ˝ n´τ n ‚ γ t τ u 2α ´ σpαq Aαpnq σpαq n n “ ¨ ` ¨ φpγq ¨ ´ ` χγ piq α n αn ¨ τ τ ˛ i“1 ˆ ˙ ´ ! )¯ iPα ˚ ÿN ‹ 2α ´ σpαq Aγ pnq σpαq ¨ φ˝pγq ‚ “ ¨α ` α n τα ˆ ˙ n n´τ t τ u σpαq ¨ φpγq n σpαq ` ´ ¨ ` ¨ χγ piq ¨ αn τ αn ˛ i“1 !) iPα ˚ ÿN ‹ ˝ ‚ O 1 p n q looooooooooooooooooooooooooooomooooooooooooooooooooooooooooon2α ´ σpαq Aγ pnq σpαq ¨ φpγq 1 “ ¨ α ` ` O . p2q α n τα n ˆ ˙ ˆ ˙

15 Using p1q and p2q, we have that

Aγ pn q φpγq 1 π2 log n α ď ¨ i ¨ 1 ´ ´ σpαq ` O , n τ p2α ´ σpαqq »¨ p2 ˛6 fi n i|α p|γï ÿ ź ˆ ˙ ˆ ˙ and hence for sufficiently large–˝ n ‚ fl

φpγq 1 π2 ¨ i ¨ 1 ´ ´ σpαq p3q τ p2α ´ σpαqq »¨ p2 ˛6 fi i|α p|γï ÿ ź ˆ ˙ –˝ ‚ Aγ pflnq Apnq will serve as an upper bound on the contribution made by α to . This n n result is better than Behrend’s findings[3] that

Aγ pnq σ pαq 1 φpγq π2 1 α ď ¨ ¨ ¨ 1 ´ ´ 1 . p4q n 2α ´ σ p α q α γ ¨ 6 p2 ˛ p|γ ź ˆ ˙ ˝ 1 ‚π2 qφpγ Inspection of p3q yields that in the case where i ¨ 1 ´ ¨ ą 2α, p2 6 τ i|α p|γ¨i ˆ ˙ is a better bound than p3q. This bound is a trivialÿ bound ź since it is arrived at by

γ assuming that every element in the set Aαpnq is non-deﬁcient.

1 π2 Proposition 2.1. If i ¨ 1 ´ ¨ ą 2α and α is non-deﬁcient, then p2 6 i|α p|γ¨i ˆ ˙ φpγq ÿ ź bound is a better estimate than p3q. p5q τ

Proof. Let τ be a square-free product of primes, τ “ αγ, 1 π2 i ¨ 1 ´ ¨ ą 2α and consider the following calculation. p2 6 i|α p|γï ˆ ˙ ÿ ź φpγq 1 π2 φpγq ¨ i ¨ 1 ´ ´ σpαq ą ¨ 2α ´ σpαq τ p2α ´ σp α qq »¨ p2 ˛6 fi τ p2α ´ σpαqq i|α p|γï ˆ ˙ ÿ ź “‰ –˝ ‚ fl φpγq “ . τ

16 Note that the trivial bound must be used is if α is a perfect number. Notice that the

quantity 2α ´ σpαq is a factor of the denominator in p3q. This bound is undefined φpγq for the case that σpαq “ 2α and hence the trivial bound will be used if α is τ a perfect number. Another justification of the use of this bound is the fact that if α is a perfect number then every multiple of α will be non-deficient and every

γ γ element in the set Aαpnq will contribute to the value of Aαpnq. The trivial bound will also be used in case that α is abundant. If α is abundant,

then every multiple of α will be abundant likewise. Thus every element in the set

γ γ Aαpnq will contribute to the value of Aαpnq.

2.3 Calculating the Density of Non-Deﬁcients

Apnq Consider the density of non-deﬁcient numbers not exceeding n, , which can be n estimated using a few prime numbers in the following fashion. Let τ be a square-

free product of prime numbers, α a divisor of τ and α ¨ γ “ τ, then I will show that Apnq Aγ pnq “ α . n n α|τ ÿ

γ and use their implied cardinality to show this result. Recall that Aαpnq denotes the set of non-deﬁcient numbers that are multiples of α, relatively prime to γ, and less than or equal to n with τ a square-free product of prime numbers and τ “ αγ.

Further, recall that Apnq is the set of non-deﬁcient numbers up to and including n.

γ γ First, I will show that Apnq “ Aαpnq. Let a P Aαpnq, then we have that α|τ α|τ a is a non-deﬁcient number that isď a multiple of α andď is relatively prime to γ, for some α, γ such that α|τ, αγ “ τ. Further a, is less than or equal to n. Hence a P Apnq.

Now suppose a P Apnq, then a is a non-deﬁcient natural number. Choose τ α “ pa, τq and γ “ . By our choices for α and γ it must be the case that pa, τq τ a P αN, α|τ, γ|τ and pa, γq “ 1. Further α ¨ γ “ pa, τq ¨ “ τ. Now a P pa, τq γ γ Aαpnq Ă Aαpnq. α|τ ď γ γ γ Thus we have that Aαpnq “ Aαpnq and Apnq “ |Apnq| “ Aαpnq . It α|τ α|τ γi ď γj ˇ ď ˇ remains to show that A pn q and A pnq are disjoint for distinct ˇαi and αj.ˇ To αi αj ˇ ˇ show this suppose to the contrary that r Aγi n Aγj n . Then r has the P αi p q X αj p q following four properties:

1. r “ lαi where l P N, 3. pr, γiq “ 1,

2. r “ kαj where k P N, 4. pr, γj q “ 1.

Since r “ lαi and τ “ αiγi “ αj γj , we have the following statement about r:

τ αj γj r “ lαi “ l ¨ “ l ¨ . γi γi

Now since αj γj “ τ, αj γj must be a square-free product of prime numbers. Also, since γi|τ , γi must be a square-free product of prime numbers. Lastly since γi|αj γj

and pαj , γj q “ 1, we have that γi “ pαj , γiqpγj , γiq. This allows us to express r as follows: α γ α γ r “ l ¨ j j “ l ¨j ¨ j . pαj , γiqpγj , γiq pαj , γiq pγj , γiq

18 γj Clearly divides γj and from the above statement it also divides r. Since pγi, γj q γj pr, γj q “ 1, it must be the case that “ 1. That is, pγi, γj q “ γj . A similar pγi, γj q calculation yields that pγj , γiq “ γi and by the uniqueness of the GCD, γi “ γj , which contradicts α α . Hence Aγi n Aγj n for all i j. i ‰ j αi p q X αj p q “ H ‰

In this section I will further investigate the condition required for the use of the

trivial bound, present alternate forms of this condition, clarify the diﬀerence be

tween Behrend’s bound and my bound, and present the ﬁnal form on the bound Aγ pnq that I will use to bound the contributing α . n The following calculation takes the condition for using the trivial bound and uses the fact that α, a divisor of τ , is a square-free product of prime numbers.

1 π2 1 1 π2 i ¨ 1 ´ ¨ “ i ¨ 1 ´ ¨ 1 ´ ¨ p2 6 p2 p2 6 i|α p|γ¨i i|α p|i p|γ ÿ ź ˆ ˙ ÿ ź ˆ ˙ ź ˆ ˙ 1 1 1 1 π2 “ i ¨ 1 ´ 1 ` ¨ 1 ´ 1 ` ¨ p p p p 6 i|α p|i p|γ ÿ ź ˆ ˙ ˆ ˙ź ˆ ˙ ˆ ˙ σpiq φpγq σpγq π2 “ φpiq ¨ ¨ ¨ ¨ i γ γ 6 i|α ÿ φp γ q ¨ σ p γ q φ p i q ¨ σ p i q π2 “ ¨ ¨ . γ2 i 6 i|α ÿ

19 Thus we have an equivalent condition for using the trivial bound,

φpi ¨ γq ¨ σpi ¨ γq 12 ą . i ¨ γ ¨ τ π2 i|α ÿ This is of particular interest because as a consequence of the calculation of this

new condition, I was able to see a valuable piece of information that led me to be able to quantify, in general, the diﬀerence between Behrend’s ﬁndings and my own

ﬁndings.

Before I discuss this diﬀerence, I will present an alternate form for the sum φpiq ¨ σpiq φpnq ¨ σpnq . Deﬁne fpnq “ and 1pnq “ 1 for n P . The func i n N i|α ÿtion f is a multiplicative function since it is a product and quotient of mul

tiplicative functions and clearly the constant function 1 is as well. Now con

sider the function F pnq deﬁned by the Dirichlet product of f and 1. Notice

that F pnq “ pf ˚ 1qpnq “ fpaq1pbq and by the deﬁnition of ˚ we can write ab“n ÿ φpiq ¨ σpiq F pnq “ pf ˚ 1qpnq “ fpiq1pn/iq “ . Now we have that f ˚ 1 is a i i|n i|n multiplicative functionÿ being the Dirichletÿ product of two multiplicative functions. k k ei ei Hence F pnq is a multiplicative function. So F p i “ F pp i q for distinct ˜i ¸ i prime numbers p1, . . . , pk and natural numbers e1ź, . . . , ek. Lastly,ź since α is a divi sor of τ which is square-free product of prime numbers, we have that

φpiq ¨ σpiq φpiq ¨ σpiq 1 “ “ p ` 1 ´ . p6q i i p i|α p|α i|p p|α ÿ ź ÿ ź ˆ ˙

On a small tangent, this presents another equivalent condition for using using the trivial bound, φpγq ¨ σpγq 1 12 ¨ p ` 1 ´ ą . γ ¨ τ p π2 p|α ź ˆ ˙

20 Amending the bound p3q from section 2.2 with p6q, we have the following expression Aγ pnq to bound each of the contributing α , n

φpγq 1 1 π2 ¨ p ` 1 ´ 1 ´ ´ σpαq . τ p2α ´ σpαqq »¨ p p2 ˛6 ﬁ p|α p|γ ź ˆ ˙ź ˆ ˙ –˝ ‚ ﬂ In order to compare this with Behrend’s bound p4q from section 2.2, we will factor

out σpαq. Since σ is a multiplicative function and since α is a square-free product of prime numbers, we have that

Aγ pnq φpγq ¨ σpαq 1 1 π2 α ď ¨ 1 ´ 1 ´ ´ 1 (7) n τ p2α ´ σpαqq »¨ ppp ` 1q p2 ˛6 fi p|α p|γ ź ˆ ˙ź ˆ ˙ –˝ ‚ fl In this form the difference between Behrend’s and my bounds is clear. It is the 1 presence of the expression 1 ´ in my calculation of the contribut ppp ` 1q p|α ź ˆ ˙ Aγ pnq 1 ing α and the fact that 1 ´ ď 1 for all α. n ppp ` 1q p|α ź ˆ ˙

21 Chapter 3

Results and Interpretations

In section 2.2, it was stated that if one were to use a set of initial primes, one could estimate from above the cardinality of the set of non-deﬁcient numbers up to n, Apnq, denoted by Apnq. It follows that this result can be used to produce an Apnq estimate of the density of non-deﬁcient numbers up to n,. n

3.1 Initial Bounds: Behrend’s Result

Behrend used the prime numbers 2 and 3 to prove that the quantity of non-deﬁcient

numbers below n is less than the quantity of deﬁcient numbers. Behrend examined 1 σpvq 1 σp2vq 1 σp3vq the sums ¨ , ¨ and ¨ and found the following: n v n 2v n 3v vďn 2vďn 3vďn pv,ÿ6q“1 pv,ÿ3q“1 pv,ÿ2q“1

A6pnq π2 1 log n 1 ă ´ ` O, n 27 3 n ˆ ˙ A3pnq 10π2 log n 2 ă ´ 1 ` O, n 81 n ˆ ˙ A2pnq 11π2 1 log n 3 ă ´ ` O, n 288 3 n ˆ ˙

22 A1pnq 1 6 ď , n 6 1 A1pnq where is used to bound 6 since every multiple of 6 is abundant. 6 n

This yields the result: Apnq A6pnq ` A3pnq ` A2pnq ` A1pnq “ 1 2 3 6 n n 1 10 11 1 1 1 log n ă ` ` ¨ π2 ´ ´ 1 ´ ` ` O 27 81 288 3 3 6 n ˆ ˙ ˆ ˙ 515 3 log n “ ¨ π2 ´ ` O, 2592 2 n ˆ ˙ Apnq log n ă 0.461 ` O. n n ˆ ˙

Apnq Hence for sufficiently large n, ă 0.462. With this information on the density n of non-deficient numbers, Behrend was able to assert that the quantity of deficient numbers is greater than that of the non-deficient numbers.

Extending Behrend’s process to include the ﬁrst three prime numbers and using the bounds developed in section 2.2 yields the following estimates for the contribut Aγ pnq ing α for suﬃciently large n. n

A30 pnq 32π2 4 A15 pnq 64π2 4 1 ď ´ 2 ď ´ n 1125 15 n 675 5 A10pnq 11π2 4 A6pnq 29π2 1 3 ď ´ 5 ď ´ n 375 15 n 2700 10 A2 pnq 319π2 2 A3 pnq 1 15 ď ´ 10 ď n 21600 15 n 15 A5 pnq 2 A1 pnq 1 6 ď 30 ď . n 15 n 30 Note that for α “ 6, 10, 30 the trivial bound is used in accordance with the discus sion at the end of section 2.2. Thus, harnessing the method asserted in section 2.2,

23 Apnq 3847π2 4 we have a bound lower than Behrend’s, ď ´ “ 0.42447. n 21600 3

3.2 An Illustration of the Bounding Method

γ If we deﬁne Aα to be the set of non-deﬁcient numbers that are multiples of α and

γ relatively prime to γ as well as Dα for deﬁcient numbers, the following diagram illustrates the partitioning of the natural numbers that accompanies this method

of estimating the density of non-deﬁcient numbers using the initial primes 2 and 3.

This diagram also alludes to the fact that this method can be extended to include

more primes, a tactic that is used in this thesis. It should be noted that the union

of the sets horizontally across the diagram is N.

N ÖŒ A D

ÖÓ ÓŒ

2 1 2 1 A1 A2 D1 D2 ÖÓÓŒ ÖÓÓŒ

6 2 3 1 6 2 3 1 A1 A3 A2 A6 D 1 D3 D2 D6 ......

Figure 3.1: Partition Scheme

24 This partitioning scheme also extends to subsets of the natural numbers con tained in r1, ns. In this case, the union of the sets horizontally is Npnq. Further, if one takes the cardinality of every element in the diagram above intersected with n, we have the following diagram. It should be noted that summing the cardinalities horizontally across the diagram results in a value of n.

|Npnq| Ö Œ

Apnq Dpnq

Ö Ó Ó Œ

2 1 2 1 A1 pnq A 2pnq D 1pnq D2pnq ÖÓÓŒ ÖÓ ÓŒ

6 2 3 1 6 2 3 1 A 1pnq A 3pnq A 2pnq A 6pnq D 1pnq D 3pnq D2pnq D6 pnq ......

Figure 3.2: Cardinality Partition Scheme up to n

3.3 The Reﬁning Process

From the density calculation method depicted in section 3.2, the inclusion of an

other distinct prime number ρ in the factorization of τ produces the set equivalence

γ ργ γ Aαpnq “ Aα pnq Y Aραpnq. However, the results shown in section 3.1 suggest that the inclusion of another prime number yields a lower upper bound on the density Aργpnq Aγ pnq Aγ pnq of abundant numbers. Mathematically that is α `ρα ď α . This n n n inequality deals with the cardinalities of the aforementioned sets.

25 Aργpnq I will show the claimed reﬁnement of the density by verifying inequality α ` n Aγ pnq Aγ pnq Aγ pnq ρα ď α generally. If we let T pα, γq denote the the trivial bound for α n n n and Npα, γq denote the non-trivial bound, the following list itemizes the conditions that must be met in order to assert that the inclusion of another prime number ρ

will produce a better, lower bound.

Table 3.1: Conditions for improvement

1. T pρα, γq ` T pα, ργq ď T pα, γq

2. Npρα, γq ` T pα, ργq ď T pα, γq

3. T pρα, γq ` Npα, ργq ď T pα, γq

4. Npρα, γq ` Npα, ργq ď T pα, γq

5. T pρα, γq ` T pα, ργq ď Npα, γq

6. Npρα, γq ` T pα, ργq ď Npα, γq

7. T pρα, γq ` Npα, ργq ď Npα, γq

8. Npρα, γq ` Npα, ργq ď Npα, γq

It was discovered that only a few key conditions from the list above required

rigour to be veriﬁed. Recall the condition for the use of the trivial bound:

φpγqσpγq 1 12 p ` 1 ´ ą γτ p π2 p|α ź ˆ ˙ Aγ pnq If this is met, the trivial bound must be used to bound α . The following n examines how a particular set requiring the trivial bound implies the necessity of

the trivial bound for another set. To do this I will investigate the trivial bound

condition for each set.

26 φpγqσpγq 1 Let Cpα, γq “ p ` 1 ´ . Now consider Cpρα, γq and Cpα, ργq. γτ p p|α ź ˆ ˙

φpγqσpγq 1 Cpρα, γq “ p ` 1 ´ γρτ p p|ρα ˆ ˙ 1 ź ρ ` 1 ´ φpγqσpγq 1 “ ρ ¨ p ` 1 ´ ρ γτ p p|α ź ˆ ˙ 1 1 φpγqσpγq 1 “ 1 ` ´ ¨ p ` 1 ´ ρ ρ2 γτ p p|α ˆ ˙ ź ˆ ˙ φpγqσpγq 1 ą p ` 1 ´ γτ p p|α ź ˆ ˙ “ Cpα, γq.

φpργqσpργq 1 Cpα, ργq “ p ` 1 ´ ργρτ p p|α ź ˆ ˙ φpρqσpρq φpγqσpγq 1 “ ¨ p ` 1 ´ ρ2 γτ p p|α ź ˆ ˙ ρ2 ´ 1 φpγqσpγq 1 “ ¨ p ` 1 ´ ρ2 γτ p p|α ź ˆ ˙ φpγqσpγq 1 ă p ` 1 ´ γτ p p|α ź ˆ ˙ “ Cpα, γq.

The above calculations yield the inequality for all ρ P P:

Cpα, ργq ă Cpα, γq ă Cpρα, γq.

This inequality provides the following information which in turn reduces the num ber of conditions that must be checked to assert the result.

27 Condition reducing information:

12 12 12 1. If Cpα, ργq ą then Cpα, γq ą and Cpρα, γq ą . π2 π2 π2 12 12 2. If Cpα, γq ą then Cpρα, γq ą . π2 π2

The revised list of conditions is provided below.

Table 3.2: Revised conditions for improvement

1. T pρα, γq ` T pα, ργq ď T pα, γq

3. T pρα, γq ` Npα, ργq ď T pα, γq

7. T pρα, γq ` Npα, ργq ď Npα, γq

8. Npρα, γq ` Npα, ργq ď Npα, γq

There is another consideration that also reduces the number of conditions to check. This deals with the deﬁnition of T pα, γq and Npα, γq. By their deﬁnition, when a calculation calls for the use Npα, γq to bound a particular set, we have that

Npα, γq ă T pα, γq. This last consideration provides the ﬁnal list of conditions that must be proven to assert the desired result.

Table 3.3: Final conditions for improvement

1. T pρα, γq ` T pα, ργq ď T pα, γq

7. T pρα, γq ` Npα, ργq ď Npα, γq

28 Aργ pnq Aγ pnq Aγ pnq Proposition 3.1. α `ρα ď α where ρ P and ρ τ. n n n P f γ Proof. Recall that Aαpnq denotes the set of non-deﬁcient numbers that are multiples of α, relatively prime to γ, and less than or equal to n with τ a square-free product

of prime numbers and τ “ αγ with T pα, γq and Npα, γq the trivial and non-trivial Aγ pnq Aγ pnq φpγq bound for α , respectively. Recall that the trivial bound for α is and n n τ

φp γ q ¨ σ p α q 1 1 π2 ¨ 1 ´ 1 ´ ´ 1 the non-trivial. τ p2α ´ σpαqq »¨ ppp ` 1q p2 ˛6 ﬁ p|α p|γ ź ˆ ˙ź ˆ ˙ –˝ ‚ ﬂ Condition 1. T pρα, γq ` T pα, ργq ď T pα, γq. φpγq φpργq T pρα, γq ` T pα, ργq “ ` ρτ ρτ

φpγq 1 φpρq “ ` τ ρ ρ ˆ ˙ φpγq 1 ρ ´ 1 “ ` τ ρ ρ ˆ ˙ φpγq “ τ “ T pα, γq

Condition 7. T pρα, γq ` Npα, ργq ď Npα, γq. Aγ pnq For this condition, recall that the use of T pρα, γq to bound ρα implies that n 12 1 1 Cpρα, γq ą . Also, we will need the fact that Cpρα, γq “ 1 ` ´ Cpα, γq π2 ρ ρ2 ˆ ˙ σpαq φpγqσpγq 1 and Cpα, γq “ ¨ 1 ´ . α γ2 ppp ` 1q p|α ź ˆ ˙

T pρα, γq ` Npα, ργq

φpγq φpργqσpαq 1 1 π2 “ ` 1 ´ 1 ´ ´ 1 ρτ ρτp2α ´ σpαqq » ppp ` 1q p2 6 fi p|α p|ργ ź ˆ ˙ź ˆ ˙ – fl φpγq φpργqσpαq φpργqσpργq 1 π2 “ ` 1 ´ ´ 1 ρτ ρτp2α ´ σpαqq » ρ2γ2 ppp ` 1q 6 fi p|α ź ˆ ˙ – fl 29

φpγq φpργqσpαq φpρqσpρq φpγqσpγq 1 π2 “ ` ¨ 1 ´ ´ 1 ρτ ρτp2α ´ σpαqq » ρ2 γ2 ppp ` 1q 6 fi p|α ź ˆ ˙ φpγq φpργqσpαq –φpρqσpρq αCpα, γq π2 fl “ ` ¨ ¨ ´ 1 ρτ ρτp2α ´ σpαqq ρ2 σpαq 6 „ j φpγqσpαq 2α ´ σpαq φpρq φpρqσpρq αCpα, γq π2 “ ` ¨ ¨ ´ 1 τp2α ´ σpαqq ρσpαq ρ ρ2 σpαq 6 „ „ jj φpγqσpαq 2α ´ σpαq 1 1 αCpα, γq π2 “ ` 1 ´ 1 ´ ¨ ¨ ´ 1 τp2α ´ σpαqq ρσpαq ρ ρ2 σpαq 6 „ ˆ ˙ „ˆ ˙ jj φpγqσpαq 2α ´ σpαq 1 1 1 αCpα, γq π2 1 “ ` 1 ´ ´ ` ¨ ¨ ´ 1 ` τp2α ´ σpαqq ρσpαq ρ2 ρ ρ3 σpαq 6 ρ „ ˆ ˙ j φpγqσpαq 2α 1 1 1 1 αCpα, γq π2 1 “ ´ ` 1 ´ ´ ` ¨ ¨ ´ 1 ` τp2α ´ σpαqq ρσpαq ρ ρ2 ρ ρ3 σpαq 6 ρ „ ˆ ˙ j φpγqσpαq 2α 1 1 1 αCpα, γq π2 “ ` 1 ´ ´ ` ¨ ¨ ´ 1 τp2α ´ σpαqq ρσpαq ρ2 ρ ρ3 σpαq 6 „ ˆ ˙ j φpγqσpαq 2α 1 1 1 αCpα, γq π2 αCpα, γq π2 “ ´ ` ´ ¨ ¨ ` ¨ ´ 1 τp2α ´ σpαqq ρσpαq ρ ρ2 ρ3 σpαq 6 σpαq 6 „ ˆ ˙ j φpγqσpαq 2α α 1 1 π2 αCpα, γq π2 “ ´ 1 ` ´ ¨ Cpα, γq ¨ ` ¨ ´ 1 τp2α ´ σpαqq ρσpαq ρσpαq ρ ρ2 6 σpαq 6 „ ˆ ˙ j φpγqσpαq 2α α π2 αCpα, γq π2 “ ´ ¨ Cpρα, γq ¨ ` ¨ ´ 1 τp2α ´ σpαqq ρσpαq ρσpαq 6 σpαq 6 „ j φpγqσpαq 2α α 12 π2 αCpα, γq π2 ă ´ ¨ ¨ ` ¨ ´ 1 τp2α ´ σpαqq ρσpαq ρσpαq π2 6 σpαq 6 „ j φpγqσpαq αCpα, γq π2 “ ¨ ´ 1 τp2α ´ σpαqq σpαq 6 „ j φpγqσpαq α σpαq φpγqσpγq 1 π2 “ ¨ ¨ 1 ´ ¨ ´ 1 τp2α ´ σpαqq »σpαq α γ2 ppp ` 1q 6 fi p|α ź ˆ ˙ – fl φpγqσpαq φpγqσpγq 1 π2 “ 1 ´ ¨ ´ 1 τp2α ´ σpαqq » γ2 ppp ` 1q 6 fi p|α ź ˆ ˙ “ Npα, γq – fl

Aργ pn q Aγ pnq Aγ pnq Thus, it must be the case that α `ρα ď α . n n n

30 Chapter 4

The Schnirelmann Density of

Deﬁcient Numbers

As conjectured in section 2.1, the Schnirelmann density of deﬁcient numbers could be found by observing that there is a local minimum value for the density up to n in an initial interval and developing a lower bound over the rest of the natural numbers.

Harnessing the bounding method presented in section 2.2, I attempted to calculate an appropriate upper bound on the density of non-deﬁcient numbers. The following table shows the results for τ being the product of the ﬁrst n prime numbers. These upper bounds were calculated using the program from the Appendix B.1.

Table 4.1: τ Speciﬁc Densities of Non-deﬁcient Numbers

n Upper Bound Calc. Time (sec) n Upper Bound Calc. Time (sec)

2 0.4609746399 0.000073 10 0.3832637915 0.003826

3 0.4244614876 0.000102 20 0.3773657121 7.708861

5 0.4011362415 0.000412 28 0.3760421118 2581.383063

31 4.1 Improvements on Bound and Execution Time

As table 4.1 suggests, the computing time becomes unmanageable for my purpose.

The number of computations this program requires is O pσ0pτ qq. This program calculates a contribution for each divisor of τ, α. Hence it will perform Op2nq calculations where n is the number of prime numbers in the factorization of τ. This is an issue since for my purpose, I need τ to contain enough prime numbers to produce a calculated upper bound of the density of non-deﬁcient numbers below

0.254386. To remedy this, I used a method extracted from Marc Del´eglise’s paper

[6] to reduce the number calculations required. The following chain of inequalities validates the use of this method. Recall that αγ “ τ. We will also need the result

32 Thus we have an alternate means of calculating an upper bound. Note that choosing

z ă τ will reduce the number of computations required to arrive at an upper bound. Aγ pnq Improvements on the bound of the contributing α were made by adapting n methods used in Dr. Mitsuo Kobayashi’s dissertation [7]. Dr. Kobayashi found Aγ pnq that the contributing α could be bounded in the following fashion. n

t 1

γ T q T t Aαpnq φpγq φpτq ¨ ¨ ˜qPJt ¸ pUT `1q ˛˛ ď ´ 1 ´ ÿ ` exp pUT `1q ´ n τ α ¨ τ ˚ ˚ t! t! ‹ ‹ ˚ ˚t “1 t“0 ‹‹ ˚ ˚ÿ ÿ ‹‹ ˚ ˚ ‹‹ ˚ ˚ ‹‹ ˝ ˝ ‚‚ where

logp3.222q 1 4 Ut “ log 1 ` ` ` log p3.222t´1c´ 1q 10 log2 p3.222tc´ 2q 15 log3 p3.222tc´ 1q ˆ ˙ 1 4 ` ` , 10 log2 p3.222t´1c´ 1q 15 log3 p3.222tc´ 1q and

´1 t ´1 t`1 ´1 ´1 log p3 expp6q ¨ c q Jt “ 3.222 c , 3.222 c , c “ max tp P P : p|τu, T “ . logp4q Z ^ ` ‰

The following tables show the results for τ being the product of the ﬁrst n prime numbers and using parameter z to bound the divisors of τ used to bound the density of non-deﬁcient numbers. These upper bounds were calculated using the program from the Appendix B.2. Table 4.2 demonstrates the impact the expedition process has had on the calculations performed in table 4.1 using parameter z “ 1014 . Table 4.3 demonstrates the impact of using Dr. Kobayashi’s improvements on the general

bound while attempting to follow an optimal path for the calculation by increasing

the number of primes used and increasing parameter z.

33 Table 4.2: Improved τ Speciﬁc Densities of Non-deﬁcient Numbers

n z Upper Bound Calc. Time (sec)

2 1014 0.4609746398770 0.007355778625465

3 1014 0.4244614875459 0.006928833108640

5 1014 0.4011362414247 0.007718682314767

10 1014 0.3832637914969 0.037821214922731

20 1014 0.3773657120793 16.944343957160942

28 1014 0.3760421140075 286.6183579182794

Table 4.3: Further Improved τ Speciﬁc Densities of Non-deﬁcient Numbers

n z Upper Bound Calc. Time (sec)

50 105 0.37055959193772 16.887428226360317

60 106 0.36064377664238 3.846549836470841

70 107 0.35552564747666 11.130322497482508

80 108 0.35284032771044 42.307565773944283

90 109 0.3513271084730 190.5851330709040

100 1010 0.3507389251057 963.2794605980319

110 1010 0.350626937241 1075.741798397919

120 1010 0.350541878258 1280.549511752465

130 1011 0.350095409772 6054.742540200833

Apnq The goal of section 4.1 was to develop an upper-bound for the sequence , n which we will denote K. The desired result is that K ă 0.25439.

34 4.2 Second-Order Terms and Further Research

Suppose that the value of K was calculated, then the Schnirelmann density of

deﬁcient numbers would be found by located the inﬁmum in the set

Dpnq 8 Dpnq N0 Dpnq 8 “ Y n n n "*1 "*1 "*N0 Dpnq In the above expression, the values for are calculated for the interval r1,N s n 0 and the values for the interval rN , 8q would be lower-bounded by 1 ´ K. Since we 0 Dpnq N0 Dpnq have min “ 0.74561 ... , ě 1 ´ K for n ě N and since we suspect n n 0 "*1 K ă 0.25438 ... , we would have δpDq “ 0.74561 ... . Further work would have to be done to assert the value of K.

As mentioned above, given K, we have that for large enough n, the density of

non-deﬁcient numbers up to n is bounded above by K. This upper bound on the

density of non-deficient numbers up to n is equivalent to the lower bound 1 ´ K on the density of deficient numbers up to n since the non-deficient numbers and

deficient numbers partition the natural numbers. So there exists an N0 P N such that if n ě N0 the lower bound 1 ´ K on the density of deficient numbers is guaranteed to be accurate. This line of reasoning yields two questions. The first question is “What is the value of N0?” and the second question is “For n ă N0, does the bound still hold?” The source of this potential error arises from the general Aγ pnq bound developed for the contributing α . Recall from section 2.2 that n

Aγ pnq φpγq 1 π2 log n α ď ¨ i ¨ 1 ´ ´ σpαq ` O n τ p2α ´ σpαqq »¨ p2 ˛6 fi n i|α p|γï ÿ ź ˆ ˙ ˆ ˙ –˝ ‚ fl for all α that require a non-trivial bound and from section 2.3 that

Apnq Aγ pnq “ α . n n α|τ ÿ

35 It follows that the terms responsible for this potential error are encapsulated in the log n expression O which goes to 0 in the limit. This error was accounted for n α|τ ˆ ˙ by Behrend[2].ÿ Behrend calculates his error by evaluating

1 φpγq i ¨ χ pdq ¨ i¨γ n »2α ´ σpαq ¨ d ˛ﬁ α|τ dďn i|α αÿRA ÿ ÿ – ˝ ‚ﬂ and then using the approximation

γ n χ pdq χ pdq φpγq γ ă γ ` plog n ` 1q . d d γ d“1 ˜d“1 ¸ ÿ ÿ

log n If we reveal the expressions from the contributing α’s that O conceals in n ˆ ˙ our calculation and label them Epα, nq, then

1 φpγq i ¨ χ pdq Epα, nq “ ¨ ¨ i¨γ n 2α ´ σpαq » ¨ d ˛ﬁ dď n i|α ÿα ÿ – ˝ ‚ﬂn´τ t n u 1 σpαq ¨ φpγq n 1 σpαq τ ` ¨ ¨ ´ ¨ ¨ χ piq . n 2α ´ σpαq τ n 2α ´ σpαq γ i“1 !) iÿPαN

36 The following calculation bounds Epα, nq from above.

1 φpγq i ¨ χ pdq Epα, nq “ ¨ ¨ i¨ γ n 2α ´ σpαq » ¨ d ˛fi dď n i|α ÿα ÿ – ˝ ‚fln´ τ t n u 1 σpαq ¨ φpγq n 1 σpαq τ ` ¨ ¨ ´ ¨ ¨ χ piq n 2α ´ σpαq τ n 2α ´ σpαq γ i“1 !) iPα ÿN 1 φpγq χ pd q 1 σp α q ¨ φ p γ q n ď ¨ i ¨ i¨γ ` ¨ ¨ n 2α ´ σpαq » d fi n 2α ´ σpαq τ i|α dď n ÿ ÿα ! ) – fl 1 φpγq φpi ¨ γq n 1 ď ¨ i ¨ ¨ log ` γc ` n 2α ´ σpαq » i ¨ γ α n fi i|α ÿ ˆ ´ ¯ ˙ 1 σpαq ¨ φpγq – n fl ` ¨ ¨ n 2α ´ σpαq τ 1 φpγq α! ¨ φ)pγq n 1 ď ¨ ¨ ¨ log ` γ ` n 2α ´ σpαq γ α c n ˆ ˙ 1 σpαq ¨ φpγq 1 ´ ¯ ` ¨ ¨ 1 ´ n 2α ´ σpαq τ ˆ ˙ α ¨ φ2pγq log n γ α ¨ φ2pγq “ ¨ α `c ¨ γ ¨ p2α ´ σpαqq n n γ ¨ p2α ´ σpαqq ` ˘ 1 σp α q ¨ φ p γ q 1 1 α ¨ φ2pγq ` ¨ ¨ 1 ´ ` ¨ . n 2α ´ σpαq τ n2 γ ¨ p2α ´ σpαqq ˆ ˙ Apnq Using this formula to bound the error in the calculation of for τ “ 2 ¨ 3 n 49 logpnq 8.03 4.09 yields Epα, nq ă ¨ ` ` . This result is better than the error 12 n n n2 α|6 ÿ 49 logpnq 20.25 bound Behrend calculated, Epα, nq ă ¨ ` , using his method for 12 n n α|6 τ “ 2 ¨ 3 and Dickson’s tablesÿ [2]. Further research will include a discussion on how

well-behaved the error term Epα, nq is for n ą N0. α|τ ÿ

37 Bibliography

[1] D’Ooge (tr.), Nicomachus, Introduction to Arithmetic, New York, 1926.

[2] F. Behrend, Uber¨ Numeri Abundantes, Sitzungsberichte Akad. Berlin, 1932.

[3] F. Behrend, Uber¨ Numeri Abundantes, II, Sitzungsberichte Akad. Berlin, 1933.

[4] J. De Koninck and F. Luca, Analytic Number Theory: Exploring the Anatomy

of Integers, American Mathematical Society, Providence, 2012.

[5] L. E. Dickson, History of the Theory of Numbers. Vol. I: Divisibility and Pri

mality, Chelsea Publishing Co., New York, 1966.

[6] M. Del´eglise, Bounds for the Density of Abundant Integers, Experimental

Mathematics 7 (1998), no. 2, 137-143.

[7] M. Kobayashi, On the Density of Abundant Numbers. Diss. Dartmouth,

Hanover, 2010 Print.

[8] T. L. Heath (tr.), Euclid, The Elements, New York, 1956.

38 Appendix A

Deﬁcient Data

Table A.1: Density of Deﬁcient Numbers up to n

n Density n Density n Density n Density n Density

1 1 11 0.909090909 21 0.80952381 31 0.774193548 41 0.780487805

2 1 12 0.833333333 22 0.818181818 32 0.78125 42 0.761904762

3 1 13 0.846153846 23 0.826086957 33 0.787878788 43 0.76744186

4 1 14 0.857142857 24 0.791666667 34 0.794117647 44 0.772727273

5 1 15 0.866666667 25 0.8 35 0.8 45 0.777777778

6 0.8333334 16 0.875 26 0.807692308 36 0.777777778 46 0.782608696

7 0.8571429 17 0.882352941 27 0.814814815 37 0.783783784 47 0.787234043

8 0.875 18 0.833333333 28 0.785714286 38 0.789473684 48 0.770833333

9 0.8888889 19 0.842105263 29 0.793103448 39 0.794871795 49 0.775510204

10 0.9 20 0.8 30 0.766666667 40 0.775 50 0.78

39 Table A.2: Density of Deﬁcient Numbers up to n continued n Density n Density n Density n Density

51 0.784313725 71 0.774647887 91 0.758241758 111 0.756756757

52 0.788461538 72 0.763888889 92 0.760869565 112 0.75

53 0.79245283 73 0.767123288 93 0.76344086 113 0.752212389

54 0.777777778 74 0.77027027 94 0.765957447 114 0.745614035

55 0.781818182 75 0.773333333 95 0.768421053 115 0.747826087

56 0.767857143 76 0.776315789 96 0.760416667 116 0.75

57 0.771929825 77 0.779220779 97 0.762886598 117 0.752136752

58 0.775862069 78 0.769230769 98 0.765306122 118 0.754237288

59 0.779661017 79 0.772151899 99 0.767676768 119 0.756302521

60 0.766666667 80 0.7625 100 0.76 120 0.75

61 0.770491803 81 0.765432099 101 0.762376238 121 0.752066116

62 0.774193548 82 0.768292683 102 0.754901961 122 0.754098361

63 0.777777778 83 0.771084337 103 0.757281553 123 0.756097561

64 0.78125 84 0.761904762 104 0.75 124 0.758064516

65 0.784615385 85 0.764705882 105 0.752380952 125 0.76

66 0.772727273 86 0.76744186 106 0.754716981 126 0.753968254

67 0.776119403 87 0.770114943 107 0.757009346 127 0.755905512

68 0.779411765 88 0.761363636 108 0.75 128 0.7578125

69 0.782608696 89 0.764044944 109 0.752293578 129 0.759689922

70 0.771428571 90 0.755555556 110 0.754545455 130 0.761538462

40 Appendix B

Programs

B.1 Program 1 pcount=input(’Input number of primes: ’); e=zeros(1,pcount+1); p=primes(1987); format long; tic; h=1; tau=1; phitau=1; sigmatau=1; while h<=pcount;

tau=tau*p(h);

phitau=phitau*(p(h)-1);

sigmatau=sigmatau*(p(h)+1);

h=h+1; end k=1;nddensity=0;hault=0; while hault<=pcount;

j=1;alpha=1;phialpha=1;sigmaalpha=1;

while j<=pcount;

41 alpha=alpha*p(j)^e(j);

if e(j)==1

phialpha=phialpha*(p(j)-1);

end

if e(j)==1

sigmaalpha=sigmaalpha*(p(j)+1);

end

j=j+1; end gamma=tau/alpha;phigamma=phitau/phialpha; sigmagamma=sigmatau/sigmaalpha;bcontri=phigamma/tau; if 2*alpha-sigmaalpha~=0

produ=pi^2/6*phigamma*sigmagamma/gamma^2;

z=1;

while z

if e(z)==1

produ=produ*(p(z)+1-1/p(z));

end

z=z+1;

end

if (2*alpha-sigmaalpha)>0

ntbound=phigamma/tau/(2*alpha-sigmaalpha)

*(produ-sigmaalpha);

if ntbound<(phigamma/tau)

bcontri=ntbound;

42 end end end

i=1;

while i

if e(i)==1

e(i)=0; i=i+1;

end

if e(i)==0

e(i)=e(i)+1;break;

end end

k=k+1;nddensity=nddensity+bcontri;ec=0;stop=0;

while ec

stop=e(ec+1)+stop;

ec=ec+1;

stopcrit=[ec,stop];

if hault==pcount

hault=hault+1;

end

if stop==pcount

hault=pcount; end end end disp(nddensity);toc

43 B.2 Program 2

global ptt stt ab pcount p nndensity num pcount=input(’Input number of primes: ’); spower=input(’Input alpha bound: 10^’); p=primes(10000000); nddensity=1; best=[0,0,1,0]; upab=0; while nddensity>0.25438597 format long; tic; h=1;ptt=1;stt=1;

while h<=pcount;

ptt=ptt*(1-1/p(h));

stt=stt*(1+1/p(h));

h=h+1;

end

pmax=p(pcount); pbound=3000000; epsin=pmax; eps=1/pmax;

Jr=[];

T=floor(log(pbound/pmax)/log(3.222));

a=primes(3.222^T*pmax);

num=0;

for b=1:T

Jt=a((a>=3.222^(b-1)*pmax&a<=3.222^b*pmax));

for zz=1:length(Jt)

Jr=[Jr,1/Jt(zz)];

end

num=num+(sum(Jr))^b/factorial(b);

44 Jr=[];

end

num=num+exp(UofBT(T+1,epsin));

for c=0:T

num=num-UofBT(T+1,epsin)^c/factorial(c);

end

nndensity=0; ab=10^(spower);

nndensity=inc4(1,1,1,1,1,1,1);

nddensity=1+nndensity*ptt;

if (best(3)<=nddensity)&&(upab<=1)

spower=spower+1;

upab=upab+1;

else

result=[(pcount),(spower),nddensity,toc]

if (nddensity

best=result;

end

upab=0;

pcount=pcount+10;

end function [Uval] = UofBT(t,epsin)

Uval=log(1+log(3.222)/log(3.222^(t-1)*epsin))

+1/10/(log(3.222^t*epsin))^2+4/15/(log(3.222^(t)*epsin))^3

+1/10/(log(3.222^(t-1)*epsin))^2

+4/15/(log(3.222^(t-1)*epsin))^3;

45 end

function [nndensity] =inc4(alpha,asa,k,produ,apa,opa,oa) global ptt stt ab pcount p nndensity num

if asa>0.5

ntbound=(((stt*ptt)*asa*apa*(pi^2/6*produ)-1)/(2*asa-1)-1)

*opa;

madj=oa*(-1+num);

if madj

ntbound=madj;

end

if ntbound<0

nndensity=nndensity+ntbound;

end end

if k<=pcount

for i=k:pcount

if(alpha*p(i)/ab<=1)

nndensity=inc4(alpha*p(i),asa/(1+1/p(i)),i+1,

produ*(1-1/(p(i)*(p(i)+1))),apa*p(i)/(p(i)-1),

opa/(p(i)-1),oa/p(i));

else

break; end end end end