THE UNLIKELY UNION OF PARTITIONS AND DIVISORS


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1 THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative umber theory we decompose a atural umber ito prime factors = p p 2 p 3... p k ad cosider the cosequeces. I the additive theory we decompose a atural umber ito a sum of elemets from some set. For eample we could try to epress as a sum of squares. I [3], ad [4], the authors treated the properties of the partitio fuctio, which is a good eample of additive umber theory. A good eample of fuctios studied i multiplicative theory is the divisor fuctio σ ( ). σ ( ) is defied as the sum of the positive divisors of. For eample the divisors of 6 are, 2, 3, ad 6. Thus the sum of the divisors of 6 is σ ( 6) = = 2. Now divisors of umbers are related to primes, ad primes seem urelated to partitios. We are ot surprised that partitios satisfy a recursio relatio. We do ot epect σ ( ) to satisfy a recursio relatio. What do the divisors of have to do with the divisors of, 2,...? Yet Euler showed that σ ( ) satisfies the same recursio relatio as does p( ), the partitio of.(see []) Oly σ ( ) is differet from p(). Euler was astoished at this result, ad you ca read a traslatio of his ow words i Polya [5] ad i Youg [8]. There are eve relatios marryig the two fuctios such as (Schroeder [6]) p ( ) = σ ( k) p ( k). k = I this paper we will eamie this ad other properties of σ ( ) ad p().
2 . BASIC PROPERTIES OF THE DIVISOR FUNCTION We use the covetio that σ () =. It is clear that σ ( p) = p + for ay prime umber p, sice the oly positive divisors of p are ad p. Also the oly divisors of 2 p are, p ad 2 p. 2 2 Thus σ ( p ) = p + p+ ad little bit of algebra shows that this ca be epressed as 2 2 σ ( p ) = p + p+ = 3 p. It should ow easy to prove p Lemma. If p is a prime ad k is a oegative iteger, the p k k k p + σ ( p ) = p + p+ = The followig is a stadard theorem i Number Theory tetbooks. (See [9]). Lemma 2. σ ( ) is multiplicative, that is, if m ad are relatively prime( they have o commo divisor other tha ), the σ ( m) = σ( ) σ( m). A immediate cosequece of Lemma ad 2 is the followig formula that ca be used to evaluate the sum of the divisors of a give iteger. k k2 k m Propositio. If = p p2 p m, where p, p2,, p m are primes ad k, k2,, k m are oegative itegers, the k+ k2+ km + k k2 k p p2 p m m 2 m p p2 pm σ( ) = σ( p p p ) = Propositio has a misleadig simplicity. Suppose we take 23 = 2 +. What are the prime factors of? I other words, if is a large umber whose prime factors caot easily be obtaied, the formula becomes less useful. It is i this sese that we ow look at the recurrece relatios that σ ( ) satisfies.
3 2. RECURENCE RELATIONS INVOLVING THE DIVISOR FUNCTION Let p( ) be the partitio of, that is, the umber of ways we ca write as a sum of positive itegers. We defie p ( ) = if. Let k(3k ) f( k) =. Euler proved the 2 followig recurrece relatio for p( ). { )} k + (.) ( ) ( p( ) = ( ) p f( k) + p f( k) k = The mai tool that Euler used to prove this formula was the fuctio give by the ifiite product properties: g ( ) = ( ). He showed that this fuctio has the followig two remarkable = ( ) ( ) (A) ( ) ( f k f k g= ) = ( ) (B) = = = g ( ) k = = ( ) = p( ) For more o this, see [3]. We ow defie the socalled Lambert series ad look at the relatio of this series with partitio ad divisor fuctios. Let { a ( )} be a sequece of real umber. The Lambert s series associated with{ a ( )} is defied by La ( ) = a( ) = This Lambert series is Taylor series epasio give by where L ( ) = A( ), a = A ( ) = ad ( ) (the summatio is take over the positive divisors of.) Note the that if A ( ) = σ ( ). Thus we have the followig result. d a ( ) = the
4 Lemma 3. = σ ( ) = = For more iterestig properties ad the may other applicatios of Lambert series the reader is hereby ivited to idulge i the classic book of Kopp [2]. We are ow i a positio to prove our first recurrece relatio that coects σ ( ) ad p( ). Let us rewrite the ifiite ( ) f ( k) f ( k) product of Euler as g ( ) = = ( ) = e. Theorem. (.2) = σ ( ) = ke p( k) k= = The sum of divisor ad partitio fuctio are related by the followig formula k = k Proof: If we take the logarithmic differetiatio of g ( ) =, ad multiply the resultig equatio by, we get (.3) O the other had, g ( ) = e = g'( ) = g ( ) implies that = = ( = g'( ) = e. We ow recall that k = p( ). Usig Lemma 3 ad substitutig these last two series for g ( ) = g '( ) ad i (.3), we get g ( ) (.4) k p( ) e = = σ ( ) = = = = Multiplyig out the two series o the left side of (.4)ad comparig coefficiets proves the theorem. I the same lie of reasoig we ca show the followig recurrece. Theorem 2. The divisor fuctio satisfies the recurrece relatio (.5) σ( ) = e e kσ ( k) Proof: We have oted that = k = = g '( ) = σ ( ). Multiply both sides by g() to get g ( ) g'( ) = g( ) σ ( ). Now use the fact that g ( ) = e ad g'( ) = e to get e = e σ ( ). Sice e =, the theorem follows from comparig = = = coefficiets of the two power series. = ) =
5 Theorem 3. The partitio fuctio ad the divisor fuctio are also related by the formula (.6) Proof: (.7) p( ) = σ ( k) p( k) k = Defie F( ) = = = p( ). The g ( ) = ( g ( )) 2 ( ) = g'( ) g'( ) g'( ) F'( ) = = F( ) g ( ) g ( ) = g ( ) O the other had, we have F'( ) = p( ) = g'( ) g ( ) = σ. Also F( ) p( ) implies that = ( ). Substitutig all these i (.7), we get (.8) ( ) p = σ ( ) p( ) = = = Multiply (.8) by to get ( ) p = σ ( ) p( ) = =. = Multiplyig the ifiite series o the right ad compare coefficiet to get the formula i the theorem. Remark: [3] cotais a combiatorial proof of Theorem 3. He first shows by a combiatorial argumet that m m= k= = p( ) = m p( km). Covertig the double sum ito a sigle sum the gives the formula i Theorem QBASIC PROGRAM FOR THE SUM OF DIVISORS Formula (.) ca be rewritte as p ( ) = p ( ) + p ( 2) p ( 5) p ( 7) p ( 2) + p ( 5) p ( 22) p ( 26) + = The authors used this recurrece formula i [3] to write a simple QBASIC program t costruct a table for the values of p(). We ca use the same program with a slight modificatio to geerate a table of values for σ ( ). Here is the modified program of [Osler ] that uses formula (.2). Program : Calculate Partitios ad Sum of Divisor
6 'Calculate partitios of N, P()=p(N), ad the sum of divisors DN ( ) = σ ( N) 'eactly up to P(3) ad D(3). 'Set double precisio, dimesio array P ad D, iitialize P ad D 'Mai loop, for each N fid P(N) ad D(N). 9 CLS DEFDBL AZ DIM P(4) DIM D(4) 2 P() = 2 D() = INPUT "ENTER A POSITIVE INTEGER", J 'Mai loop, for each N fid P(N)ad D(N) 2 FOR N = TO J 2 SIGN = 25 P(N) = 26 D(N) = 22 FOR K = TO N 'Calculate two terms i recursio relatio for P(N)ad D(N) 23 F = K * (3 * K  ) / 2 'Eit loop if argumet egative 24 IF N  F < THEN GOTO 4 25 P(N) = P(N) + SIGN * P(N  F) 25 D(N) = D(N) + F * SIGN * P(N  F) 26 F = K * (3 * K + ) / 2 'Eit loop if argumet egative 27 IF N  F < THEN GOTO 4 28 P(N) = P(N) + SIGN * P(N  F) 28 D(N) = D(N) + F * SIGN * P(N  F) 29 SIGN = SIGN 3 NEXT K 'Prit results 4 PRINT N, P(N), D(N) 'Pause after pritig 2 lies o the scree 45 IF 2 * INT(N / 2) = N THEN INPUT A$: CLS 5 NEXT N REFERENCES
7 . Grosswald, Emil, Topics from the Theory of Numbers, MacMilla Co., N. Y., 966, p Kopp, K. The Theory ad Applicatios of Ifiite Series. 3. Hasse, Abdul ad Osler, Thomas, Playig With Partitios O The Computer Mathematics ad Computer Educatio. Vol. 35 No., page 57 (Witer 2) 4. Chadrupatla, T R; Hasse, Abdul; ad Osler, Thomas, A Table of Partitio Fuctio. Mathematical Spectrum, Vol. 34 No 3, page (2/22) 5. Polya, G., Mathematics ad Plausible Reasoig, Vol., Iductio ad Aalogy i Mathematics, Priceto Uiversity Press, Priceto, NJ, 954, pp Schroeder, M. H., Number Theory i Sciece ad Commuicatio, (Secod Ed.), SprigerVerlag, New York, 986, p Specer, Doald D., Eplorig Number Theory with Microcomputers, Camelot Publishig, Orlado Beach, Florida, Youg, Robert M., Ecursios i Calculus, A Iterplay of the Cotiuous ad the Discrete, Mathematical Associatio of America, 992, pp Rose, Keeth, Elemetary Number Theory ad Its Applicatios, 4th ed., Addiso Wesley Pub. Co.
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