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INNOVATIVE ARITHMETICAL FUNCTIONS: INFINITE PRODUCTS -------Version 2 By Garimella Rama Murthy, Associate Professor ABSTRACT Various known arithmetical functions are briefly reviewed. Several novel arithmetical functions are defined. Motivated by the Euler product, several new infinite products in unknown variable “x” are defined. These infinite products are related to power series type functions associated with novel arithmetical functions. Also some important constants associated with the sequence of primes/integers are defined. 1. Introduction: Ever since the dawn of civilization, natural numbers stimulated the curiosity of homo-sapiens. Geometric investigations led to the discovery of hidden relationships among natural numbers. For instance, the Pythagorean Theorem led to an interesting Diophantine equation. Books 7,8,9 of Euclid’s elements were devoted to number theoretic investigations [3]. Investigations on relationship between divisors of an integer and the integer itself resulted in the definition of perfect numbers (A number which is equal to the sum of its proper divisors e.g 6, 28). Thus mathematicians defined an interesting function, an arithmetical function: Sum of proper divisors of an integer. This arithmetical function is related to the specification of so called amicable numbers. These are a pair of numbers with the following property. The sum of all proper divisors of the first number exactly equals the second number while the sum of all proper divisors of the second like wise equals the first. For instance, it can easily be verified that 220 and 284 are amicable numbers. Prime numbers were also subjected to serious investigations from the period of Euclid (who provided an elegant proof that they are infinite). Efforts to utilize “analysis” techniques in number theoretic investigations led to the field of analytic number theory. Efforts of many mathematicians focused on one important arithmetical function: Number of prime numbers until an integer ‘n’ i.e. (n). This arithmetical function is associated with the celebrated prime number theorem. Also Euler discovered the famous Totient function: the number of integers relatively prime to “n”. He also studied the partition function p(n): Number of ways in which an integer “n” can be written as the sum of other integers. Mathematicians like Hardy, Lambert, Ramanujan et. al introduced and studied the properties of the following arithmetical functions (using the techniques of analytic number theory) [1, 2] (n) : Number of divisors of an integer “n” s (n) : Sum of ‘s’th powers of divisors of an integer “n”. In this research paper, we define new arithmetical functions. It is shown how these functions naturally arise in connection with some infinite products. These results are summarized in Sections 2 and 3. The technical report concludes in Section 4. 2. Novel Arithmetical Functions: Consider the “unique” factorization of an arbitrary integer. Define the following functions related to the “prime” divisors occurring in the unique factorization. (a) (n) : Number of distinct prime divisors occurring in the unique factorization of “n”. This arithmetical function was studied extensively in the field of probabilistic number theory. For instance Ramanujan studied this function in his famous paper with Hardy [5] (the normal number of prime factors of a number n ). (b) (n) : Sum of distinct prime divisors occurring in the unique factorization of “n”. © k (n) : Sum of “k” th powers of prime divisors of n. It should be noted that 1 ( n) ( n ) . (D) (n) : Product of exponents of primes (distinct) occurring in the unique factorization of ‘n’. This arithmetical function is vaguely related to the n number of divisors of an integer “N”. Let N = p1n1 p 2n2 .... p l l . It is easy to see that (n) = (1 n1 )(1 n2 )....(1 nl ) whereas (n) n1 n2 ...nl (E) (n) : Product of primes occurring in the unique factorization of “n” (F) Consider the unique factorization of an integer, N i.e. n n n N = p1 1 p 2 2 .... p l l . ……………………(2.1) Define a novel arithmetical function, g(N) as n n n g(N) = p1 1 p 2 2 .... p l l . (G) Consider the above unique factorization of integer N, Define a new arithmetical function, (N ) as ( N ) n1 n2 ... nl This function can be interpreted as the total number of divisors based on distinct prime divisors. (H) Consider the above unique factorization of an integer, N. Define a novel arithmetical function, h(N) as h(N) = n1 p1 n2 p 2 ... nl pl Remark: It is well accepted that Mobius function, Euler Totient function, Mangoldt function have deeper significance in number theory. Also, as in the case of (n) , we expect that detailed investigation of the above arithmetical functions will be of considerable utility. Thus, we expect large number of research papers will be written discussing the properties of the above arithmetical functions. Remark: It should be kept in mind that a large number of arithmetical functions can be easily defined. For instance, in [4], several novel arithmetical functions are studied. Some of these functions have deeper significance in number theory, other mathematical disciplines and even in physics. In the following, we define some more arithmetical functions. (I) Consider the primes until an integer “n”. Define function as the sum of primes until n i.e. say A(n). a new arithmetical (J) Sum of say “s” th power of primes until n i.e. B(n). (K) Consider the following arithmetical functions: (i) Sum of pairwise products of primes, (ii) Sum of products of primes (THREE at a time) and so on. Remark: One of the approaches to define new arithmetical function is to generalize the basis on which existing arithmetical function is defined. For instance the arithmetical functions studied in [4] are generalizations of Euler’s totient function (N ). Specifically, we consider Given an integer “N”, the number of integers below N which share 1, 2, …,i,…,m prime divisors with N., say F (i ) ( N ). It is easy to see that N = (N ) L F (i ) ( N ) ….The sum is a finite sum and denotes j 1 the number of integers which share atleast one prime divisor with N. Similarly, given integer N, the number of integers below N which share 1,2,… composite divisors with N. (L) Given an integer n, the number of integers below “n” which are relatively prime to each other (note that these integers are “LIKE” prime numbers) leads to the definition of a new arithmetical function, say J(n). It is clear that J(n) (n ) . It should be possible to derive tigher bounds using the results in [4]. It is easily noted that the celebrated prime number theorem says that ( n) n . log n Thus a similar Theorem on the asymptotic order of J(n) could be stated and proved. In this design the following notation: connection, we (M) Consider the unique factorization of an integer, N. Define a new arithmetical function, M(N) as the total number of single prime based divisors (divisor monomial in primes contains single prime) of an integer. It is easy to see that M(N) = ( n1 n2 ....nl ) . In a similar manner, 2,3, ….prime divisor based arithmetical functions can be defined and studied. The definitions can consider the “number”, “Sum”, “sum of “s”th powers of divisors (i.e. divisor monomials are based on 2,3,…prime divisors). The goal of investigations in this technical report is to associate functions (power series) related to these arithmetical functions. These functions are connected with some infinite products. Also the eventual goal is to study the properties of these novel arithmetical functions. In this direction we briefly study the multiplicativity of some of the arithmetical functions. We expect that well developed techniques of analytic number theory are of considerable utility. 3. Infinite Products: Novel Arithmetical Functions: The main motivation for the investigations in this technical report is the following Euler Product: n 1 1 nk 1 (1 p pP k 1 ...) , p 2k where P is the set of all prime numbers. Now utilizing the fundamental Theorem of Arithmetic (i.e the unique factorization of any integer using the prime numbers), the following identities are derived. It is clear that the identities hold true from the point of view Of SYMBOLIC ALGEBRA. Using standard arguments from analysis, it can easily be shown that most of the identities hold true from the point of view of convergence. Detailed analytic arguments are avoided for the sake of brevity. LEMMA A: Consider the following infinite product in unknown variable ‘x’ x x x ( n ) ( 1 k 2 k ...) = 1 k , …………………(1) p p n n 1 pP where P is the set of prime numbers and (n) is the number of distinct prime divisors of “n”. Proof: The Left Hand Side (LHS) of the above equation, (1) becomes L.H.S = pP x / pk (1 ) 1 1 k p Using the unique factorization of any integer, it is easy to see that the above equality holds true from the point of view of symbolic algebra. Now we consider its truth from the convergence view point. Typical Convergence Argument: Consider the power series on the right hand of equation (1). Upper bound this power series by the following power series i.e. (1 n 1 x ( n ) ) < (1 + nk x n )………(1.5) n 1 It is easy to see that the power series defined on RHS of (1.5) converges absolutely on the interval (-1,1). Thus, invoking standard Theorem from real analysis, it can be seen that the power series on the right hand side of (1) converges. Q.E.D. LEMMA B. Now let us consider another infinite product in unknown variable ‘x’ (1 pP xp xp x ( n ) ...) = , 1 pk p 2k nk n 1 ………………..(2) where P is the set of prime numbers and (n ) is the sum of prime divisors on ‘n’. The Left Hand Side (LHS) of the above equation, (2) becomes L.H.S = (1 pP x p / pk ) 1 1 k p Typical Convergence Argument: Consider the power series on the right hand of equation (2). Upper bound this power series by the following power series i.e. (1 n 1 x ( n) ) < (1 + nk 2 x n )………(2.5) n 1 It is easy to see that the power series defined on RHS of (2.5) converges absolutely on the interval (-1,1). It represents a theta function. Thus, invoking standard Theorem from real analysis, it can be seen that the power series on the right hand side of (2) converges. Q.E.D. LEMMA C: Generalizing the above logical approach, we have the following infinite product in unknown variable ‘x’. pP x (n) xp xp , ( 1 k 2 k ...) = 1 nk p p n 1 l l l Where P is the set of prime numbers and l (n) of prime divisors of “n”. ……..(3) is the sum of “l”th power Note: Equation (2) is a special case of equation (3). The (LHS) of the above equation, (3) becomes L.H.S = pP Left Hand Side l x p / pk (1 ) 1 1 k p LEMMA D: Another interesting infinite product in unknown “x” is given by pP x x2 x ( n ) ( 1 k 2 k ...) = 1 ,………..(4) p p nk n 1 l Where n p1n1 p2n2 ......... pl l k and (n) n1 n2 ..... nl i.e sum of exponents occurring in unique n factorization. The Left Hand Side (LHS) of the above equation,(4) becomes L.H.S = (1 pP x / pk ) x 1 k p Remark: From symbolic algebra point of view (based on the factorization of an integer), the equations (1), (2), (3), (4) hold true. unique LEMMA E. The most general form of identity is given by (1 pP x h( p) x h( p) x ( n ) ...) = ,………….(5) 1 pk p 2k nk n 1 Where (n) h( p1 ) h( p 2 ) ..... h( pl ) . h(.) is an arbitrary function With n p1n1 p2n2 ......... pl The Left Hand Side (LHS) of the above equation, (5) becomes nl L.H.S = (1 pP x h( p) / p k ). 1 1 k p LEMMA F. Consider the following infinite product in unknown variable ‘x’ (n) x ( n ) x 2x 3x = , …………………(6) . ..) 1 p k p 2k p 3k nk n 1 pP where P is the set of prime numbers, (n) is the product of exponents of primes occurring in the unique factorization of ‘n’ and (n) is the number of (1 distinct prime divisors of “n”. The most general term becomes (n1 x)( n 2 x)...( nl x) p1n1 p 2n2 k ... p lnl l = in the infinite product on the LHS of (6) ( n) x ( n ) nk Thus the identity in equation (6) is readily obtained. LEMMA G. Consider the following infinite product in unknown variable ‘x (n) x ( n ) x p 2x p 3 x p ( 1 k 2 k 3k ......) = 1 …………(7) p p p nk pP n 1 Where the arithmetical functions (n) , (n) are as defined previously. The most general term in the infinite product on the LHS of (7) becomes (n1 x p1 )( n2 x p2 )...( nl x pl ) p1n1k p 2n2 k ... plnl k ( n) x ( n ) = nk Thus the identity in equation (7) is readily obtained. LEMMA H. Using similar argument as in (7), the following identity follows: pP 3xp ( n) x xp 2x p ( 1 k 2 k 3k ...) = 1 nk p p p n 1 l l l l (n) The arithmetical functions arising in the above identity are as defined previously. LEMMA I. Consider the following infinite product in unknown variable ‘x’: pP ( n) x x 2 x 2 3x 3 ( 1 k 2 k 3k ...) = 1 p p p nk n 1 l (n) . By considering the general term on the left handside, it can be easily Seen that the above identity follows. LEMMA J. Consider the following infinite product in unknown variable ‘x’ (1 pP ( n ) ( n) x ( n ) px 2 px 3 px = . Where (n) is ...) 1 pk p 2k p 3k nk n 1 the product of primes occurring in the unique factorization of “n”. By considering the general term on the LHS, it can easily be seen that the identity follows. LEMMA K. Consider the following infinite product in unknown variable “x” pP px px px ( 1 k 2 k 3k ..) p p p (n) x ( n ) n 1 nk 1 = , Once again by considering the general term on the LHS, it can easily be seen that the identity follows. It is easily seen that along the lines of derivation of identities (B), (C); identities associated with infinite sums, ( n) x ( n ) n 1 nk 1 , 1 n 1 ( n) x l (n) nk are easily derived. Details are avoided for brevity. LEMMA L. Consider the following infinite product in unknown variable “x” pP px p x2 p x3 ( n) x ( n ) ( 1 k 2 k 3k ...) = 1 . p p p nk n 1 By considering the general identity follows. term on the LHS, it can easily be seen that the LEMMA M: Consider the following infinite product in unknown variable “x” 2 pP 3 xp xp xp x g (n) . ( 1 k 2 k 3k ...) = 1 p p p nk n 1 Once again, by considering the general term on the LHS, the identity follows. LEMMA N: Consider the following infinite product in unknown varaiable “x” (1 pP xp x2 p x3 p x h( n) ...) = 1 pk p 2k p 3k nk n 1 By considering an arbitrary term on the LHS, it can be easily seen that the identity follows. It is seen that all the above identities (leading to novel arithmetical functions) are effectively derived based on the unique factorization in the integer ring. By abstracting the essential idea (in all the above identities), the following most general identity is derived. THEOREM O: Consider the following infinite product (1 ( p,1) x K ( p , 1) pk pP ( p, 2) x K ( p , 2) p 2k = 1 + ( ( p , n ) …. ( p , n ) n 1 1 1 l l ( p, 3 ) x K ( p , 3 ) p 3k ...) x K ( p1 ,n1 ) ...... K ( pl ,nl ) ) / n k , Where the integer “n” has the following unique factorization n p1n1 p 2n2 ..... p lnl As in the previous identities, by considering an arbitrary term on LHS, It can be seen that the identity holds true. REMARK: Utilizing the logical procedure adopted in this identity (and thus all previous identities), various new arithmetical identities can be defined and associated with infinite products (in variable x). Detailed enumeration of all possible identities are avoided for brevity. Multiplicativity of Novel Arithmetical Functions: In existing discussions on arithmetical functions, various properties of the function (under consideration) are studied. One of the very basic properties studied is the multiplicative nature of the function [6]. Definition: An arithmetical function f is called multiplicative if identically zero and if f(mn) = f(m) f(n) whenever (m,n) =1 i.e f is not when integers “m” and “n” are relatively prime. A multiplicative function f is called completely multiplicative if we also have f(mn) = f(m) f(n) for all m, n. Now let us consider the arithmetical function: (n) i.e number of distinct prime divisors of “n”. It is easy to see that if integers “m”, “n” are relatively prime, then (m n) = (m) (n) i.e (.) is multiplicative, but not completely multiplicative The counter example follows. Let and N 2 p 2 p3 and N 1 N 2 p1 p 22 p 3 . N1 p1 p2 It is clear that ( N1 ) 2 , ( N 2 ) 2 . But ( N1 N 2 ) 3 ( N1 ) ( N 2 ) . Now for the sake of convenience, consider the arithmetical function, (n ). It is easily seen that this arithmetical function is not multiplicative. For instance, let N1 p1 p2 , N 2 p 3 p 4 ( N1 , N 2 N 1 N 2 p1 p 2 p 3 p 4 . It is clear that are relatively prime) and ( N 1 ) ( p1 p 2 ) , ( N 2 ) ( p 3 p 4 ) . But ( N1 N 2 ) ( N1 ) ( N 2 ) . For the sake of completeness, let us investigate the completeness of arithemetical function (N ). Consider integers m, n. For the sake of concreteness (logic is easily generalized), let m p12 p 23 ; n p 32 p 43 . Clearly (m, n ) =1 i.e relatively prime. (m) p1 p 2 , (n) p 3 p 4 and (mn) (m) (n). Thus (.) is multiplicative. By considering ( m, n) 1 , it is easy to see that (.) is not completely multiplicative. By using the same procedure, it is easy to see that (.) is multiplicative but not completely multiplicative. In the spirit of the above efforts, multiplicativity of other arithmetical functions defined above can easily be determined. Details are avoided for brevity. Some Important Constants Associated with Primes/Integers: In the following numbers. For the sake Integers. It should be defined. But some of sciences. we list some interesting constants associated with prime of completeness, we also list constants associated with kept in mind that whole set of such constants could be them will have applications in physics and other natural Infinite Sequence based Constants: 1 1 1 1 1 3 5 7 ... p ...... = C1 2 2 3 5 7 p 1 1 1 1 1 (2) 1 + 2 2 2 2 ..... 2 ..... = C2 2 3 5 7 p (1) 1 + In the same spirit of (2), we have (3) 1 + 1 1 1 1 1 k k k ..... k ..... = C 3 (k ) for any arbitrary integer ‘k’. k 2 3 5 7 p As in the case of Euler sums, it will be nice if these constants can be specified. In the spirit of Zeta function, we can define a function of complex variable “s”, (s ). (s) 1 + p K 1 , where “K” is the set of ps prime numbers. Study of properties of this function may lead to new results in analytic number theory. (4) 1 + 1 1 1 4 5 ... ………………. = C4 3 2 3 4 (5) 1 + 1 1 1 3 4 ............. = C5 . 2 2 3 4 4. Arithmetical Functions of multiple variables: Traditionally, mathematicians have defined arithmetical functions of single integer variable e.g Euler’s totient function. A question naturally arises as to why we cannot define arithmetical functions of multiple independent variables. In research literature, various arithmetical functions of multiple independent variable are defined and studied. We add to that list in the following: Consider Definition 1: a function of “k” integer variables i.e. say ( n1 , n2 ,...., nk ) . Let the function be the number of primes common/shared in the unique factorization of the integers { ni }li 1 . It should be noted that if the integers l { ni }i 1 are either primes or they are relatively prime, then (.,.,....,.) 0. Also, it easily seen Specifically that (.,.,....,.) is a symmetric function in the arguments/variables. ( n1 , n2 ) ( n2 , n1 ) . In this case consider “ n1 ” and its prime divisors ( n1 , n2 ) ( p ,n i 2 ) where the sum extends over the divisors of n1 . pi In this spirit some trivial arithmetical functions can easily be defined i.e (i) Number of primes between n1 , n2 , say the function be K( n1 , n2 ). (ii) Sum of “s”th powers of primes between n1 , n2 . (iii) Number of relatively prime integers between n1 , n2 (iV) Sum of “l”th degree composite divisors between n1 , n2 . Definition 2: Given integers, { ni }li 1 , let a new arithmetical function be the composite divisors shared by them. 5. Interesting Investigation: The above discussion leads to the definition and study of functions of the following form: n 1 f1 (n) x f 2 ( n ) ………………………………(6) Where f1 (n) , f 2 (n) are arithmetical functions like (n), s (n) ….etc. 6.Conclusions: Several well known arithmetical functions are summarized. Various novel arithmetical functions are defined. Motivated by the Euler product, several infinite products in unknown variable “x” are defined. These infinite products are related to “power series type functions” associated with the novel arithmetical functions. REFERENCES: [1] S. Ramanujan, “On Certain Arithmetical Functions, “ Transactions Cambridge Philosophical Society, “XXII, No. 9, 1916, Pages 159-184 of the [2] R. Kanigel, “The Man Who Knew Infinity: A Life of the Genius Ramanujan, “ Washington Square Press, Published by Pocket Books [3] W.Dunham, “Journey through Genius, The Great Theorems of Mathematics ,” Penguin Books, 1990 [4] A. Sangameshwar and G. Rama Murthy, “Novel Arithmetical Technical Report: IIIT/TR/2004/18 Functions, “ IIIT [5] G. H. Hardy and E.M.Wright, “An Introduction to the Theory of Numbers, “, Oxford (at the Clarendon Press ) [6] Tom M. Apostol, “Introduction to Analytic Number Theory, “Narosa Publishing House, New Delhi, 1989.