Some properties and applications of a new arithmetic function in analytic number theory

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1 NNTDM 17 (2011), 3, Some properties applications of a new arithmetic function in analytic number theory Ramesh Kumar Muthumalai Department of Mathematics, D.G. Vaishnav College Chennai , Tamil Nadu, India ramjan 80@yahoo.com Abstract: We introduce an arithmetic function study some of its properties analogous to Möbius function in analytic number theory. In addition, we derive some simple expressions to connect infinite series infinite products through this new arithmetic function. Moreover, we study applications of this new function to derive simple identities on partition of an integer some special infinite products in terms of multiplicative functions. Keywords: Möbius function, Infinite product, Lambert series, Arithmetic function, Partition, Divisor function. AMS Classification: 11A25; 11K65; 05A17; 11P82; 40A20 1 Introduction Analytic number theory is the branch of number theory which uses real complex analysis to investigate various properties of arithmetic functions prime numbers [1]. There is a variety of truly interesting arithmetic functions such as Euler s totient function, Möbius function, divisor function, etc. Möbius function is a simple multiplicative function that arises in many different places in number theory. It plays an important role in the study of divisibility properties of integers. The classical Möbius function [1, 4, 5] is defined by µ(n) 1, If n 1 ( 1), If n is square free with distinct primes 0, Otherwise (1.1) Some of its fundamental properties are a remarably simple formula for the divisor sum d n µ(d), extended over the positive divisors of n, Möbius inversion formula [1, 4, 5]. There are many generalizations of the Möbius function, as well as analogous functions described by different authors in number theory. The survey of many generalizations of Möbius function in Group theory, Lattice theory, Partially ordered sets, etc., are found in [7]. Infinite products play an important role in many branches of mathematics. In fact, they provide an elegant way of encoding manipulating combinatorial identities. The product expansion of generating function of partition function is a familiar example [2, 3]. In this paper, we introduce an arithmetic function study some of its properties analogous to Möbius function. Further more, we incorporate this new function applying it to infinite products, partition of an integer expressions connecting with divisor functions in analytic number theory. 38

2 2 The new arithmetic function its properties Let us begin by defining the new arithmetic function on positive integers. Definition 2.1. For any two positive integers n p, define an arithmetic function υ p (n) as follow as 1, If n 1 2 r 1, If n p r, r N, p > 1 ( 1), If either p n or p 1 n is υ p (n) square-free with distinct primes (2.1) ( 1) 2 r 1, If n p r m, r N, p m m is square-free with distinct primes 0, Otherwise Remar 2.2. The following results are immediately derived from the definition of υ p (n) Möbius function, 1. If p n n is square free, then υ p (n) µ(n). 2. If n p r m, p m m is square free, then υ p (n) 2 r 1 µ(m). 3. In particular υ 1 (n) µ(n), n N. Theorem 2.3. If p is any prime or p 1, then υ p is multiplicative. Proof. It is clearly true when p 1. Assume that p is any prime. Suppose that m n are any two square-free integers with (m, n) 1 p mn, then by Remar 2.2 υ p (mn) µ(mn). Since µ is multiplicative, υ p is also multiplicative. In case (p r, n) 1 n is square free with distinct primes, then by (2.1) υ p (p r n) 2 r 1 ( 1) υ p (p r )υ p (n). If (p r m, n) 1, p mn m, n are square free with 1, 2 distinct primes, then υ p (p r mn) 2 r 1 ( 1) 1+ 2 υ p (p r m)υ p (n). It is also true for other cases. Hence, υ p is multiplicative. Definition 2.4. For any two positive integers n p, define δ p (n) { 1, If p n p > 1 1, If either p n or p 1 (2.2) The following theorem gives an interesting property of υ p (n) analogous to Möbius function. Theorem 2.5. If p is any prime or p 1 n N, then d n δ p(n/d)υ p (d) { 1, if n 1 0, if n > 1 (2.3) Proof. The formula is true if n 1 or p 1, assume that n > 1 p is any prime. If p n n is square free, then n p 1 p 2... p (p 1, p 2,..., p are distinct primes different from p). Since p n, then δ p (n/d) 1 using Remar 2.2, we find that δ p (n/d)υ p (d) d n d p 1 p 2...p υ p (d) d p 1 p 2...p µ(d) 0 39

3 If n p, then by Definition 2.4, we find that δ p (p i ) 1, δ p (1) 1 using Definition 2.1, we obtain 1 1 δ p (p /d)υ p (d) 1 υ p (p i ) + υ p (p ) 1 2 i d p i1 i1 If n p m, where m is square free integer p m. Let d runs through divisors of p m. The summation can be splitted as follows d p m 1 δ p (p m/d)υ p (d) δ p (p w m/d)υ p (p w d) + δ p (m/d)υ p (p d) w0 d m d m Since p m using (2.1) (2.4), we find that d m µ(d) 1 w1 2 w 1 d m Further d n δ p(n/d)υ p (d) 0, for other values of n. µ(d) d m µ(d) 0 Theorem 2.6. For any prime p or p 1, δ p υ p are Dirichlet inverse to each other, i.e. υ p υp 1 δ p. δ 1 p Proof. If I denotes identity function for all values of n, then using Theorem 2.5 we find that I(n) d n δ p (n/d)υ p (d) (2.4) In the notation of Dirichlet multiplication [1, 5], this becomes Hence, δ p υ p are Dirichlet inverse to each other. I δ p υ p (2.5) The above property of δ p υ p, along with the associative property of Dirichlet multiplication, enables us to give a simple proof of the next theorem. Theorem 2.7. Suppose that f(n) d n δ p(n/d)g(d) if only if g(n) d n υ p(d)f(n/d) for prime p or p 1. Proof. Assume that f(n) d n δ p(n/d)g(d) is true. So that f g δ p. Multiplying by υ p on both sides gives f υ p (g δ p ) υ p g (δ p υ p ) g I g Conversly, multiplication of f υ p g by δ p gives the proof. Theorem 2.8. If f is completely multiplicative, then (δ p f) 1 (n) υ p (n)f(n) for all n 1 (2.6) 40

4 Proof. Given that f is completely multiplicative let g(n) υ p (n)f(n) (g (δ p f))(n) ( n ) ( n ) υ p (d)f(d)δ p f d d d n f(n) ( n ) υ p (d)δ p d d n I(n) Since f(1) 1 I(n) 0 for n > 1. Hence, g (δ p f) 1. The following lemma connects the new arithemtic function υ p (n) infinite products. Lemma 2.9. For any positive integer p > 1, x 2 1 x cos θ 1, 2, 3,... cos θ δ p ()x Proof. The LHS of (2.7) can be splitted as follow as cos θ δ p ()x 1 2 log (1 2xp cos pθ + x 2p ) 2/p 1 2x cos θ + x 2 (2.7),p Now adding subtracting cos pθ xp p following identity [6, pp-52], cos θ x after simplification, we obtain (2.7). cos θ x p cos pθ x p to the RHS of the above equation, using the 1 2 log ( 1 2x cos θ + x 2) Theorem If p is any prime, x 2 1 x cos θ 1, 2, 3,..., then e 2x cos θ (( 1 2x p cos pθ + x 2p) 2/p 1 2x cos θ + x 2 ) υp() (2.8) Proof. Let us consider the following infinite series υ p () log ( 1 2x p cos pθ + x 2p) 2/p 1 2x cos θ + x 2 where all υ p are defined from (2.1). Now using the lemma 2.9, we find that ( ) υ p () cos iθ 2 δ p (i) x i i i1 Rearranging in ascending powers of x, using Theorem 2.5 after simplification, we obtain (2.8). 41

5 Corollary If x 2 1 x cos θ 1, 2, 3,..., then ( e 2x cos θ 1 2x cos θ + x 2) υ 1 () (2.9) Proof. This can be easily found by using (2.7) Theorem 2.5. Remar For any prime p, the following infinite products are obtained as some special cases of Theorem 2.10 Corollary If θ 0 in (2.8), then e x (( ) 1 x p 2/p 1 x ) υp() 2. If θ 0 in (2.9), then we obtain following infinite product which can be rewritten as Lambert series of Möbius function e x (1 x ) υ 1 () 3 Connecting infinite series infinite products through ν p (n) In this section, we derive simple expressions through ν p (n) that connect infinite series infinite products. Theorem 3.1. For x 2 1, x cos θ 1, N P (x, θ) i1 c(i)xi cos iθ, if p is any prime, then (( 1 2x p e 2P (x,θ) cos pθ + x 2p) ) 2/p a() (3.1) 1 2x cos θ + x 2 If p 1, then where e 2P (x,θ) ( 1 2x cos θ + x 2) a() a() d c(d) υ p(/d) /d Proof. Given that P (x, θ) i1 c(i)xi cos iθ, taing logarithm on both sides using (2.9), we obtain 1 ( 1 2x ip cos ipθ + x 2ip) 2/p c(i) υ p () log 2 1 2x i cos iθ + x 2i i1 Rearranging the above, we have 1 c(d) υ ( p(/d) 1 2x p cos pθ + x 2p) 2/p log 2 /d 1 2x cos θ + x 2 d (3.2) (3.3) Assuming a() d /d we easily obtain (3.2) from (2.9). c(d) υp(/d) after simplification, we obtian (3.1). In a similar manner, 42

6 Remar 3.2. Putting θ 0 in (3.1) (3.2), for any prime p, we obtain, (( ) ) 1 x p 2/p a() e P (x,0) (3.4) 1 x if p 1, Corollary 3.3. If a() d e P (x,0) ( ) 1 x a() υp(/d) c(d), then c() δp(/d) /d d a(d) /d Proof. From Theorem 2.7, we find that f() d υ p(/d)g(d). Comparing this with a() d υp(/d) c(d), we have f() a() g(d) dc(d), where a() is function in /d c(d) is function in d. Hence, c() d a(d)dδ p(/d). Now, simplifying this, we obtain the corollary. Example 3.4. We can express cosine function on 0 < x < 1 as follow as (3.5) cos x (1 x 2 ) a(1) (1 x 4 ) a(2) (1 x 6 ) a(3)... (3.6) where c() 22 1 (2 2 1) B.2! 2. It can be found by using (3.2) the following infinite series [6] ( ) 1 log 2(22 1) B 2 x (2 4 1) B 4 x (2 6 1) B 6 x cos x 1.2! 2.4! 3.6! Theorem 3.5. For any prime p or p 1, x 0, x < θ < 2π, if T (θ) t() cos θ then (( 1 2x p e 2T (θ) cos pθ + x 2p) ) 2/p t () (3.7) 1 2x cos θ + x 2 where t () d t(d) υ p(/d) x d /d Proof. Given that T (θ) i1 t(i) cos iθ, using (2.7), we get T (θ) 1 ( t(i) 1 2x ip cos ipθ + x 2ip) 2/p υ 2 x i p () log 1 2x i cos iθ + x 2i i1 After simplification, we obtain (( 1 1 2x p cos pθ + x 2p) ) 2/p log 2 1 2x cos θ + x 2 Setting t () d obtain (3.7). d t(d) υ p (/d) x d /d t(d) υ p(/d) transforming from logarithmic form to exponential form, we x d /d Example 3.6. We can write the infinite series cos x log 2 sin x 2 in [6], x 0 as infinite produts as follow as csc 2 θ (( x p cos pθ + x 2p) 2/p 1 2x cos θ + x 2 where t () 1 υ p(/d) d x d 43 ) t (i) on [0 < θ < 2π] found (3.8)

7 4 Applications of the new arithmetic function ν p Let us start from the following lemmas that connect infinite series containing divisor sum that extends over positive integers with infinite products. Lemma 4.1. For any prime p, x 2 1 x cos θ 1, N, if ϱ (p) v () d v dv δ p (/d v ) α p (x, θ) () x cos θ, then ϱ(p) v e 2αp(x,θ) ( 1 2x p cos pθ + x 2p) 2/p (4.1) 1 2x cos θ + x 2 Proof. Setting a() 1, for 1 v, 2 v, 3 v,... a() 0 for other s then by Corollary 3.3, we find that c() δ p (/d v ) 1 d v δ /d v p (/d v ) 1 ϱ(p) v () (say) d v d v Using Theorem (3.1) setting P (x, θ) α p (x, θ), we obtain (4.1). Lemma 4.2. For x 2 1, x cos θ 1, N, if ρ v () d v dv ρ v() x cos θ, then e 2β(x,θ) 1 2x cos θ + x 2 (4.2) Proof. Setting a() 1, for 1 v, 2 v, 3 v,... a() 0 for other s then by Corollary 3.3, we find that ρ v () d v dv. Using Theorem 3.2, we obtain (4.2). 4.1 Partition of an interger Let us assume α p (x) α p (x, θ) β(x, θ) at θ 0. ϱ(p) v () x β(x) ρ v() x which are special cases of Theorem 4.3. Let p (v) (n) denote number of partition of n into perfect v-th power, for x < 1 v > p (v) (n)x n e β(x) (4.3) Proof. We now that 1 + n0 p (v) (n) 1 n! p (v) (n)x n Putting θ 0 setting β(x, 0) β(x) in Lemma 4.2, we find that e β(x) d ( n e β(x)) (4.4) dx n x x v 1 1 x v (4.5)

8 Equating above equations, we get (4.3). Now, differentiating n times with respect to x setting x 0, we obtain (4.4). Theorem 4.3 clarifies that e β(x) is a generating function for partition of an integer into perfect v-th power in terms of divisor functions. Corollary 4.4. For v, N np (v) (n) n ρ v ()p (v) (n ) (4.6) Proof. Taing log on both sides of (4.3) differentiating with respect to x, ( ) ) p v (n)nx n 1 ρ v (n)nx (1 n 1 + p v (n)x n Simplifying the above expression equating coefficients of x, we obtain (4.6). Theorem 4.5. For v, N p (v) (n) 1 ρ v (1) i ρ v (2) j ρ v (3) h... ρ v (l) (4.7) i!j!h!...! 1 i 2 j 3 h... l where ρ v () d v dv the symbol indicates summation over all solutions in non negative integers of the equation i + 2j + 3h l n. Proof. This is immediatly from (4.3), by exping through the exponential function, simplifying equating respective coefficients of x n. Remar 4.6. The following identity is obtained by putting v 1 in (4.7) p(n) 1 σ1(1)σ i 1(2)σ j 1 h (3)... σ1(l) i!j!h!...! 1 i 2 j 3 h... l The symbol indicates summation over all solutions in non negative integers of the equation i + 2j + 3h l n. Theorem 4.7. If r < 1, β c (r, α) ρ v() cos α β s (r, α) ρ v() sin α, then p (v) (m) 1 2π e βc(r,α) β πr m s (r, α) cos mαdα (4.8) 1 πr m 0 2π Proof. If we tae x re iα r < 1 then β(re iα ) 0 e βc(r,α) β s (r, α) sin mαdα (4.9) ρ v ()r cos α + i ρ v ()r sin α Setting β s (r, α) ρ v()r sin α β c (r, α) ρ v()r cos α, then we have β(re iα ) β c (r, α) + iβ s (r, α). Using this in (4.5) comparing real imaginary parts, we obtain 1 + p (v) (n)r n cos nα e βc(r,α) cos β s (r, α) (4.10) 45

9 p (v) (n)r n sin nα e βc(r,α) sin β s (r, α) (4.11) Multiplying by cos mα sin mα, m 1, 2, 3,..., integrating on the interval (0, 2π). We obtain (4.8) (4.9). Remar 4.8. Now, squaring adding (4.10) (4.11), we have ( 2 ( ) p (v) (n)r n cos nα) + p (v) (n)r n sin nα e 2βc(r,α) Simplifying (4.12) integrating on (0, 2π), we have p (v) (n) 2 r 2n 1 2 2π 2π 0 (4.12) e 2βc(r,α) dα (4.13) We can find similar generating functions identities for various partition functions. For example, if β(x) x ρ() ρ() d,odd d d, then eβ(x) is the generating function for the number of partitions of n (say p(n)) into parts which are odd. Similarly, ρ() d,even d d, then e β(x) generates the number of partitions of n into parts which are even so on. Hence, we find generalized identities for the partition function, as follows n np(n) ρ()p(n ) p(n) 1 ρ i (1)ρ j (2)ρ h (3)... ρ (l) i!j!h!...! 1 i 2 j 3 h... l The symbol indicates summation over all solutions in non negative integers of the equation i + 2j + 3h l n. To find similar generating functions identities for number of partitions of n into parts which are unequal, set p 2, θ 0 replace α 2 (x, 0) by α 2 (x). Hence, we have e α2(x) 1 + x (where α 2 (x) 0 ϱ(2) v () x ϱ(2) v () d dδ 2(/d)). Now, e α2(x) generates the number of partitions of n into parts which are unequal. Similarly, we can find generating functions of other cases such as distinct primes, odd, etc. So, we generalize the following identities n np(n) ϱ (2) v ()p(n ) p(n) 1 ϱ (2) v (1) i ϱ (2) v (2) j ϱ (2) v (3) h... ϱ (2) v (l) i!j!h!...! 1 i 2 j 3 h... l 4.2 Some special infinite products If p is any positive integer, then varying the values of a() by Corollary 3.3, we find some interesting infinite products connecting with divisor sum extended over positive integers. Let us consider for any prime p, (( 1 2x p e 2αp(x,θ) cos pθ + x 2p) ) 2/p a() (4.14) 1 2x cos θ + x 2 46

10 when p 1 ( e 2β(x,θ) 1 2x cos θ + x 2p) a() The values of α(x, θ) β(x, θ) for various a s are listed below: (4.15) 1. If a() 1 s, N α p (x, θ) x cos θ d δ p (/d) d s 1 σ s 1 (x) x cos θ. 2. Let a() 1 s, if is prime a() 0 for other s α p (x, θ) x cos θ x cos θ d prime p d prime p δ p (/d) d s 1 1 d s Let a() φ()/ s, where φ is Euler s totient function α p (x, θ) x cos θ d φ(d) δ p(/d) d s 1 x cos θ d φ(d) d s 1. In particular when s 1, using the identity d n φ(d) n, we have x cos θ 4. Let a() Λ()/ (where Λ is Mangoldt s function) using the identity d n Λ(d) log n, we have log x cos θ. 5. Let a() µ()/ 2 using the identity φ(n)/n d n µ(d), we have d φ() 2 x cos θ. 47

11 6. Let a() µ2 () using the identity n φ() φ(n) d n µ 2 (d), we have φ(d) x cos θ. φ() 7. Let a() µ2 () using the identity 2 v(n) d n µ2 (d) where v(1) 0 v(n) if n p a p a, we have 5 Conclusion 2 v() x cos θ. In conclusion, we note that some properties applications of a new arithmetic function υ p (n) have been developed in this article. Firstly, we have introduced υ p (n) proved some of its properties analogous to Möbius function. Secondly, we have derived some simpler infinite products of exponential function some new expressions to connect infinite series infinite products in terms of υ p (n). Finally, we have developed some applications to generate functions of various partition functions of an integer, simple partition identities special infinite products in terms of multiplicative functions. References [1] Andrews, G. E. Number Theory, W. B. Saunders Co., Philadelphia, [2] Andrews, G. E, The Theory Partitions, Cambridge University Press, UK, [3] Andrews, G. E, Erisson K, Integer Partitions, Cambridge University Press, UK, [4] Apostal, T. M. Introduction to Analytic Number Theory, Springer International Student Edition, New Yor (1989). [5] Bateman, P. T, H. G. Diamond. Analytic Number Theory, World Scientific Publishing Co Ltd, USA, [6] Gradshteyn, I. S, I. M. Ryzhi. Tables of Integrals, Series Products, 6 Ed, Academic Press (2000), USA [7] Sándor, J., B. Crstici. Hboo of Number Theory II, Springer, Kluwer Acedemic Publishers, Netherl,