On the Dirichlet Convolution of Completely Additive Functions

Transcription

1 Journal of Integer Sequences, Vol , Article On the Dirichlet Convolution of Copletely Additive Functions Isao Kiuchi and Makoto Minaide Departent of Matheatical Sciences Yaaguchi University Yaaguchi Japan kiuchi@yaaguchi-u.ac.p inaide@yaaguchi-u.ac.p Abstract Let k and l be non-negative integers. For two copletely additive functions f and g, we consider various identities for the Dirichlet convolution of the kth powers of f and the lth powers of g. Furtherore, we derive soe asyptotic forulas for sus of convolutions on the natural logariths. 1 Stateents of results Let f and g be two arithetical functions that are copletely additive. That is, these functions satisfy fn = f+fn and gn = g+gn for all positive integers and n. We shall consider the arithetical function D k,l n;f,g := n f k dg l, 1 d which represents the Dirichlet convolution of the kth power of f and the lth power of g for non-negative integers k and l. The above function provides a certain generalization of the classical nuber-of-divisors function dn. In fact, D 0,0 n;f,g = dn. 1

2 The first purpose of this study is to investigate soe recurrence forulas for D k,l n;f,g with respect to k and l. Since where f is a copletely additive function, we have fd = 1 dnfn, D 1,1 n;f,g = 1 dnfngn fdgd. 3 Siilarly, as in 3, we use 1 for D k,l+1 n;f,g to obtain D k,l+1 n;f,g = n n f k dg l g d d = gn n f k dg l d f k dgdg l n d. Hence, we deduce the following two recurrence forulas. Theore 1. Let k and l be non-negative integers, and let f and g be copletely additive functions. Then we have D k,l+1 n;f,g+ n f k dg l gd = gnd k,l n;f,g, 4 d D k+1,l n;f,g+ n n f k df g d l = fnd k,l n;f,g. 5 d Now, we put f = g in 4 or 5, and set D k,l n;f := D k,l n;f,f. Then, we deduce the following corollary. Corollary. Using the sae notation given above, we have Particularly, if k = l, we have D k,l+1 n;f+d k+1,l n;f = fnd k,l n;f. 6 D k+1,k n;f = D k,k+1 n;f = 1 fnd k,kn;f. 7 Because the syetric property D k,l n;f = D l,k n;f, we only consider the function D k,k+ n,f for = 1,,...

3 Exaple 3. The forulas 6 and 7 iply that D k,k+ n;f = 1 f nd k,k n;f D k+1,k+1 n;f, D k,k+3 n;f = 1 f3 nd k,k n;f 3 fnd k+1,k+1n;f, D k,k+4 n;f = 1 f4 nd k,k n;f f nd k+1,k+1 n;f+d k+,k+ n;f, D k,k+5 n;f = 1 f5 nd k,k n;f 5 f3 nd k+1,k+1 n;f+ 5 fnd k+,k+n;f, D k,k+6 n;f = 1 f6 nd k,k n;f 3f 4 nd k+1,k+1 n;f+ 9 f nd k+,k+ n;f D k+3,k+3 n;f, D k,k+7 n;f = 1 f7 nd k,k n;f 7 f5 nd k+1,k+1 n;f+7f 3 nd k+,k+ n;f 7 fnd k+3,k+3n;f, D k,k+8 n;f = 1 f8 nd k,k n;f 4f 6 nd k+1,k+1 n;f+10f 4 nd k+,k+ n;f 8f nd k+3,k+3 n;f+d k+4,k+4 n;f. Next, we shall deonstrate that the explicit evaluation of the function D k,k+ n;f =,3,... can be expressed as a cobination of the functions D k,k n;f, D k+1,k+1 n;f, D k+,k+ n;f,..., D k+,k+ n;f. Hence, we shall give a recurrence forula between D k,k n;f,..., D k+,k+ n;f and D k,k+ n;f. Theore 4. Let k and be positive integers, and let D k,k+ n;f be the function defined by the above forula. Then we have D k,k+ n;f = c k, f nd k+,k+ n;f, 8 where c k, = 1, if = 0;, if = 1; 1 1 +i, if!. i=1 Proof. By 7 in Corollary, the equality 8 holds for = 1 and all k N. Now, we assue that 8 is true for = 1,,...,l and k N. Using this assuption and 6 in 3

4 Corollary, we observe that For even l = q, we have and l D k,k+l+1 n;f = c l k, fl+1 nd k+,k+ n;f l 1 c l 1 k+1, fl 1 nd k+1+,k+1+ n;f. D k,k+q+1 n;f = 1 fq+1 nd k,k n;f q + c q k, c q 1 k+1, 1 f q+1 nd k+,k+ n;f c q k, c q 1 +1 k+1, 1 = 1q q +iq = c q+1 k,.! For odd l = q 1, we observe that By our assuption, since c q 1 k, i=1 D k,k+q n;f = 1 fq nd k,k n;f q 1 + c q 1 k, c q k+1, 1 f q nd k+ n;f c q k+1, 1 = q 1! + 1 q D k+ q,k+ q n;f. i=1 we obtain the assertion 8 for all k and N. q 1 +iq 1 = c q k,, Now, we consider another expression for D k,l n;f,g using the arithetical function H k, n;f,g := f k dg d. 9 4

5 If f = g, we set H k+ n;f = H k, n;f,f. The right-hand side of 9 iplies the Dirichlet convolution of 1 and f k g. Since g is a copletely additive function, we have D k,l n;f,g = = f k dgn gd l l l f k d 1 g l ng d. =0 Fro 9 and the above, we obtain the following theore. Theore 5. Let k and l be non-negative integers, and let f and g be copletely additive functions. Then we have D k,l n;f,g = l l 1 g l nh k, n;f,g, =0 where the function H k, n;f,g is defined by 9. We iediately obtain the following corollary. Corollary 6. Let k and l be non-negative integers, and let f and g be copletely additive functions. Then we have D k,l n;f = l l 1 f l nh k+ n;f. 10 =0 Note that H k, n;f,g = f k n d n g d = k i=0 k 1 i+ f k i ng n i f i dg d. 11 Applying 11 to Theore 5, we have the following theore. Theore 7. Let k and l be non-negative integers, and let f and g be copletely additive functions. Then we have D k,l n;f,g l k l k = 1 +i+ i =0 i=0 f k i ng l n f i dg d. 5

6 In the case where f = g, note that H k+ n;f = fn fd k+ k+ k + = 1 f k+ n f d. Fro 10 and the above, we obtain the following corollary. Corollary 8. Let k and l be non-negative integers, and let f be a copletely additive function. Then we have D k,l n;f = l k+ l k =0 f k+l n f d. 1 Recurrence forula connectingd k,l n;f with f d The second purpose of this study is to derive another expression for D k,l n;f that involves the divisor function dn. Before stating Theore 10, we prepare the following lea. Lea 9. Letf beacopletelyadditivefunction. Thereexistthe constantse q,q, e q,q 1,...,e q,1 q = 1,,... that satisfy the equation q 1 f q 1 d = e q,q dnf q 1 n+ e q,q f q 1 n f d. 13 Moreover, the relations aong sequences e q,q q are as follows. e q,q = 1 q 1 q 1 1 e, = q 1B q, 14 1 q e q,q = 1 q 1 q 1 q 1 e i,i, i 1 i= i 1 where B n denotes the nth Bernoulli nuber, which is defined by the Taylor expansion z e z 1 = n=1 B n n! zn, z < π. 6

7 Proof. By, the case q = 1 in 13 is trivial. Assue that there exist e p,p, e p,p 1,...,e p,1 p q such that p 1 f p 1 d = e p,p dnf p 1 n+ e p,p f p 1 n f d. 15 Since q+1 q +1 f q+1 d = 1 f q+1 n f d, we have f q+1 d = 1 dnfq+1 n+ 1 1 q q q +1 f q+1 n q +1 f q+ n 1 f d f 1 d. 16 Applying 15 to 16, we obtain f q+1 d = 1 q q +1 1 e, dnf q+1 n q q +1 q q +1 i 1 i= i 1 e i,i f q +1 n f d. By induction, this copletes the proof, except for the second ter on the right-hand side of 14. The first ter on the right-hand side of 14 iplies e q,q = 1 q k=1 Here we put ak = ke k,k. Then we have q 1 e k,k = 1 k 1 aq = q q k=1 q k=1 q k k q e k,k. q ak. 17 k Since a1 = e 1,1 = 1/ and 1B = 1/, we only need to show that k 1 B k k = 1,...,q satisfies the recurrence forula 17. Consider the nth Bernoulli polynoial 7

8 B n x, which is defined by the following Taylor expansion: ze xz e z 1 = n=0 B n x z n n! z < π. The following relations are known aong B n 1, B n 1/ and B n, 1 B n 1 = B n, B n = 1 1 n B n. By the forula [1, Th. 1.1, p. 64] we observe that B n y = n k=0 n B k y n k, 18 k q q B q y = y q qy q 1 + B k y q k. k In this equation, we consider y = 1 and y = 1/; then q q B q = 1 q + B k 19 k and Subtracting 0 fro 19, we obtain k=1 k= 1 q B q = q B q q q = 1 q + B k k. 0 k q 1 B q = q q k=1 k=1 q k 1 B k. k This recurrence forula for k 1 B k s is equivalent to 17. This copletes the proof of 13. Applying Lea 9 to 1 in Corollary 8, we have D k,l n;f = l l 1 =0 l l 1 =0 k+ k++1 k + f l+k n f d k + f l+k+1 n 1 f 1 d. 1 8

9 The second ter on the right-hand side of 1 gives us l l 1 =0 l =0 k++1 1 k+ k + e, f l+k ndn 1 l k + 1 e, i f l+k i n 1 i=1 f i d using 13. Fro 14, 1 and, we have the following theore. Theore 10. Let k and l be non-negative integers, and let f be a copletely additive function. There exist the constants e,, e, i = 1,,..., k++1, 1 i < and A k,l such that where D k,l n;f =A k,l f l+k ndn l k+ l k f l+k n f d 3 =0 l k+ 1 l k + 1 e, i f l+k i n f i d, 1 =0 i=1 A k,l = = l l 1 1 =0 k++1 1B k + 1 l l 1 k++1 1 k ++1 B k =0 Proof. We only need to show 4 to coplete the proof of Theore 10. We set A k,l = l l 1 1 J k,, J k, = =0 Fro the identity 1 k+ 1 = k++1 k++1, we have J k, = k ++1 k++1 k++1 1B 1B k ++1. k

10 Since B +1 = 0 if 1 and B 1 = 1, we have k++1 k ++1 J k, = 1B k ++1 Taking y = 1 and y = 1 in 18, we get 1 n B n = n n B and B n = n n B, respectively. Hence we have 1 n B n B n = n 1 n B. 6 On the other hand, it is known that 1 B n x = n 1 See [1, p. 75]. Taking x = 0 and = in 7, we obtain 1 n B n B n = n 1B n. B n x+. 7 Then we have, fro 6 and the above, n 1B n = n 1 n B. Substituting this relation in 5, we have Hence we have J k, = k ++1 k++1 1B k A k,l = l l 1 k++1 1 k ++1 B k++1. =0 Fro 3, we obtain the following corollary. 10

11 Corollary 11. For every positive integer k, there are constants c k,c k 1,...,c 0 such that D k,k n;f = c k dnf k n+ k c k f k n f d. 8 Fro 8 and 8, we obtain the following corollary. Corollary 1. For every positive integer k and, there are constants c 0,c 1,...,c k,...,c k+ as in Corollary 11 and c k,1 =, c k, = 3, c 4 k,3,..., c k, as in Theore 4 such that D k,k+ n;f = 1 c k + c dnf k+ n p=1 p=1 k,p c k+p k c k f k+ n f d c k,p Exaple 13. Corollaries 11 and 1 give us k+p c k+p f k+ n f d. D 1,1 n;f = 1 dnf n f d, D,1 n;f = 1 4 dnf3 n 1 fn f d, D 3,1 n;f = 1 4 dnf4 n+ 3 f n f d f 4 d, D, n;f = 1 dnf4 n f n f d+ f 4 d, D 4,1 n;f = 1 dnf5 n+ 5 f3 n f d 3 fn f 4 d, D 3, n;f = 1 4 dnf5 d f n f d+ 1 fn f 4 d. 11

12 3 Applications The second author [, p. 330] showed an asyptotic forula for n x D 1,1n;log D 1,1 n;log = 1 6 xlog3 x 1 xlog x+1 A 1 xlogx n x +A 1 4A 1x+O ε x 1 3 +ε 9 for x > and all ε > 0. Here the constants A 1 and A are coefficients of the Laurent expansion of the Rieann zeta-function ζs in the neighbourhood s = 1: ζs = 1 s 1 +A 0 +A 1 s 1+A s 1 +A 3 s We use 7, 9 and Abel s identity [1, Th. 4., p. 77] to obtain n x D,1 n;log = 1 1 xlog4 x 1 3 xlog3 x+1+a 1 xlog x 1+A xlogx+1+a x+o ε x 1 3 +ε. Furtherore, a generalization of 9 for the partial sus of D k,k n;log for positive integers k was considered by the second author [, Th. 1., p. 36], who deonstrated that there exists a polynoial P k+1 of degree k +1 such that D k,k n;log = xp k+1 logx+o k,ε x 1 3 +ε 30 n x for every ε > 0. Applying Theore 4 and the above forula 30, we have the following theore. Theore 14. There exists a polynoial U k++1 of degree k ++1 such that D k,k+ n;log = xu k++1 logx+o k,,ε x 1 3 +ε n x for =,3,... and every ε > 0. 4 Acknowledgent The authors deeply thank the referee for carefully reading this paper and indicating soe istakes. 1

13 References [1] T. M. Apostol. Introduction to Analytic Nuber Theory. Springer-Verlag, [] M. Minaide. The truncated Voronoï forula for the derivative of the Rieann zetafunction. Indian J. Math , Matheatics Subect Classification: Priary 11A5; Secondary 11P99. Keywords: copletely additive function, Dirichlet convolution. Received April 3 014; revised versions received May ; July 9 014; July 0 014; July Published in Journal of Integer Sequences, August Return to Journal of Integer Sequences hoe page. 13