Haukkanen, Pentti

Merikoski, Jorma

Tossavainen, Timo

English

Trepo - Institutional Repository of Tampere University

MATHEMATICAL COMMUNICATIONS 107Math. Commun. 25(2020), 107{115Asymptotics of partial sums of the Dirichlet series of thearithmetic derivativePentti Haukkanen1,, Jorma K. Merikoski1 and Timo Tossavainen21 Faculty of Information Technology and Communication Sciences, FI-33 014 TampereUniversity, Finland2 Department of Arts, Communication and Education, Lulea University of Technology,SE-97 187 Lulea, SwedenReceived January 10, 2019; accepted November 25, 2019Abstract. For ; 6= P  P, let DP be the arithmetic subderivative function with respectto P on Z+, let DP be the function dened by the Dirichlet series of DP , and let DPdenote its abscissa of convergence. Under certain assumptions concerning s and P , wepresent asymptotic formulas for the partial sums of DP (s) and show that DP = 2. Wealso express DP (s), s > 2, using the Riemann zeta function.AMS subject classications: 11N37, 11N56Key words: Abscissa of convergence, arithmetic derivative, Dirichlet series1. IntroductionLet n 2 Z+. There exists a unique sequence of nonnegative integers (with onlynitely many positive terms)(p(n))p2P;where P stands for the set of primes, such thatn =Yp2Ppp(n):We use the approach mostly from [1, 3, 5, 7]. Let ; 6= P  P. The arithmeticsubderivative of n with respect to P isDP (n) = n0P :=Xp2Pn0p;where n0p is the arithmetic partial derivative of n with respect to p 2 P, dened byDp(n) = n0p = n0fpg :=p(n)pn:Corresponding author. Email addresses: pentti.haukkanen@tuni.fi (P.Haukkanen),jorma.merikoski@tuni.fi (J.K.Merikoski), timo.tossavainen@ltu.se (T.Tossavainen)http://www.mathos.hr/mc c2020 Department of Mathematics, University of Osijek108 P.Haukkanen, J.K.Merikoski and T.TossavainenThe arithmetic derivative of n isD(n) = n0 := n0P =Xp2Pn0p:We dene the (arithmetic) logarithmic subderivative, logarithmic partial derivative,and logarithmic derivative of n, respectively, as follows:ldP (n) =n0Pn; ldp(n) =n0pn; ld(n) =n0n:Let f be an arithmetic function. There exists f 2 R [ f1g such that itsDirichlet series 1Xn=1f(n)ns; s 2 C;converges if <(s) > f (< denotes the real part) and diverges if <(s) < f (see [6,p. 108, Theorem 3]). We call f the abscissa of convergence of this series and denethe function f byf (s) =1Xn=1f(n)ns; <(s) < f :For example, let the function u be identically one. The Riemann zeta function is(s) =1Xn=11ns= u(s); and u = 1:Our paper originates from three results due to Barbeau [1]. The rst one givesan upper bound for n0 using n:Lemma 1 (see [1, p. 118] or [7, Theorem 9]). Let n 2 Z+. Thenn0  n log n2 log 2:In the next theorem, the rst and second formula describe the asymptotic be-havior of X1nxld(n) andX1nxn0 :Theorem 1 (see [1, pp. 119{121] or [7, Theorem 24]). Asymptotically,X1nxld(n) = Cx+O(log x log log x)and X1nxn0 = Cx22+O(x1+):HereC =Xp2P1p(p  1) = 0:749 : : : ; (1)and  > 0 is arbitrary.Asymptotics of Dirichlet series of arithmetic derivative 109In the proofs cited above, actually x 2 Z+, but they can easily be extended tohold for x 2 R, x  1.Our goal is to nd asymptotic formulas for the partial sums of DP (s), in otherwords, for the sumsX1nxn0nsandX1nxn0pnsand; more generally; forX1nxn0Pns; (2)where s 2 R. As a corollary, we will see that D = Dp = DP = 2. For s = 1 ands = 0, the formulas concerning the rst sum are already given in Theorem 1. Lastly,we express DP (s), s > 2, using the values of .Our main tool is the following Abel's summation formula:Lemma 2 (see [6, p. 3, Theorem 1]). Let (an) be a sequence of complex numbers,let x > 1, and let g : [1; x]! C be a continuously dierentiable function. ThenX1nxang(n) = X1nxang(x) Z x1 X1ntang0(t)dt:2. Partial sums of D(2)In this section, we consider the rst sum of (2) with s = 2. We obtain the followingresult:Theorem 2. Asymptotically,X1nxn0n2= C log x+O(1):Proof. Applying Lemma 2 toan =n0n; g(x) =1x;we obtain X1nxn0n2=X1nxn0n1n= H(x) +K(x);whereH(x) = X1nxn0n 1x; K(x) =Z x1 X1ntn0n 1t2dt:By Theorem 1,H(x) = C +O(x 1 log x log log x) = O(1) (3)110 P.Haukkanen, J.K.Merikoski and T.TossavainenandK(x) =Z x1(Ct+O(log t log log t))1t2dt=Z x1C1tdt+Z x1O(t 2 log t log log t)dt= C log x+OZ x1t 2 log t log log t dt:Further, sincelog t log log t = O(t)for any  2 (0; 1), we haveK(x) = C log x+OZ x1t 2dt= C log x+O(x 1) +O(1)= C log x+O(1): (4)Now, the claim follows from (3) and (4).Corollary 1. It holds that D = 2.Proof. By Lemma 1,0  n0ns n log n2ns log 2=log n2ns 1 log 2: (5)If s > 2, then the series1Xn=1log nns 1converges. By using (5), we conclude that the series1Xn=1n0nsconverges, too. Hence D  2. On the other hand, since by Theorem 2 the series1Xn=1n0n2diverges, we have D  2.3. Partial sums of D(s), 1 6= s < 2Next, we study the rst sum of (2) in the case of 1 6= s < 2.Asymptotics of Dirichlet series of arithmetic derivative 111Theorem 3. Let 1 6= s < 2. Asymptotically,X1nxn0ns=C2  sx2 s +R(x);where R(x) is dened as follows: If 1 < s < 2, then R(x) = O(1). If s < 1, thenR(x) = O(x (s 1)) for any  > 0.Proof. Assume rst that 1 < s < 2. We proceed as in the proof of Theorem 2 buttakeg(x) =1xs 1:Then X1nxn0ns= H(x) +K(x);whereH(x) = Cx2 s +O(x1 s log x log log x) = Cx2 s +O(1)andK(x) =Z x1(Ct+O(log t log log t))s  1tsdt= C(s  1)Z x1dtts 1+ (s  1)Z x1O(t s log t log log t)dt= Cs  12  sx2 s +O(1) +OZ x1t s log t log log t dt= Cs  12  sx2 s +O(1) +OZ x1t sdt= Cs  12  sx2 s +O(1) +O(x (s 1)) +O(1):We can restrict ourselves to 0 <   s  1. Then   (s  1)  0, which implies thatK(x) = Cs  12  sx2 s +O(1)and further,H(x) +K(x) = C1 +s  12  sx2 s +O(1) =C2  sx2 s +O(1);completing the proof in this case.If s < 1, thenK(x) = Cs  12  sx2 s +O(x (s 1));and we can proceed as above.Note that this theorem is a generalization of the latter part of Theorem 1; justset s = 0.112 P.Haukkanen, J.K.Merikoski and T.Tossavainen4. Partial sums of Dp(1)We show that the asymptotic formulas for the partial sums of D(s) given in Theo-rems 1{3 have variants for those of Dp(s). In these variants, the coecient C givenin (1) is replaced by Cp dened asCp =1p(p  1) ; p 2 P:Note that C =Pp2P Cp.We begin the study of the partial sums of Dp(s) with s = 1.Theorem 4. Let p 2 P. Asymptotically,X1nxldp(n) = Cpx+O(log x):Proof. It is easy to see that it is enough to consider the sumnXk=1ldp(k) = ldpnYk=1k = ldp(n!):We modify the proof of the rst part of Theorem 1. By [2, Theorem 416],n! =Yq2Pqq(n); (6)whereq(n) =1Xm=1nqm=(n)Xm=1nqm; (n) =log nlog 2: (7)Now, denoting by(i)= that the equation follows from the formula (i), we obtainldp(n!)(6)= ldpYq2Pqq(n) =p(n)p(7)=1p(n)Xm=1npm=1p(n)Xm=1npm+1p(n)Xm=1O(1)(7)= n(n)+1Xm=21pm+O(log n)= n1Xm=21pm  n1Xm=(n)+21pm+O(log n)= Cpn  np(n)+1(p  1) +O(log n):It remains to study the complexity ofA(n) =np(n)+1(p  1) :Sincep(n)+1  2(n)+1 > nby (7), it follows that A(n) = O(1), and the proof is complete.Asymptotics of Dirichlet series of arithmetic derivative 1135. Partial sums of Dp(s) and DP (s)In this section, we continue by studying the second sum of (2), where 1 6= s  2.We rst assume that s = 2.Theorem 5. Let p 2 P. Asymptotically,X1nxn0pn2= Cp log x+O(1):Proof. The proof is analogous to that of Theorem 2. We apply Lemma 2 toan =n0pn; g(x) =1x;and use Theorem 4.Corollary 2. Let p 2 P. Then Dp = 2.Proof. Clearly, 0  n0p  n0 for all n 2 Z+. Since D = 2 by Corollary 1, we haveDp  2. On the other hand, since by Theorem 5 the series1Xn=1n0pn2diverges, it follows that Dp  2.Next, we consider the case of 1 6= s < 2.Theorem 6. Let p 2 P and 1 6= s < 2. Asymptotically,X1nxn0pns=Cp2  sx2 s +R(x);where R(x) is as in Theorem 3.Proof. The proof is a simple modication of that of Theorem 3.Corollary 3 (see Theorem 1). Let p 2 P. ThenX1nxn0p = Cpx22+O(x+1)for any  > 0.Our results about Dp(s) can be extended to concern DP (s) if P  P is nonemptyand nite (or if P = P, see Theorem 3). Then Cp is replaced byCP =Xp2P1p(p  1) :For example, Theorem 4 and Theorem 6 (s = 0) extend toX1nxldP (n) = CPx+O(log x);X1nxn0P = CPx22+O(x+1);and Corollary 2 extends to DP = 2.114 P.Haukkanen, J.K.Merikoski and T.Tossavainen6. Reducing DP to It is natural to expect that DP has a close relation to the Riemann zeta function .For Dp , this relation is already known in the following lemma (originally withdierent terminology and notation):Lemma 3 (see [4, Lemma 6]). Let p 2 P and s > 2. ThenDp(s) =(s  1)ps   p :We extend this to DP .Theorem 7. Let ; 6= P  P and s > 2. ThenDP (s) = (s  1)Xp2P1ps   p :Proof. We haveDP (s) =1Xn=1n0Pns=1Xn=1nPp2Pp(n)pns=1Xn=1Xp2Pp(n)pns 1: (8)Since the series (8) converges and all its terms are nonnegative, we can change theorder of summation. Therefore, by the simple calculation and applying Lemma 3we obtainDP (s) =Xp2P1Xn=1p(n)pns 1=Xp2P1Xn=1np(n)pns=Xp2P1Xn=1n0pns=Xp2PDp(s) =Xp2P(s  1)ps   p ;completing the proof.In particular,D(s) = (s  1)Xp2P1ps   p :7. Three further questionsIn the case of s  2, Theorems 1{3 give asymptotic formulas for the rst sumof (2), and Theorems 4{6 give those for the second. What about the case of s > 2?Theorems 3 and 6 with R(x) = O(1) hold also then, but since the main term hasa smaller complexity than the error term, we get nothing reasonable out of them.The question about a nontrivial asymptotic formula for the second (and third) sumof (2) in the case of s > 2 therefore remains open.Asymptotics of Dirichlet series of arithmetic derivative 115As noted at the end of Section 5, our results about Dp(s) can be extendedto DP (s) if P  P is nonempty and nite or if P = P. Can they be extended alsoif P  P is innite? This question remains open, too.Using advanced number-theoretic methods, the error terms of our asymptoticformulas can probably be improved, i.e., their complexity can be decreased. Howcould this be done? This is our third question.8. AcknowledgmentWe thank the referees. Their suggestions improved signicantly the presentation ofour paper.References[1] E. J. Barbeau, Remarks on an arithmetic derivative, Canad. Math. Bull. 4(1961),117{122.[2] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, FifthEdition, Oxford University Press, Oxford, 1983.[3] J. Kovic, The arithmetic derivative and antiderivative, J. Integer Seq. 15(2012), Article12.3.8.[4] N. Kurokawa, H. Ochiai, M. Wakayama, Absolute derivations and zeta functions,Documenta Math., Extra Volume Kato (2003), 565{584.[5] J. K. Merikoski, P. Haukkanen, T. Tossavainen, Arithmetic subderivatives andLeibniz-additive functions, Ann. Math. Informat. 50(2019), 145{157.[6] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, CambridgeUniversity Press, Cambridge, 1995.[7] V. Ufnarovski, B. Ahlander, How to dierentiate a number, J. Integer Seq. 6(2003),Article 03.3.4.

Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

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Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

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