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

MATHEMATICAL COMMUNICATIONS 107 Math. Commun. 25(2020), 107–115 Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative Pentti Haukkanen1,∗, Jorma K. Merikoski1and Timo Tossavainen2 1Faculty of Information Technology and Communication Sciences, FI-33 014 Tampere University, Finland 2Department of Arts, Communication and Education, Lulea University of Technology, SE-97 187 Lulea, Sweden Received January 10, 2019; accepted November 25, 2019 Abstract. For ∅ =P⊆P, let DPbe the arithmetic subderivative function with respect to Pon Z+, let ζDPbe the function defined by the Dirichlet series of DP, and let σDP denote its abscissa of convergence. Under certain assumptions concerning sand P, we present asymptotic formulas for the partial sums of ζDP(s) and show that σDP= 2. We also express ζDP(s), s > 2, using the Riemann zeta function. AMS subject classifications: 11N37, 11N56 Key words: Abscissa of convergence, arithmetic derivative, Dirichlet series 1. Introduction Let n∈Z+. There exists a unique sequence of nonnegative integers (with only finitely many positive terms) (νp(n))p∈P, where Pstands for the set of primes, such that n=∏ p∈P pνp(n). We use the approach mostly from [1, 3, 5, 7]. Let ∅ =P⊆P. The arithmetic subderivative of nwith respect to Pis DP(n) = n′ P:= ∑ p∈P n′ p, where n′ pis the arithmetic partial derivative of nwith respect to p∈P, defined by Dp(n) = n′ p=n′ {p}:= νp(n) pn. ∗Corresponding author. Email addresses: [email protected] (P. Haukkanen), [email protected] (J. K. Merikoski), [email protected] (T. Tossavainen) http://www.mathos.hr/mc c 2020 Department of Mathematics, University of Osijek 108 P. Haukkanen, J. K. Merikoski and T. Tossavainen The arithmetic derivative of nis D(n) = n′:= n′ P=∑ p∈P n′ p. We define the (arithmetic)logarithmic subderivative,logarithmic partial derivative, and logarithmic derivative of n, respectively, as follows: ldP(n) = n′ P n,ldp(n) = n′ p n,ld(n) = n′ n. Let fbe an arithmetic function. There exists σf∈R∪ {±∞} such that its Dirichlet series ∞ ∑ n=1 f(n) ns, s ∈C, converges if ℜ(s)> σf(ℜdenotes the real part) and diverges if ℜ(s)< σf(see [6, p. 108, Theorem 3]). We call σfthe abscissa of convergence of this series and define the function ζfby ζf(s) = ∞ ∑ n=1 f(n) ns,ℜ(s)< σf. For example, let the function ube identically one. The Riemann zeta function is ζ(s) = ∞ ∑ n=1 1 ns=ζu(s),and σu= 1. Our paper originates from three results due to Barbeau [1]. The first one gives an upper bound for n′using n: Lemma 1 (see [1, p. 118] or [7, Theorem 9]).Let n∈Z+. Then n′≤nlog n 2 log 2 . In the next theorem, the first and second formula describe the asymptotic behavior of ∑ 1≤n≤x ld(n) and ∑ 1≤n≤x n′: Theorem 1 (see [1, pp. 119–121] or [7, Theorem 24]).Asymptotically, ∑ 1≤n≤x ld(n) = Cx +O(log xlog log x) and ∑ 1≤n≤x n′=Cx2 2+O(x1+δ). Here C=∑ p∈P 1 p(p−1) = 0.749 . . . , (1) and δ > 0is arbitrary. Asymptotics of Dirichlet series of arithmetic derivative 109 In the proofs cited above, actually x∈Z+, but they can easily be extended to hold for x∈R,x≥1. Our goal is to find asymptotic formulas for the partial sums of ζDP(s), in other words, for the sums ∑ 1≤n≤x n′ nsand ∑ 1≤n≤x n′ p nsand,more generally,for ∑ 1≤n≤x n′ P ns,(2) where s∈R. As a corollary, we will see that σD=σDp=σDP= 2. For s= 1 and s= 0, the formulas concerning the first sum are already given in Theorem 1. Lastly, we express ζDP(s), s > 2, using the values of ζ. 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]→Cbe a continuously differentiable function. Then ∑ 1≤n≤x ang(n) = (∑ 1≤n≤x an)g(x)−∫x 1(∑ 1≤n≤t an)g′(t)dt. 2. Partial sums of ζD(2) In this section, we consider the first sum of (2) with s= 2. We obtain the following result: Theorem 2. Asymptotically, ∑ 1≤n≤x n′ n2=Clog x+O(1). Proof.Applying Lemma 2 to an=n′ n, g(x) = 1 x, we obtain ∑ 1≤n≤x n′ n2=∑ 1≤n≤x n′ n 1 n=H(x) + K(x), where H(x) = (∑ 1≤n≤x n′ n)1 x, K(x) = ∫x 1(∑ 1≤n≤t n′ n)1 t2dt. By Theorem 1, H(x) = C+O(x−1log xlog log x) = O(1) (3) 110 P. Haukkanen, J. K. Merikoski and T. Tossavainen and K(x) = ∫x 1 (Ct +O(log tlog log t)) 1 t2dt =∫x 1 C1 tdt+∫x 1 O(t−2log tlog log t)dt =Clog x+O(∫x 1 t−2log tlog log tdt). Further, since log tlog log t=O(tδ) for any δ∈(0,1), we have K(x) = Clog x+O(∫x 1 tδ−2dt)=Clog x+O(xδ−1) + O(1) =Clog x+O(1).(4) Now, the claim follows from (3) and (4). Corollary 1. It holds that σD= 2. Proof.By Lemma 1, 0≤n′ ns≤nlog n 2nslog 2 =log n 2ns−1log 2.(5) If s > 2, then the series ∞ ∑ n=1 log n ns−1 converges. By using (5), we conclude that the series ∞ ∑ n=1 n′ ns converges, too. Hence σD≥2. On the other hand, since by Theorem 2 the series ∞ ∑ n=1 n′ n2 diverges, we have σD≤2. 3. Partial sums of ζD(s),1=s < 2 Next, we study the first sum of (2) in the case of 1 =s < 2. Asymptotics of Dirichlet series of arithmetic derivative 111 Theorem 3. Let 1=s < 2. Asymptotically, ∑ 1≤n≤x n′ ns=C 2−sx2−s+R(x), where R(x)is defined as follows: If 1< s < 2, then R(x) = O(1). If s < 1, then R(x) = O(xδ−(s−1))for any δ > 0. Proof.Assume first that 1 < s < 2. We proceed as in the proof of Theorem 2 but take g(x) = 1 xs−1. Then ∑ 1≤n≤x n′ ns=H(x) + K(x), where H(x) = Cx2−s+O(x1−slog xlog log x) = Cx2−s+O(1) and K(x) = ∫x 1 (Ct +O(log tlog log t))s−1 tsdt =C(s−1) ∫x 1 dt ts−1+ (s−1) ∫x 1 O(t−slog tlog log t)dt =Cs−1 2−sx2−s+O(1) + O(∫x 1 t−slog tlog log tdt) =Cs−1 2−sx2−s+O(1) + O(∫x 1 tδ−sdt) =Cs−1 2−sx2−s+O(1) + O(xδ−(s−1)) + O(1). We can restrict ourselves to 0 < δ ≤s−1. Then δ−(s−1) ≤0, which implies that K(x) = Cs−1 2−sx2−s+O(1) and further, H(x) + K(x) = C(1 + s−1 2−s)x2−s+O(1) = C 2−sx2−s+O(1), completing the proof in this case. If s < 1, then K(x) = Cs−1 2−sx2−s+O(xδ−(s−1)), and we can proceed as above. Note that this theorem is a generalization of the latter part of Theorem 1; just set s= 0. 112 P. Haukkanen, J. K. Merikoski and T. Tossavainen 4. Partial sums of ζDp(1) We show that the asymptotic formulas for the partial sums of ζD(s) given in Theorems 1–3 have variants for those of ζDp(s). In these variants, the coefficient Cgiven in (1) is replaced by Cpdefined as Cp=1 p(p−1), p ∈P. Note that C=∑p∈PCp. We begin the study of the partial sums of ζDp(s) with s= 1. Theorem 4. Let p∈P. Asymptotically, ∑ 1≤n≤x ldp(n) = Cpx+O(log x). Proof.It is easy to see that it is enough to consider the sum n ∑ k=1 ldp(k) = ldp n ∏ k=1 k= ldp(n!). We modify the proof of the first part of Theorem 1. By [2, Theorem 416], n! = ∏ q∈P qµq(n),(6) where µq(n) = ∞ ∑ m=1 ⌊n qm⌋= α(n) ∑ m=1 ⌊n qm⌋, α(n) = ⌊log n log 2 ⌋.(7) Now, denoting by (i) = that the equation follows from the formula (i), we obtain ldp(n!) (6) = ldp∏ q∈P qµq(n)=µp(n) p (7) =1 p α(n) ∑ m=1 ⌊n pm⌋ =1 p α(n) ∑ m=1 n pm+1 p α(n) ∑ m=1 O(1) (7) =n α(n)+1 ∑ m=2 1 pm+O(log n) =n ∞ ∑ m=2 1 pm−n ∞ ∑ m=α(n)+2 1 pm+O(log n) =Cpn−n pα(n)+1(p−1) +O(log n). It remains to study the complexity of A(n) = n pα(n)+1(p−1). Since pα(n)+1 ≥2α(n)+1 > n by (7), it follows that A(n) = O(1), and the proof is complete. Asymptotics of Dirichlet series of arithmetic derivative 113 5. Partial sums of ζDp(s)and ζDP(s) In this section, we continue by studying the second sum of (2), where 1 =s≤2. We first assume that s= 2. Theorem 5. Let p∈P. Asymptotically, ∑ 1≤n≤x n′ p n2=Cplog x+O(1). Proof.The proof is analogous to that of Theorem 2. We apply Lemma 2 to an=n′ p n, g(x) = 1 x, and use Theorem 4. Corollary 2. Let p∈P. Then σDp= 2. Proof.Clearly, 0 ≤n′ p≤n′for all n∈Z+. Since σD= 2 by Corollary 1, we have σDp≥2. On the other hand, since by Theorem 5 the series ∞ ∑ n=1 n′ p n2 diverges, it follows that σDp≤2. Next, we consider the case of 1 =s < 2. Theorem 6. Let p∈Pand 1=s < 2. Asymptotically, ∑ 1≤n≤x n′ p ns=Cp 2−sx2−s+R(x), where R(x)is as in Theorem 3. Proof.The proof is a simple modification of that of Theorem 3. Corollary 3 (see Theorem 1).Let p∈P. Then ∑ 1≤n≤x n′ p=Cp x2 2+O(xδ+1) for any δ > 0. Our results about ζDp(s) can be extended to concern ζDP(s) if P⊂Pis nonempty and finite (or if P=P, see Theorem 3). Then Cpis replaced by CP=∑ p∈P 1 p(p−1). For example, Theorem 4 and Theorem 6 (s= 0) extend to ∑ 1≤n≤x ldP(n) = CPx+O(log x),∑ 1≤n≤x n′ P=CP x2 2+O(xδ+1), and Corollary 2 extends to σDP= 2. 114 P. Haukkanen, J. K. Merikoski and T. Tossavainen 6. Reducing ζDPto ζ It is natural to expect that ζDPhas a close relation to the Riemann zeta function ζ. For ζDp, this relation is already known in the following lemma (originally with different terminology and notation): Lemma 3 (see [4, Lemma 6]).Let p∈Pand s > 2. Then ζDp(s) = ζ(s−1) ps−p. We extend this to ζDP. Theorem 7. Let ∅ =P⊆Pand s > 2. Then ζDP(s) = ζ(s−1) ∑ p∈P 1 ps−p. Proof.We have ζDP(s) = ∞ ∑ n=1 n′ P ns= ∞ ∑ n=1 n∑p∈P νp(n) p ns= ∞ ∑ n=1 ∑ p∈P νp(n) pns−1.(8) Since the series (8) converges and all its terms are nonnegative, we can change the order of summation. Therefore, by the simple calculation and applying Lemma 3 we obtain ζDP(s) = ∑ p∈P ∞ ∑ n=1 νp(n) pns−1=∑ p∈P ∞ ∑ n=1 nνp(n) pns=∑ p∈P ∞ ∑ n=1 n′ p ns =∑ p∈P ζDp(s) = ∑ p∈P ζ(s−1) ps−p, completing the proof. In particular, ζD(s) = ζ(s−1) ∑ p∈P 1 ps−p. 7. Three further questions In the case of s≤2, Theorems 1–3 give asymptotic formulas for the first sum of (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 has a 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) sum of (2) in the case of s > 2 therefore remains open. 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