Lampret, V.

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 21 (2020), No. 1, pp. 241–247 DOI: 10.18514/MMN.2020.2877EFFICIENT ESTIMATE OF THE REMAINDER FOR THEDIRICHLET FUNCTION η(p) FOR p ∈ R+V. LAMPRETReceived 03 March, 2019Abstract. For any k,n,q ∈N, where n≥ 2k+1≥ 3, and p ∈R+ the nth remainder ρ(n, p) of theDirichlet eta function η(p), known also as the alternating Riemann zeta function,η(p) :=∞∑i=1(−1)i+1 1ip,is given in the formρ(n, p) = ∆q(n, p)+δq(k, p),where k and q are parameters controlling the magnitude of the error term δq(k, p). The function∆q(n, p) consists of bq/2c+1 simple summands and δq(k,n, p) is estimated as∣∣δq(k, p)∣∣< 53 · p(q−1)piq−1 · 1(2k)p+q−1 ,where p(q−1) is the rising Pochhammer product.2010 Mathematics Subject Classification: 41A60; 65B10; 33E20; 33F05; 40A25Keywords: alternating generalized harmonic number, alternating p-series, approximation, Di-richlet’s eta function, estimate, remainder, slow convergence1. INTRODUCTIONTo obtain the numerical values of the sequence n 7→ H∗(n, 11000),H∗(n, 11000):=n∑j=1(−1) j+1 1j1/1000= 1.0000 . . .−0.9993 . . .+0.9989 . . .−0.9986 . . .+0.9983−0.9982 . . . + · · · + (−1)n+1n1/1000(n ∈ N),c© 2020 Miskolc University Press242 V. LAMPRETthe Dirichlet eta function,1η(p) :=∞∑j=1(−1) j+1 1jp(p > 0),is not a very useful tool although H∗(n, 11000) ≈ limn→∞H∗(n, 11000) = η( 11000) ≈limp↓0η(p) = 12 [1, Proposition 5.3]. The approximation H∗(n, 11000)≈ η( 11000) is goodonly for “very large” n due to slow convergence of H∗(n, 11000)illustrated2 in Fig-ure 1.50 100 150 2000.20.40.60.81.0H*Hn,11000L » 0 Hn evenLH*Hn,11000L » 1 Hn oddLΗH11000L » 12FIGURE 1. The graph of the sequence n 7→ H∗(n, 11000).We also remark that for any (fixed) n ∈ N we havelimp↓0n∑j=1(−1) j+1jp= limp↓01−∑(n−1)/2i=1(1(2i)p − 1(2i+1)p), n odd∑n/2i=1(1(2i−1)p − 1(2i)p), n even={1, n odd0, n even.Motivated by the results obtained by L. To´th – J. Bukor [5], V. Timofte [4], and theSıˆnta˘ma˘rian’s papers [2, 3], we shall accurately estimate the remainderρ(n, p) := η(p)−H∗(n, p) (n ∈ N, p ∈ R+), (1.1)whereH∗(n, p) :=n∑j=1(−1) j+1 1jp(1.2)is called the nth alternating generalized harmonic number of order p.1known also as the alternating Riemann zeta function2In this article all figures are plotted by Stephen Wolfram’s Mathematica Software, Version 7.0.THE REMAINDER OF THE DIRICHLET η(p) 2432. AUXILIARY RESULTSIn the sequel we shall use special sums3σ∗q(x, p) :=bq/2c∑i=1(4i−1) B2i · p(2i−1)xp+2i−1 · (2i)! (q ∈ N, p,x ∈ R), (2.1)where Bk is the kth Bernoulli coefficient (or Bernoulli number)4, defined by the iden-tity tet−1 ≡ ∑∞k=0 Bk tkk! (|t| < 2pi). Here, for x ∈ R and k ∈ N0 := {0}∪N the symbolx(k) means the upper (rising) Pochhammer product defined asx(0) := 1 and x(k) :=k−1∏i=0(x+ i) = x(x+1) · . . . · (x+ k−1) (k ∈ N). (2.2)According to [1, Theorem 3.2] there holds the following lemma.Lemma 1. For p ∈ R+ and for any k,n,q ∈ N, where n≥ 2k+1≥ 3, we haveH∗(n, p) = H∗q (k,n, p)+ r∗q(k, p), (2.3)whereH∗q (k,n, p) := A∗q(k, p)+B∗q(n, p), (2.4)A∗q(k, p) := H∗(2k−1, p)− 12(2k)p −σ∗q(2k, p), (2.5)B∗q(n, p) :=(−1)n+12(2bn+12 c)p +σ∗q(2bn+12 c, p), (2.6)∣∣r∗q(k, p)∣∣≤+ p(q−1) ·pip+13(2kpi)p+q−1 (2.7). (2.8)Referring to [1, Theorem 5.2] we have also the next lemma.Lemma 2. For p ∈ R+ and for any k,q ∈ N, there holds the following relation(see (2.1)),η(p) = H∗(2k−1, p)− 12(2k)p −σ∗q(2k, p)+ εq(k, p), (2.9)where ∣∣εq(k, p)∣∣< p(q−1)3piq−2(2k)p+q−1 . (2.10)3By definition, ∑ni=m xi = 0 if m > n.4B1 =− 12 , B2 = 16 , B3 = B5 = B7 = . . .= 0, B4 = B8 =− 130 , B6 = 142 , . . .244 V. LAMPRET3. THE MAIN RESULTFrom Lemmas 2 and 1 immediately follows the next result.Theorem 1. For p ∈ R+ and every k,n,q ∈ N, with n ≥ 2k + 1 ≥ 3, the nth re-mainder ρ(n, p) := η(p)−∑nj=1(−1) j+1 1jp is given in the formρ(n, p) = ∆q(n, p)+δq(k, p), (3.1)where∆q(n, p) =(−1)n2(2bn+12 c)p −σ∗q(2bn+12 c, p), (3.2)∣∣δq(k, p)∣∣< 5p(q−1)3piq−1 · 1(2k)p+q−1 . (3.3)Proof. We haveρ(n, p) = η(p)−H∗(n, p)=(((((((H∗(2k−1, p)− 12(2k)p −XXXXXσ∗q(2k, p)+ εq(k, p))−(((((((H∗(2k−1, p)− 12(2k)p −XXXXXσ∗q(2k, p)+ (−1)n+12(2bn+12 c)p+σ∗q(2bn+12 c, p)+ r∗q(k, p))= εq(k, p)− (−1)n+12(2bn+12 c)p −σ∗q(2bn+12 c, p)− r∗q(k, p),where ∣∣δq(k, p)∣∣= ∣∣εq(k, p)− r∗q(k, p)∣∣<p(q−1)3piq−2(2k)p+q−1+p(q−1)3(2kpi)p+q−1=p(q−1)3piq−2(2k)p+q−1(1+pi−(p+1))<p(q−1)3piq−2(2k)p+q−1· 5pi. In our Theorem k and q are parameters controlling the magnitude of the error termδq(k, p) = εq(k, p)−rq(k, p). We add that, for 0 < p≤ 1 and q∈N, using the Stirlingfactorial formula, we estimatep(q−1) = p(p+1) · · ·(p+q−2)≤ (q−1)! = q!q<1q(qe)q√2piq · e 112q .THE REMAINDER OF THE DIRICHLET η(p) 245Thus, for n≥ 2k+1≥ 3 and 0 < p≤ 1, we obtain the estimates∣∣δq(k, p)∣∣< 5p(q−1)3piq−1 · 1(2k)p+q−1 < 53√2piq(qe)qe112q · 1(2pik)q−1=53e√2piq( q2epik)q−1e112q < 2√q( q2epik)q−1. (3.4)Figure 2 shows the graph of the sequence n 7→ ∆5(n,1/1000).5 10 15 20-0.6-0.4-0.20.20.40.6D5Hn,11000L » 12 Hn evenLD5Hn,11000L » -12 Hn oddLΗH11000L » 12FIGURE 2. The graph of the sequence n 7→ ∆5(n, 11000).Since σ1(x, p)≡ 0, we obtain from Theorem the first corollary.Corollary 1. For p ∈ R+ and any k,n ∈ N, with n≥ 2k+1≥ 3, we haven∑j=1(−1) j+1 1jp+(−1)n2(2bn+12 c)p − 53(2k)p< η(p)<n∑j=1(−1) j+1 1jp+(−1)n2(2bn+12 c)p + 53(2k)p .Setting q = 5 in Theorem we obtain the following result.Corollary 2. For p ∈ R+ and k,n ∈ N, such that n ≥ 2k + 1 ≥ 3, the followingequation followsη(p) =n∑j=1(−1) j+1 1jp+(−1)n2(2bn+12 c)p − p4(2bn+12 c)p+1+p(p+1)(p+2)48(2bn+12 c)p+3 +δ5(k, p),246 V. LAMPRETwhere ∣∣δ5(k, p)∣∣< 5p(p+1)(p+2)(p+3)3pi4(2k)p+4 .Example 1. Since∣∣δ5(10,0.001)∣∣< 10−9 the approximationη(0.001)≈S∗n:=∑nj=1(−1) j+1 1j0.001 +(−1)n2(2b n+12 c)0.001− 0.0014(2b n+12 c)1.001 + 0.001×1.001×2.00148(2b n+12 c)3.001is very accurate for every n≥ 21 as is seen from Figure 3 where are plotted the graphsof the sequence n 7→ S∗n and the constant function n 7→ η(0.001) = H∗(∞,0.001) =0.5002257 . . .. The corrected term ∆5(n,0.001) is crucial since, for example, wehaveH∗(100000,0.001) = 0.005949215 . . ., H∗(100001,0.001) = 0.994502300 . . .,∆5(100000,0.001) = 0.494276544 . . . and ∆5(100001,0.001) =−0.494276539 . . ..20 25 30 35 40ΗH11000L = 0.500 225 760 834 . . .Sn*0.5002257608320.5002257608400.500225760848FIGURE 3. The graph of the sequence n 7→ S∗n.Obviously, using the approximation η(0.001) ≈ S∗n we directly obtain accurateapproximation of H∗(n, 11000)from Introduction:H∗(n, 11000)≈η(0.001)+ (−1)n+12(2b n+12 c)0.001 + 0.0014(2b n+12 c)1.001− 0.001×1.001×2.00148(2b n+12 c)3.001 (n∈N).CONCLUSIONThe paper provides an estimate of the remainder of the series representing Dirich-let’s function η(p) for p ∈R+. The presented estimate is effective also in the case ofslowly convergent series occurring for p≈ 0.THE REMAINDER OF THE DIRICHLET η(p) 247REFERENCES[1] V. Lampret, “Efficient approximations of finite and infinite real alternating p-series.” J. NonlinearSci. Appl., vol. 11, no. 11, pp. 1250–1261, 2018, doi: http://dx.doi.org/10.22436/jnsa.011.11.05.[2] A. Sıˆnta˘ma˘rian, “A new proof for estimating the remainder of the alternating harmonic series.”Creat. Math. Inform., vol. 21, no. 2, pp. 221–225, 2012.[3] A. Sıˆnta˘ma˘rian, “Sharp estimates regarding the remainder of the alternating harmonic series.” Math.Inequal. Appl., vol. 18, no. 1, pp. 347–352, 2015, doi: 10.7153/mia-18-24.[4] V. Timofte, “On Leibniz series defined by convex functions.” J. Math. Anal. Appl., vol. 300, no. 1,pp. 160–171, 2004, doi: 10.1016/j.jmaa.2004.06.038.[5] L. To´th and J. Bukor, “On the alternating series 1− 12 + 13 − 14 + · · · .” J. Math. Anal. Appl., vol.282, no. 1, pp. 21–25, 2003, doi: 10.1016/S0022-247X(02)00344-X.Author’s addressV. LampretUniversity of Ljubljana, Faculty of Civil and Geodetic Engineering, 2 Jamova St., 1000 Ljubljana,SloveniaE-mail address: vito.lampret@guest.arnes.si

Efficient estimate of the remainder for the Dirichlet function η(p) for p ∈ R+

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