Journal of Mathematical Inequalities Volume, Number 07, 09 5 doi:0.753/jmi--0 A RECIPROCAL SUM RELATED TO THE RIEMANN ζ FUNCTION LIN XIN AND LI XIAOXUE Communicated by J. Pečarić Abstract. This paper, we using the elementary method and several new inequalities to study the computational problem of one kind reciprocal sums related to the Riemann zeta-function at the point s, and give an explicit computational formula for it.. Introduction Let complex number s σ + it,ifσ >, then the famous Riemann zeta-function ζs is defined as the Dirichlet series ζs n and by analytic continuation to the whole complex plane except for a simple pole at s with residue. About the various properties of ζs, many mathematicians had studied them, and obtained abundant research results, for example, see [], [] and [3]. Very recently, the first author [] studied the computational problem of the reciprocal sum related to the Riemann ζ -function, and used the elementary method and some new inequalities to prove that for any positive integer n, one has the identities k n n s, and k 3 nn, where function [x] denotes the greatest integer x. Mathematics subject classification 00: B3, M06. Keywords and phrases: Riemann zeta-function, inequality, function [x], computational formula, elementary method. This work is supported by the N. S. F. 379 of P. R. China, G. I. C. F. YZZ5009 of Northwest University, S. P. D. E. S. F. 5JK7 of P. R. China. c D l,zagreb Paper JMI--0 09
0 L. XIN AND L. XIAOXUE At the same time, Lin Xin [] also proposed the following open problem: Whether there exists an explicit computational formula for, k s where s is an integer with s. This problem is interesting, because it is actually an effective approximation for the partial sum of the Riemann zeta-function ζs. More contents related to other second-order linear recurrence sequences and polynomials had also been studied by many authors, see references [5] []. For example, H. Ohtsuka and S. Nakamura [5] studied the properties of the sum of reciprocal Fibonacci numbers, and proved the identities { Fn, if n is an even number; F n, if n is an odd number, F k F k { Fn F n, if n is an even number; F n F n, if n an odd number. Zhang Wenpeng and Wang Tingting [] considered the computational problem of Pell numbers, and proved the identity { Pn + P n, if n is an even number; P n + P n, if n is an odd number. P k where Pell numbers P n are defined as P 0 0, P, and P n+ P n + P n for all integers n. The main purpose of this paper is to study the computational problem of for s, and use the elementary method and some new inequalities to give an interesting computational formula for with s. That is, we shall prove the following conclusion: THEOREM. For any positive integer n, we have the identity ] k m 3 m + m 3 5m + [ 35m [ 35m 7, if n m; ], if n m. For general integers s 5, whether there exists an explicit computational formula for is an open problem. Perhaps using our method one can study it for the cases s 5, but it is very complicated to construct those new inequalities when s is larger. With the aid of computer and mathematical software Mathematica, I believe that we can solve this problem completely.
A RECIPROCAL SUM RELATED TO THE RIEMANN ζ -FUNCTION. Several Lemmas In this section, we shall give some lemmas which are necessary in the proof of our theorem. First we have the following: LEMMA. For any integer n >, we have the inequality n 3 n + 5 n 9 n + 3 n + + 5 n + 9 n + n +. Proof. In fact, the inequality in Lemma is equivalent to the inequality or Let 3 n 3 6n + 5 n 3 3 n 3 + n + 9 n + 33 3n + 3n 3 + n + n + 56n + 5n 7 + 3n 6 + n 5 + 6n n + n + 3 n 3 6n + 5 n 3 n 3 + n + 9 n + 33 3n + 3n 3 + n + n + 56n + 5n 7 + 3n 6 + n 5 + 6n. f n n + n + 3 56n + 5n 7 + 3n 6 + n 5 + 6n, and gn Then n 3 6n + 5 n 3 n 3 +n + 9 3n n+33 +3n 3 +n +n+. f n0n 0 + 50n 9 + 550n + 33n 7 + 6n 6 + 56n 5 + n. 3 gn 6n 6 + 96n 5 + n n 3 + 9 n + 39 n 99 6 3n + 3n 3 + n + n + 0n 0 + 50n 9 + 550n + 33n 7 + 7n 6 + 30n 5 + 305 n. From 3 and we knowthat f n gn. So inequality is correct. This proves Lemma.
L. XIN AND L. XIAOXUE LEMMA. For any integer n, we also have the inequality n + n + n 3 n + 5 n 3 n + 3 n + + 5 n + 3. Proof. First we have the identity n 3 n + 5 n 3 n + 3 n + + 5 n + 3 n + n + 3 n 3 6n + 5 n n 3 + n + 9 n + n + n + 3 6n 6 + 96n 5 + n n 3 + 3 n + 5 n. Let hn 6n 6 + 96n 5 + n n 3 + 3 n + n 3n + 3n 3 + n + n +,thenwehave hn 0n 0 + 50n 9 + 550n + 36n 7 + 60n 6 + 76n 5 +n + n 3 9 n 53 n. 6 From 3 and 6 we know that f n > hn. Thus, from the definition of f n we have the inequality n + n + n 3 n + 5 n 3 n + 3 n + + 5 n + 3. This proves Lemma. LEMMA 3. For any integer n, we have the inequality 3 n 3 n + 9 n 7 > n + n. Proof. It is clear that 3 n + 3 n + + 9 7 n + n + n 3n 3n 3 + n n + 66n 3n 7 + n 6 n 5 + n. 7
A RECIPROCAL SUM RELATED TO THE RIEMANN ζ -FUNCTION 3 3 n 3 n + 9 n 7 3 n + 3 n + + 9 7 n + n n + 3 n 3 n + 9 n 7 n 3 + 6n + 5 n + n n + 3 6n 6 96n 5 + n + 0n 3 + 5 n 7n 7. 6 For convenience, we let Un 6n 6 96n 5 + n + 0n 3 + 5 n 7n 7 6 3n 3n 3 + n n +, Vn6 6n 3n 7 + n 6 n 5 + n n n + 3. Then by computations, we have Vn Un6n 7 6n 6 + n 5 6n + 7n 3 7 n 3 n 7 6. It is easy to prove that Vn Un > 0 for all integers n. So combining 7 and we may immediately deduce Lemma 3. LEMMA. For any integer n, we have the inequality n + n > 3 n 3 n + 9 n 5 6 3 n + 3 n + + 9. 5 n + 6 Proof. From the method of proving Lemma 3 we can easily deduce this inequal- ity. 3. Proof of the theorem In this section, we shall complete the proof of our theorem. If n m is an even number, then from Lemma and Lemma we have m 3 m + 5 m 9 km km k 3 k + 5 k 9 k + 3 k + + 5 k + 9 k + k + k. 9 km
L. XIN AND L. XIAOXUE Similarly, from Lemma we can deduce that km k km km k + k + k 3 k + 5 k 3 k + 3 k + + 5 k + 3 m 3 m + 5 m 3. 0 Combining 9 and 0 we may immediately deduce that m 3 m + 5 m 3 km km k m 3 m + 5 m 9, which implies according to the parity of m: [ ] 35m k m 3 m +. If n m be an odd number, then from Lemma 3, Lemma and the method of proving 9 and 0 we have m 3 5m + 7 m 5 km km k m 3 5m + 7 m 5, which implies according to the parity of m: [ ] 7m 5 k m 3 5m +. Applying and we know that for any integer n, we have the identity [ ] m 3 m + 35m, if n m; k m 3 5m + [ ] 7m 5, if n m. This completes the proof of the theorem. Acknowledgement. The authors would like to thank the referee for very helpful and detailed comments, which have significantly improvedthe presentation of this paper.
A RECIPROCAL SUM RELATED TO THE RIEMANN ζ -FUNCTION 5 REFERENCES [] E. C. TITCHMARSH, The Theory of the Riemann Zeta-Function, Oxford UP, London, 95, rev. ed., 96. [] A. IVIC, The Riemann zeta-function, Wiley, New York, 95. [3] R. P. FERGUSSON, An application of Stieltjes integration to the power series coefficients of the Riemann zeta-function, Amer. Math. Monthly, 70 963, 60 6. [] LIN XIN, Some identities related to the Riemann zeta-function, Journal of Inequalities and Applications 06, 06: 3. [5] H. OHTSUKA AND S. NAKAMURA, On the sum of reciprocal Fibonacci numbers, The Fibonacci Quarterly, 6/6 00/009, 53 59. [6] HAN ZHANG AND ZHENGANG WU, On the reciprocal sums of the generalized Fibonacci sequences, Advances in Difference Equations 03, 03: 377. [7] ZHEFENG XU AND TINGTING WANG, The infinite sum of the cubes of reciprocal Fibonacci numbers, Advances in Difference Equations 03, 03:. [] ZHANG WENPENG AND WANG TINGTING, The infinite sum of reciprocal Pell numbers, Applied Mathematics and Computation 0, 66 667. [9] ZHENGANG WU AND HAN ZHANG, On the reciprocal sums of higher-order sequences, Advances in Difference Equations 03, 03: 9. [0] ZHENGANG WU AND WENPENG ZHANG, The sums of the reciprocals of Fibonacci polynomials and Lucas polynomials, Journal of Inequalities and Applications 0, 0: 3. [] ZHENGANG WU AND WENPENG ZHANG, Several identities involving the Fibonacci polynomials and Lucas polynomials, Journal of Inequalities and Applications, 03, 03: 05. [] T. KOMATSU, On the nearest integer of the sum of reciprocal Fibonacci numbers, Aportaciones Matematicas Investigacion 0 0, 7. [3] T. KOMATSU AND V. L AOHAKOSOL, On the sum of reciprocals of numbers satisfying a recurrence relation of orders, Journal of Integer Seqences 3, Article ID 0.5., 00. [] E. KILIC AND T. ARIKAN, More on the infnite sum of reciprocal usual Fibonacci, Pell and higher order recurrences, Applied Mathematics and Computation 9 03, 773 77. Received February 3, 06 Lin Xin School of Mathematics, Northwest University Xi an, Shaanxi, P. R. China Li Xiaoxue School of Mathematics, Northwest University Xi an, Shaanxi, P. R. China e-mail: lxx0070@63.com Journal of Mathematical Inequalities www.ele-math.com jmi@ele-math.com