×

zbMATH — the first resource for mathematics

Sharp bounds for the ratio of two zeta functions. (English) Zbl 07143660
Summary: In this paper, the authors prove that the function \[x \mapsto \frac{1}{2^x} \frac{\zeta (x) - 2^{- p} \zeta (x + p)}{\zeta (x) - \zeta (x + p)}\] is strictly increasing on \((1, \infty)\) for fixed \(p \geq 1\), which not only yields sharp lower and upper bounds of the ratio \(\zeta (x + p) / \zeta (x)\) for \(x \in (a, b) (b > a \geq 1)\), but also leads to the best lower and upper bounds for the ratio of any two Bernoulli numbers. Moreover, the authors obtain a more accurate upper estimation for \(\zeta (x + 1)\) in terms of \(\zeta (x)\) and \(\zeta (x + 2)\). Final, the authors pose two open problems.

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
26A48 Monotonic functions, generalizations
11B68 Bernoulli and Euler numbers and polynomials
39B62 Functional inequalities, including subadditivity, convexity, etc.
26D07 Inequalities involving other types of functions
Software:
Equator
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Titchmarsh, E. C., The Theory of the Riemann Zeta Function (1951), Oxford Univ. Press · Zbl 0042.07901
[2] Ivić, A., The Riemann Zeta Function: The Theory of the Riemann Zeta-Function with Applications (1985), Wiley: Wiley New York · Zbl 0556.10026
[3] Magnus, W.; Oberhettinger, F.; Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics (1966), Springer-Verlag · Zbl 0143.08502
[4] Oldham, K. B.; Myland, J. C.; Spanier, J., An Atlas of Functions: With Equator, the Atlas Function Calculator (2009), Springer-Verlag: Springer-Verlag New York · Zbl 1167.65001
[5] Wang, K. C., The logarithmic concavity of (1−21−r)ζ(r), J. Changsha Comm. Univ., 14, 1-5 (1998), (in Chinese)
[6] Alzer, H.; Kwong, M. K., On the concavity of Dirichlet’s eta function and related functional inequalities, J. Number Theory, 151, 172-196 (2015), Available online at http://dx.doi.org/10.1016/j.jnt.2014.12.009 · Zbl 1311.11087
[7] Adell, J. A.; Lekuona, A., Dirichlet’s eta and beta functions: Concavity and fast computation of their derivatives, J. Number Theory, 157, 215-222 (2015), Available online at http://dx.doi.org/10.1016/j.jnt.2015.05.006 · Zbl 1355.11090
[8] Zhu, L.; Hua, J.-K., Sharpening the Becker-Stark inequalities, J. Inequal. Appl., 2010, Article 931275 pp. (2010), 4 pages; Available online at https://doi.org/10.1155/2010/931275 · Zbl 1185.26059
[9] Cerone, P.; Dragomir, S. S., Some convexity properties of Dirichlet series with positive terms, Math. Nachr., 282, 7, 964-975 (2009), Available online at https://doi.org/10.1002/mana.200610783 · Zbl 1181.26030
[10] S. Hu, M.-S. Kim, On Dirichlet’s lambda functions, arXiv:1806.07762v2 [math.NT]. · Zbl 07096807
[11] Apostol, T. M., Introduction To Analytic Number Theory, Undergrad (1976), Texts Math: Texts Math New York-Berlin
[12] (Abramowitz, M.; Stegun, I. A., HandBook of Mathematical Functions with Formulas. HandBook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (1972), Dover: Dover New York) · Zbl 0543.33001
[13] Temme, N. M., Special Functions: An Introduction To the Classical Functions of Mathematical Physics (1996), Wiley: Wiley New York · Zbl 0856.33001
[14] Qi, F., A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, J. Comput. Appl. Math., 351, 1-5 (2019), Available online at https://doi.org/10.1016/j.cam.2018.10.049 · Zbl 1425.11043
[15] Guo, B.-N.; Mező, I.; Qi, F., An explicit formula for the Bernoulli polynomials in terms of the r-Stirling numbers of the second kind, Rocky Mountain J. Math., 46, 6, 1919-1923 (2016), Available online at https://doi.org/10.1216/RMJ-2016-46-6-1919 · Zbl 1371.11045
[16] Lv, H.-L.; Yang, Z.-H.; Luo, T.-Q.; Zheng, S.-Z., Sharp inequalities for tangent function with applications, J. Inequal. Appl., 2017, 94, 17 (2017), Available online at https://doi.org/10.1186/s13660-017-1372-5
[17] Qi, F.; Chapman, R. J., Two closed forms for the Bernoulli polynomials, J. Number Theory, 159, 89-100 (2016), Available online at https://doi.org/10.1016/j.jnt.2015.07.021 · Zbl 1400.11070
[18] Yang, Z.-H., Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function, J. Math. Anal. Appl., 441, 2, 549-564 (2016), Available online at https://doi.org/10.1016/j.jmaa.2016.04.029 · Zbl 1336.33005
[19] Zhu, L., New bounds for the exponential function with cotangent, J. Inequal. Appl., 2018, 106, 13 (2018), Available online at https://doi.org/10.1186/s13660-018-1697-8
[20] Qi, F., Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers, Turkish J. Anal. Number Theory, 6, 5, 129-131 (2018), Available online at https://doi.org/10.12691/tjant-6-5-1
[21] Srivastava, H. M., Some rapidly converging series for ζ(2n+1), Proc. Amer. Math. Soc., 127, 385-3996 (1999) · Zbl 0903.11020
[22] Srivastava, H. M., Some families of rapidly convergent series representation for the zeta function, Taiwan. J. Math., 4, 569-596 (2000) · Zbl 0964.11033
[23] Cvijović, D.; Klinowski, J., Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math., 142, 2, 435-439 (2002), Available online at https://doi.org/10.1016/S0377-0427(02)00358-8 · Zbl 1010.11047
[24] Cerone, P., Bounds for Zeta and related functions, J. Inequal. Pure Appl. Math., 6, 5, Article 134 pp. (2005), Available online at http://jipam.vu.edu.au/ · Zbl 1087.11052
[25] Alzer, H., Sharp bounds for the Bernoulli numbers, Arch. Math. (Basel), 74, 3, 207-211 (2000), Available online at https://doi.org/10.1007/s000130050432 · Zbl 0960.11016
[26] Leeming, D. J., The real zeros of the Bernoulli polynomials, J. Approx. Theory, 58, 124-150 (1989), Available online at https://doi.org/10.1016/0021-9045(89)90016-6 · Zbl 0692.41006
[27] Aniello, C. D., On some inequalities for the Bernoulli numbers, Rend. Circ. Mat. Palermo, 43, 329-332 (1994), Available online at https://doi.org/10.1007/BF02844246
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.