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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.

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
Full Text: DOI
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