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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.

##### 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
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