A note on Mathieu's inequality

Summary

The function defined by the real infinite series$S(t): = \sum\limits_{n = 1}^\infty {2n(n^2 + t^2 )^{ - 2} }$ has been the subject of a number of estimates, starting with an 1890 conjecture by Mathieu (in connection with work on elasticity of solid bodies) thatS(t) < t ⁻². More recently it appeared in work on Mercerian theorems for Cesàro summability. A combination of the best of the various published inequalities so far (for larget) is

$$\frac{1}{{t^2 }} - \frac{5}{{16t^4 }}< S(t) = \frac{1}{{t^2 }} - \frac{1}{{6t^2 }} + O\left( {\frac{1}{{t^6 }}} \right).$$

However, one has only to observe that

$$g(z): = \sum\limits_{n = 0}^\infty {\frac{1}{{(n + z)^2 }} = \left( {\frac{d}{{dz}}} \right)^2 } \log \Gamma (z) z \ne 0, - 1, - 2,...)$$

in order to see that the original series satisfies

$$ - tS(t) = \operatorname{Im} g(it)$$

and one therefore has access to all the theory of the Gamma-function. While there are some difficulties due to the fact that Re(it) = 0, so that the remainder estimates in the standard asymptotic expansion require modification, nonetheless we can prove that

$$S(t) = \frac{1}{{t^2 }} - \frac{{B_1 }}{{t^4 }} - ... - \frac{{B_r }}{{t^{2r + 2} }} + \frac{{( - 1)^r }}{{t^{2r + 2} }}\int_0^\infty {f^{(2r + 1)} (x)\cos tx dx} $$

whereB ₁,B ₂,⋯ are Bernoulli numbers andf(x): = x/(e ^x − 1). Sincef ^(2r+1) (x) = O(xe ^−x) asx → + ∞, this certainly shows that the remainder term isO(t ^−2r-2). However, more delicate analysis allows us to place bounds on the remainder term.