Marcel Riesz defined a function :

$R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$

The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$) For any $\varepsilon$

We have a relatively trivial bound $O(x^{1/2})$

Also various standard properties can be extracted from basic zeta function theory .

Questions:

(1) Does exponent between (1/4) and (1/2) for bound exists. i.e is there a bound better than $x^{1/2}$ but weaker than $x^{1/4 + {\varepsilon}}$?

(2) If such intermediate bound is achieved , what equivalent property about zeroes of Zeta function would be proved ?

(3) $R(x)$ looks lot like Taylor or any other symmetrical functional expansion . Is there any possible compact form for $R(x)$?

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