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On the distribution of values of Hardy’s Z-functions in short intervals, II : The q-aspect
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An asymptotic formula for is derived, where is Hardy’s function. The cubic moment of Z(t) is also discussed, and a mean value result is presented which supports the author’s conjecture that \displaystyle{ \int _{1}^{T}Z^{3}(t)dt\; =\; O_{\varepsilon }(T^{3/4+\varepsilon }). }
Some problems involving the classical Hardy function
Z\left( t \right) = \zeta \left( {\frac{1}
{2} + it} \right)\left( {\chi \left( {\frac{1}
{2} + it} \right)} \right)^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} , \zeta \left( s \right) = \chi \left( s \right) \zeta \left( {1 - s} \right)
, are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.
If
Z(t) = c-1/2(\frac12 + it) z( \frac12 + it) Z(t) = \chi^{-1/2}(\frac{1}{2} + it) \zeta ( \frac{1}{2} + it)
denotes Hardyz(s) = c(s)z(1 - s) \zeta(s) = \chi(s)\zeta(1 - s)
, then it is proved that
ò0T Z(t)dt = Oe(T1/4+e) \int_0^T Z(t)dt = O_{\varepsilon}(T^{1/4+\varepsilon})
.
We investigate the distribution of positive and negative values of Hardy's function In particular we prove that where denotes the Lebesgue measure and \begin{align*} { I}_+(T,H) &\;=\; \bigl\{T< t\le T+H\,:\, Z(t)>0\bigr\}, { I}_-(T,H) &\;=\; \bigl\{T< t\le T+H\,:\, Z(t)<0\bigr\}. \end{align*}
An analogue of Atkinson's formula is proved for the integral function F(T)=∫0TZ(t)dt of Hardy's function Z(t). As an application of this formula, we analyze the behavior of the function F(T) showing that it can be approximated by a simple step-function. It follows that F(T)=O(T1/4) and F(T)=Ω±(T1/4); these results were recently obtained by M.A. Korolev using an alternative method.
Let Z(t) be the classical Hardy function in the theory of Riemann’s zeta-function. An asymptotic formula with an error term O(T
1/6log T) is given for the integral of Z(t) over the interval [0,T], with special attention paid to the critical cases when the fractional part of Ö{T/2p}\sqrt{T/2\pi } is close to
\frac14\frac{1}{4} or
\frac34\frac{3}{4}.
We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call “value recycling”.