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.
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We prove an explicit formula of Atkinson type for the error term in the asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. To deal with the case when coefficients of the Dirichlet polynomial are complex, we apply the idea of the first author in his study on mean values of Dirichlet L-functions.
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Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of $$\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|$$ . If $$E^* \left( t \right) = E\left( t \right) - 2\pi \Delta ^* \left( {t / 2\pi } \right)$$ with $$\Delta ^* \left( x \right) = - \Delta \left( x \right) + 2\Delta \left( {2x} \right) - \tfrac{1}{2}\Delta \left( {4x} \right)$$ , then we obtain $$\int_0^T {\left( {E^* \left( t \right)} \right)^4 dt \ll _e T^{16/9 + \varepsilon } } $$ . We also show how our method of proof yields the bound $$\sum\limits_{r = 1}^R {\left( {\int_{tr - G}^{tr + G} {\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|^2 dt} } \right)^4 \ll _e T^{2 + e} G^{ - 2} + RG^4 T^\varepsilon } $$ , where T 1/5+ε≤G≪T, T
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Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of $$\left| {\zeta \left( {\frac{1}{2} + it} \right)} \right|$$ . If E *(t)=E(t)-2πΔ*(t/2π) with $$\Delta *\left( x \right) + 2\Delta \left( {2x} \right) - \frac{1}{2}\Delta \left( {4x} \right)$$ , then we obtain $$\int_0^T {\left| {E*\left( t \right)} \right|^5 dt} \ll _\varepsilon T^{2 + \varepsilon } $$ and $$\int_0^T {\left| {E*\left( t \right)} \right|^{\frac{{544}}{{75}}} dt} \ll _\varepsilon T^{\frac{{601}}{{225}} + \varepsilon } .$$ It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of $$\left| {\zeta \left( {\frac{1}{2} + it} \right)} \right|$$ .
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