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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 $$\left|\varsigma \left(\frac{1}{2}+it\right)\right|$$ . If $$E^{*}\left(t\right)=E\left(t\right)-2\pi {\Delta }^{*}\left(t/2\pi \right)$$ with $${\Delta }^{*}\left(x\right)=-\Delta \left(x\right)+2\Delta \left(2x\right)-\frac{1}{2}\Delta \left(4x\right)$$ , then we obtain $${\int }_0^T{\left(E^{*}\left(t\right)\right)}^{4}dt{\ll }_e{T}^{16/9+\epsilon }$$ . We also show how our method of proof yields the bound $$\sum _{r=1}^R{\left({\int }_{tr-G}^{tr+G}{\left|\varsigma \left(\frac{1}{2}+it\right)\right|}^{2}dt\right)}^4{\ll }_e{T}^{2+e}{G}^{-2}+R{G}^4{T}^{\epsilon }$$ , where T 1/5+ε≤G≪T, T

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 $${\int }_0^T{\left|E*\left(t\right)\right|}^{5}dt{\ll }_{\epsilon }{T}^{2+\epsilon }$$ and $${\int }_0^T{\left|E*\left(t\right)\right|}^{\frac{544}{75}}dt{\ll }_{\epsilon }{T}^{\frac{601}{225}+\epsilon }.$$ It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of $$\left|\zeta \left(\frac{1}{2}+it\right)\right|$$ .

Some problems involving the classical Hardy function $$Z\left(t\right)=\zeta \left(\frac{1}{2}+it\right){\left(\chi \left(\frac{1}{2}+it\right)\right)}^{-1/\phantom{12}2},\zeta \left(s\right)=\chi \left(s\right)\zeta \left(1-s\right)$$ , are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.

Let $$T\left(x\right)=K_{1}x{log}^{2}x+K_{2}xlogx+K_{3}x+\Delta \left(x\right),$$ where $T\left(x\right)$ denotes the number of subgroups of all abelian groups whose order does not exceed $x$ and whose rank does not exceed $2$, and $\Delta \left(x\right)$ is the error term. It is proved that $${\int }_1^X{\Delta }^2\left(x\right)dx\ll X^2{log}^{31/3}X,{\int }_1^X{\Delta }^2\left(x\right)dx=\Omega \left(X^2{log}^{4}X\right).$$

We have ${\sum }_{K-G\le k_j\le K+G}{\alpha }_j{H}_j^3\left(\frac{1}{2}\right){\ll }_{ϵ}G{K}^{1+ϵ}$ for $K^{ϵ}\le G\le K,\text{where}{\alpha }_j={\left|{\rho }_j\left(1\right)\right|}^2{(cosh\pi k_j)}^{-1},\text{and}{\rho }_j\left(1\right)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue ${\lambda }_j=k_j^2+\frac{1}{4}$ to which the Hecke series $H_j\left(s\right)$ is attached. This result yields the new bound $H_j(\frac{1}{2}{\ll }_{ϵ}{k}_j^{\frac{1}{3}+ϵ}.$

Several problems and results on mean values of $\zeta \left(s\right)$ are discussed. These include mean values of $\left|\zeta \right(\frac{1}{2}+it\left)\right|$ and the fourth moment of $\left|\zeta \right(\sigma +it\left)\right|$ for $1/2\<\sigma \<1$.

For a fixed integer $k\ge 3$, and fixed $\frac{1}{2}\<\sigma \<1$ we consider $${\int }_1^T{\left|\zeta (\sigma +it)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_k^2\left(n\right)n^{-2\sigma }T+R(k,\sigma ;T),$$ where $R(k,\sigma ;T)=0\left(T\right)(T\to \infty )$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k,\sigma ;T)$ are derived in the range min $({\beta }_k,{\sigma }_k^{*})\<\sigma \<1$. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.