Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\le x,y,z\le H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)=c_k{H}^3+O\left(H^{9/4+\epsilon }\right)$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O\left(H^{7/3+\epsilon }\right)$ obtained by G.-L. Zhou, Y. Ding (2022).

Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds $x/{\left(logx\right)}^{.36}$ for all large x, while for φ it is equal to $x/{\left(logx\right)}^{1+o\left(1\right)}$, an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.

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 $$\left|\zeta \left(\frac{1}{2}+it\right)\right|$$ .

Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term $E_k\left(x\right)$ where $1/k!{\sum }_{n\le x}n/\varphi \left(n\right){(1-n/x)}^k=M_k\left(x\right)+E_k\left(x\right)$. For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying $k\gg {\left(logx\right)}^{1+ϵ}$.