1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, ${}_{16}$, each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is a restricted...

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 $k\gg {\left(logx\right)}^{1+ϵ}$.

Let $p$ be a prime number. A finite Galois extension $N/F$ of a number field $F$ with group $G$ has a normal $p$-integral basis ($p$-NIB for short) when ${𝒪}_N^{\prime }$ is free of rank one over the group ring ${𝒪}_F^{\prime }\left[G\right]$. Here, ${𝒪}_F^{\prime }={𝒪}_F[1/p]$ is the ring of $p$-integers of $F$. Let $m=p^e$ be a power of $p$ and $N/F$ a cyclic extension of degree $m$. When ${\zeta }_m\in F^{\times }$, we give a necessary and sufficient condition for $N/F$ to have a $p$-NIB (Theorem 3). When ${\zeta }_m\notin F^{\times }$ and $p\nmid [F\left({\zeta }_m\right):F]$, we show that $N/F$ has a $p$-NIB if and only if $N\left({\zeta }_m\right)/F\left({\zeta }_m\right)$ has a $p$-NIB (Theorem 1). When $p$ divides $[F\left({\zeta }_m\right):F]$, we show that this descent property...