On sums of \(k\)-th powers: A mean-square bound over short intervals (Q5933536)

For a fixed integer \(k>1\), denote \(r_k(u)= \#\{(u_1,u_2)\in \mathbb{Z}^2: |u_1|^2+|u_2|^2= n\}\) and \(R_k(u)= \sum_{1\leq n\leq u^k} r_k(n)\). One is interested in finding an asymptotic formula for \(R_k(u)\). If \(k=2\), the problem is a well-known circle problem. It was investigated by many authors who improved the upper bound for \(\theta\) in the formula \(P_2(u)\equiv R_2-\pi u^2\ll u^{\theta+ \varepsilon}\) for any \(\varepsilon>0\). The best published result is due to M. N. Huxley, who proved that \(P_2(u)\ll u^{46/73}(\log u)^{315/146}\). If \(k>2\) then NEWLINE\[NEWLINER_k(u)= 2\Gamma^2 (1/k)/(k\Gamma(2/k)) u^2+ B_k\Phi_k(u) u^{1-1/k}+ P_k(u)NEWLINE\]NEWLINE where \(B_k= 3^{3-1/k} \pi^{-1-1/k} k^{1/k} \Gamma(1+1/k)\) and \(\Phi_k(u)= \sum_{n=1}^\infty n^{-1-1/k} \sin(2\pi nu- \pi/2k)\). It was proved by Kuba that Huxley's upper bound for \(P_2(u)\) holds for \(P_k(u)\). It is also conjectured that \(\inf\{\theta\in R: |P_k(u)|\ll u^\theta\}=1/2\). \textit{W. Nowak} and \textit{M. Kühleitner} [Acta Arith. 99, 191-203 (2001; Zbl 0986.11068)] proved, respectively, that NEWLINE\[NEWLINE\frac{1}{T} \int_0^T (P_k(u))^2 du\ll T \quad\text{and}\quad \frac{1}{T} \int_0^T (P_2(u))^2 du= C_kT+ O(T^{1-\omega_k+ \varepsilon})NEWLINE\]NEWLINE with some explicit constants \(C_k\), \(\omega_k\). NEWLINENEWLINENEWLINEIn this paper the author proves that for any fixed \(k>2\) and large \(T\), NEWLINE\[NEWLINE\int_{T-1/2}^{T+1/2} (P_2(u))^2 du\ll T(\log u)^2.NEWLINE\]