On sums of two \(k\)th powers: a mean-square asymptotics over short intervals (Q2759128)

Let \(k\geq 3\) be a fixed integer and let the arithmetic function \(r_k(n)\) be defined as the number of ways to write the natural number \(n\) as a sum of two \(k\)th powers of absolute values of integers. Further let \(R_k(u)= \sum_{1\leq n\leq u^k} r_k(n)\). It is well-known that NEWLINE\[NEWLINER_k(u)= A_ku^2+ B_k\Phi_k(u) u^{-1-1/k}+ P_k(u)NEWLINE\]NEWLINE with some constants \(A_k> 0\), \(B_k\neq 0\) and NEWLINE\[NEWLINE\Phi_k(u)= \sum_{n=1}^\infty n^{-1-1/k} \sin\biggl(2\pi nu- \frac{\pi}{2k}\biggr).NEWLINE\]NEWLINE It is conjectured that the error term \(P_k(u)\) is of order \(u^{1/2+\varepsilon}\), \(\varepsilon>0\). This is supported by \textit{M. Kühleitner}'s [Acta Arith. 92, 263-276 (2000; Zbl 0948.11035)] result NEWLINE\[NEWLINE\tfrac 1T \int_0^T P_k^2(u) \,du= C_kT+ O(T^{1-\varepsilon})NEWLINE\]NEWLINE with positive \(C_k\) and \(\varepsilon\). In place of this mean-square on the whole interval \([0,T],\) the authors now consider short intervals \(T-\Lambda\), \(T+\Lambda\). The main result is NEWLINE\[NEWLINE\int_{T-\Lambda}^{T+\Lambda} P_k^2(u)\, du\sim 4C_k\Lambda TNEWLINE\]NEWLINE as \(T\to\infty\). Here it is assumed that \(\Lambda= \Lambda(T)\) is an increasing function such that \(\Lambda(T)\leq \frac 12 T\) and NEWLINE\[NEWLINE\lim_{T\to\infty} \frac{\log T}{\Lambda(T)}= 0.NEWLINE\]