On the Chebyshev function \(\psi(x)\). (Q1881767)

Let \(\psi(x)=\sum_{n \leq x}\Lambda(n)\), where \(\Lambda(n)\) is the von Mangoldt function. The author proves that, for \(n\geq2\), \(0<\beta\leq1\) and \(T\to\infty\), \[ \int_1^{T^\beta}\Big(\psi\Big(x+{x\over T}\Big)-\psi(x)-{x\over T}\Big)^n{dx\over x^2}\sim{\beta^n\over n}{\log^nT\over T}, \] and, for \(\beta>1\) and assuming the Riemann Hypothesis, \[ \int_1^{T^\beta}\Big| \psi\Big(x+{x\over T}\Big)-\psi(x)-{x\over T}\Big| ^n{dx\over x^{n/2+1}}\ll_{n,\beta}{\beta^n\over n}{\log^nT\over T}. \] The two results here extend those for the case \(n=2\) which were dealt with by \textit{P.~X.~Gallagher} and \textit{J.~H.~Mueller} [J. Reine Angew. Math. 303/304, 205--220 (1978; Zbl 0396.10028)] and by \textit{A.~Selberg} [Arch. Math. Naturvid. 47, No. 6, 87--105 (1943; Zbl 0063.06869)], respectively.