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In particular, under the Lindel\\\"of hypothesis for $\\zeta(s)$, it is shown that $$ \\int_X^{2X} \\Bigl(\\Delta(x+U)-\\Delta(x)\\Bigr)^2\\,{\\roman d} x \\;\\ll_\\epsilon\\; X^{9/7+\\epsilon}U^{8/7}, $$ while under the Lindel\\\"of hypothesis for the Rankin-Selberg zeta-function the integral is bounded by $X^{1+\\epsilon}U^{4/3}$. 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