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For $n\\geq 1$ and $t>0$ we define $$ S_{n}(\\sigma,t) = \\int_0^t S_{n-1}(\\sigma,\\tau)\\,d\\tau\\, + \\delta_{n,\\sigma\\,}, $$ where $\\delta_{n,\\sigma}$ is a specific constant depending on $\\sigma$ and $n$. Let $0\\leq \\beta<1$ be a fixed real number. Assuming the Riemann hypothesis, we show lower bounds for the maximum of the function $S_n(\\sigma,t)$ on the interval $T^\\beta\\leq t \\leq T$ and near to the critical line, when $n\\equiv 1\\mod 4$. 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For $n\\geq 1$ and $t>0$ we define $$ S_{n}(\\sigma,t) = \\int_0^t S_{n-1}(\\sigma,\\tau)\\,d\\tau\\, + \\delta_{n,\\sigma\\,}, $$ where $\\delta_{n,\\sigma}$ is a specific constant depending on $\\sigma$ and $n$. Let $0\\leq \\beta<1$ be a fixed real number. Assuming the Riemann hypothesis, we show lower bounds for the maximum of the function $S_n(\\sigma,t)$ on the interval $T^\\beta\\leq t \\leq T$ and near to the critical line, when $n\\equiv 1\\mod 4$. 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