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For example, if we assume the Generalized Density Hypothesis, then for any arithmetic progression modulo $q\\leq \\log^{\\ell} x$ with $\\ell>0$ and any $\\varepsilon>0$, the prime number theorem holds in all intervals $(x-\\sqrt{x}\\exp(\\log^{\\frac{2}{3}+\\varepsilon} x),x]$ and almost all intervals $(x-\\exp(\\log^{\\frac{2}{3}+\\varepsilon} x),x]$ as $x\\rightarrow\\infty$. 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For example, if we assume the Generalized Density Hypothesis, then for any arithmetic progression modulo $q\\leq \\log^{\\ell} x$ with $\\ell>0$ and any $\\varepsilon>0$, the prime number theorem holds in all intervals $(x-\\sqrt{x}\\exp(\\log^{\\frac{2}{3}+\\varepsilon} x),x]$ and almost all intervals $(x-\\exp(\\log^{\\frac{2}{3}+\\varepsilon} x),x]$ as $x\\rightarrow\\infty$. 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