论文标题
在短时间间隔和算术进程中，无方整数的差异
On the variance of squarefree integers in short intervals and arithmetic progressions
论文作者
论文摘要
我们渐近地评估无方整数数的差异，长度为$ h <x^{6/11- \ varepsilon} $，以及无方整数数的差异，最高$ x $的Arithmetic Progressions Modulo $ Q $ Q $ Q $ q> x^^x^^5/11 + varips的差异。在分别假设Lindelöf假设和广义的Lindelöf假设的假设上，我们表明可以将这些范围分别提高到$ h <x^{2/3 - \ varepsilon} $和$ q> x^{1/3 + \ \ varepsilon} $。此外，我们表明，在整个范围内获得$ h^{\ varepsilon} $的因素$ h <x^{1 - \ varepsilon} $等同于riemann假设等同于。在霍尔（Hall）（1982）的结果中，这些结果在短时间内得到了改善，以及在算术进行的情况下，Warlimont，Vaughan，Blomer，Nunes和Le Boudec的早期结果得到了改善。
We evaluate asymptotically the variance of the number of squarefree integers up to $x$ in short intervals of length $H < x^{6/11 - \varepsilon}$ and the variance of the number of squarefree integers up to $x$ in arithmetic progressions modulo $q$ with $q > x^{5/11 + \varepsilon}$. On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $H < x^{2/3 - \varepsilon}$ and $q > x^{1/3 + \varepsilon}$. Furthermore we show that obtaining a bound sharp up to factors of $H^{\varepsilon}$ in the full range $H < x^{1 - \varepsilon}$ is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.
