几乎所有短区间内乘法函数的改进界
Improved bounds for multiplicative functions in almost all short intervals
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中文总结 AI 辅助
研究改进乘法函数短平均值的Matomäki-Radziwiłł方法，对特定函数证明最优衰减界，关键是精确处理筛法误差，还给出其他函数的改进界及局限性，应用于平均Chowla猜想获改进界。
中文摘要 AI 辅助
我们改进了乘法函数短平均值的Matomäki-Radziwiłł方法。对于Liouville函数和由光滑数支撑的乘法函数，我们证明了用Matomäki-Radziwiłł方法本质上最优的衰减界。关键新要素是对筛法误差更精确的处理，通过在限制具有典型分解的整数时引入更广泛分离的最终素数范围实现。我们还给出了任意1 - 有界乘法函数的较弱但仍有改进的界，并讨论了该方法的一些局限性。作为应用，我们给出了Matomäki-Radziwiłł-Tao平均Chowla猜想的改进界，通过他们的方法似乎本质上是最优的。
英文摘要
We refine the Matomäki-Radziwiłł method for short averages of multiplicative functions. For the Liouville function and multiplicative functions supported on smooth numbers, we prove decay bounds that are essentially optimal with the Matomäki-Radziwiłł method. The key new ingredient is a sharper treatment of the sieve error, achieved by introducing a more widely separated final prime range when restricting to integers with typical factorizations. We additionally give a weaker but still improved bound for arbitrary 1-bounded multiplicative functions and discuss some limitations of the method. As an application, we give an improved bound for the averaged Chowla conjecture of Matomäki-Radziwiłł-Tao that seems essentially best possible via their method.
