Squarefree polynomials and Mobius values in short intervals and arithmetic progressions
Squarefree polynomials and Mobius values in short intervals and arithmetic progressions
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短区间和算术级数中的无平方多项式和莫比乌斯值
DOI：
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发表时间：
2015
期刊：
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通讯作者：
Z. Rudnick
中科院分区：
文献类型：
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作者：
J. Keating;Z. Rudnick
我们计算的平均值和方差的莫比乌斯函数和指示函数的平方，在短时间间隔和算术级数，在环的多项式在有限域的$q$元素，在限制$q\到\infty$。我们这样做的问题，某些矩阵积分的酉群，使用最近的等分布结果，由于N。Katz，然后通过计算这些积分。在许多情况下，我们的结果反映了已知的或为相应的问题，涉及在整数上的总和，这有很长的历史。在某些情况下，存在微妙和令人惊讶的差异。我们的结果保持的范围是显着大于那些建立相应的问题，在数域设置。
We calculate the mean and variance of sums of the Mobius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, which have a long history. In some cases there are subtle and surprising differences. The ranges over which our results hold is significantly greater than those established for the corresponding problems in the number field setting.