A Speculation of N. Katz Concerning the Distribution of Arithmetic Quantities
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چکیده
There are many important theorems in number theory asserting that a sequence of arithmetic quantities of some kind is uniformly distributed. N. Katz has discovered a remarkable pattern in the finer aspects of distribution of many of these sequences, which we will describe below. Suppose for each n ≥ 1 we have a set Sn = {0 ≤ θ1 ≤ θ2 ≤ · · · ≤ θνn ≤ 1} of “angles” (normalized so that 2π = 1). We shall assume that limn→∞ νn = +∞. We say this set sequence is uniformly distributed if for any continuous C-valued function f(θ) on R/Z we have
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