Vaughan's identity is an important result in analytic number theory, particularly in the context of additive number theory and the study of sums of arithmetic functions. The identity provides a way to express the sum of a function over a set of integers in terms of more manageable sums and is often used in the context of problems involving the distribution of prime numbers.
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For thresholds , write and similarly for the other truncations. Vaughan's identity isIt decomposes a sum weighted by into short, Type I, and Type II bilinear sums.