Vaughan's identity

Vaughan's identity

In mathematics, Vaughan's identity is an identity found by harvs|txt=yes|authorlink=Robert Charles Vaughan (mathematician)|last=Vaughan|first=R. C. |year=1977 that can be used to simplify Vinogradov's work on trigonometric sums. It can be used to estimate sums of the form:sum f(n)Lambda(n)where "f" is some function of positive integers "n", whose values in applications are often roots of unity, and Λ is the von Mangoldt function.

Vaughan's identity has been used to simplify the proof of the Bombieri–Vinogradov theorem and to study Kummer sums.

Vaughan's identity was generalized by harvtxt|Heath-Brown|1982.

References

*springer|id=V/v130030|first=S.W.|last= Graham
*citation|id=MR|0678676
last=Heath-Brown|first= D. R.
title=Prime numbers in short intervals and a generalized Vaughan identity
journal=Canad. J. Math.|volume= 34 |year=1982|issue=6|pages= 1365-1377
*citation|first= R.C. |last=Vaughan|title=Sommes trigonométriques sur les nombres premiers|journal= C.R. Acad. Sci. Paris Sér. A |volume=285 |year=1977|pages= 981–983|id=MR|0498434


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