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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  [[I.M. Vinogradov|Vinogradov]]'s work on [[trigonometric sum]]s. It can be used to estimate sums of the form
:<math>\sum f(n)\Lambda(n)</math>
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]].


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Vaughan's identity has been used to simplify the proof of the [[Bombieri–Vinogradov theorem]] and to study [[Kummer sum]]s.
 
Vaughan's identity was generalized by {{harvtxt|Heath-Brown|1982}}.
 
==References==
*{{springer|id=V/v130030|first=S.W.|last= Graham}}
*{{citation
|authorlink=Heath-Brown
|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
|mr=0678676
|doi=10.4153/CJM-1982-095-9}}
*{{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|mr= 0498434}}
 
[[Category:Analytic number theory]]
 
 
{{numtheory-stub}}

Latest revision as of 09:15, 17 July 2013

In mathematics, Vaughan's identity is an identity found by Template:Harvs that can be used to simplify Vinogradov's work on trigonometric sums. It can be used to estimate sums of the form

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 Template:Harvtxt.

References

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