Vaughan's identity

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In mathematics and analytic number theory, Vaughan's identity is an identity found by R. C. Vaughan (1977) that can be used to simplify Vinogradov's work on trigonometric sums. It can be used to estimate summatory functions of the form

where f is some arithmetic function of the natural integers n, whose values in applications are often roots of unity, and Λ is the von Mangoldt function.

Procedure for applying the method

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The motivation for Vaughan's construction of his identity is briefly discussed at the beginning of Chapter 24 in Davenport. For now, we will skip over most of the technical details motivating the identity and its usage in applications, and instead focus on the setup of its construction by parts. Following from the reference, we construct four distinct sums based on the expansion of the logarithmic derivative of the Riemann zeta function in terms of functions which are partial Dirichlet series respectively truncated at the upper bounds of   and  , respectively. More precisely, we define   and  , which leads us to the exact identity that

 

This last expansion implies that we can write

 

where the component functions are defined to be

 

We then define the corresponding summatory functions for   to be

 

so that we can write

 

Finally, at the conclusion of a multi-page argument of technical and at times delicate estimations of these sums,[1] we obtain the following form of Vaughan's identity when we assume that  ,  , and  :

 

It is remarked that in some instances sharper estimates can be obtained from Vaughan's identity by treating the component sum   more carefully by expanding it in the form of

 

The optimality of the upper bound obtained by applying Vaughan's identity appears to be application-dependent with respect to the best functions   and   we can choose to input into equation (V1). See the applications cited in the next section for specific examples that arise in the different contexts respectively considered by multiple authors.

Applications

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In particular, we obtain an asymptotic upper bound for these sums (typically evaluated at irrational  ) whose rational approximations satisfy

 

of the form

 

The argument for this estimate follows from Vaughan's identity by proving by a somewhat intricate argument that

 

and then deducing the first formula above in the non-trivial cases when   and with  .

Generalizations

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Vaughan's identity was generalized by Heath-Brown (1982).

Notes

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  1. ^ N.b., which if you read Davenport frequently enough will lead you to conclude evident properties about the difficulty level of the complete details to proving Vaughan's identity carefully.
  2. ^ Tao, T. (2012). "Every integer greater than 1 is the sum of at most five primes". arXiv:1201.6656 [math.NT].
  3. ^ Conrey, J. B. (1989). "More than two fifths of the zeros of the Riemann zeta function are on the critical line". J. Reine Angew. Math. 399: 1–26.
  4. ^ H. L. Montgomery and R. C. Vaughan (1981). "On the distribution of square-free numbers". Recent Progress in Analytic Number Theory, H. Halberstam (Ed.), C. Hooley (Ed.). 1: 247–256.
  5. ^ D. R. Heath-Brown and S. J. Patterson (1979). "The distribution of Kummer sums at prime arguments". J. Reine Angew. Math. 310: 110–130.

References

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