Vaughan identity
In 1937, I.M. Vinogradov [a9] proved the odd case of the Goldbach conjecture (cf. also Goldbach problem); i.e., he proved that every sufficiently large odd number can be written as a sum of three prime numbers (cf. also Vinogradov method). The essential new element of his proof was a non-trivial estimate for an exponential sum involving prime numbers (cf. also Exponential sum estimates). Let denote
and let
, where
runs over the prime numbers. By simply observing that
for all
,
and using the prime number theorem (cf. de la Vallée-Poussin theorem), one immediately sees that
. Vinogradov was able to improve on this estimate on the "minor arcs" ; in other words, he obtained a better estimate for those values of
that could not be well approximated by a rational number with a small denominator. Vinogradov's estimate used the sieve of Eratosthenes (cf. also Eratosthenes, sieve of; Sieve method) to decompose the sum
into subsums of the form
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and of the form
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The sums have become known as sums of type I and type II, respectively.
Vinogradov's method is quite powerful and can be adapted to general sums of the form . However, the technical details of his method are formidable and, consequently, the method was neither widely used nor widely understood. In 1977, R.C. Vaughan [a6] found a much simpler approach to sums over prime numbers.
Vaughan's identity is most easily understood in the context of Dirichlet series. Suppose that
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are both absolutely convergent in the half-plane . Then
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in this same half-plane. One of the simplest and most useful Dirichlet series is the Riemann zeta-function (cf. also Zeta-function), which is defined as for complex numbers
with real part exceeding
. The Euler product formula states that
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where the product is over all prime numbers . Taking the reciprocal of the Euler product, one sees that
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where is the Möbius function defined by
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By looking at the coefficients of , one obtains the useful identity
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By taking the logarithmic derivative of the Euler product formula, one sees that
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where the coefficients are defined as
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This is the Mangoldt function. By computing the product
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in two different ways, one sees that
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For technical reasons, it is often simpler to work with sums of the form than with sums of the form
, and estimates for the latter sum can usually be easily derived from estimates for the former.
Let ,
be arbitrary real numbers, both exceeding
, and define
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Thus, and
are partial sums of
and
respectively. In particular,
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Now consider the Dirichlet series identity
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Comparing coefficients of on both sides of the equation, one sees that if
, then
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where
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If one multiplies this equation by and sums over
, one obtains the Vaughan identity:
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In general, the first and second sums can be treated as type-I sums, and the third sum can be treated as a type-II sum. The logarithm factor in the first sum is easily finessed with partial summation. In some applications, it is useful to divide the second sum into subsums with and
, where the first subsum is treated as type-I and the second subsum as type-II.
For a brief and very accessible account of how Vaughan's identity is applied, see Vaughan's original article [a6]. There, he proves that
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whenever . Another self-contained account of this can be found in [a1].
There are many applications of Vaughan's identity in the literature. Vaughan [a7] used it to obtain new estimates on the distribution of , and he also used it to give an elegant proof of the Bombieri–Vinogradov theorem on prime numbers in arithmetic progressions [a8]. H.L. Montgomery and Vaughan [a5] obtained a new estimate for the error term in the formula for the number of square-free integers up to
, conditional on the Riemann hypothesis (cf. Riemann hypotheses). This requires a slightly different form of Vaughan's identity. In this case, let
be as before, but take
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From the equation
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one can obtain an identity for sums of the form . D.R. Heath-Brown and S.J. Patterson [a3] used Vaughan's identity to prove a long-standing conjecture of E. Kummer about distribution of cubic Gauss sums (cf. also Kummer hypothesis; Gauss sum). Heath-Brown [a2] developed a more general and more flexible version of Vaughan's identity, and G. Harman [a4] has developed an alternative treatment that returns to Vinogradov's original idea of using the sieve of Eratosthenes (cf. also Eratosthenes, sieve of).
References
[a1] | H. Davenport, "Multiplicative number theory" , Springer (1980) (Edition: Second) |
[a2] | D.R. Heath-Brown, "Prime numbers in short intervals and a generalized Vaughan identity" Canad. J. Math. , 34 (1982) pp. 1365–1377 |
[a3] | D.R. Heath-Brown, S.J. Patterson, "The distribution of Kummer sums at prime arguments" J. Reine Angew. Math. , 310 (1979) pp. 110–130 |
[a4] | G. Harman, "Eratosthenes, Legendre, Vinogradov, and beyond" G.R.H. Greaves (ed.) G. Harman (ed.) M.N. Huxley (ed.) , Sieve Methods, Exponential Sums, and their Applications in Number Theory , London Math. Soc. Lecture Notes , 237 , Cambridge Univ. Press (1996) |
[a5] | H.L. Montgomery, R.C. Vaughan, "On the distribution of square-free numbers" H. Halberstam (ed.) C. Hooley (ed.) , Recent Progress in Analytic Number Theory , 1 (1981) pp. 247–256 |
[a6] | R.C. Vaughan, "Sommes trigonométriques sur les nombres premiers" C.R. Acad. Sci. Paris Sér. A , 285 (1977) pp. 981–983 |
[a7] | R.C. Vaughan, "On the distribution of ![]() |
[a8] | R.C. Vaughan, "An elementary method in prime number theory" Acta Arith. , 37 (1980) pp. 111–115 |
[a9] | I.M. Vinogradov, "A new estimation of a certain sum containing primes" Mat. Sb. , 44 (1937) pp. 783–791 (In Russian) |
[a10] | I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Wiley/Interscience (1954) (In Russian) |
Vaughan identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vaughan_identity&oldid=17846