View source for Vaughan identity

In 1937, I.M. Vinogradov [[#References|[a9]]] proved the odd case of the Goldbach conjecture (cf. also [[Goldbach problem|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|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|Exponential sum estimates]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300301.png" /> denote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300302.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300303.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300304.png" /> runs over the prime numbers. By simply observing that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300305.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300307.png" /> and using the prime number theorem (cf. [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]), one immediately sees that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300308.png" />. Vinogradov was able to improve on this estimate on the  "minor arcs" ; in other words, he obtained a better estimate for those values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v1300309.png" /> 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|Eratosthenes, sieve of]]; [[Sieve method|Sieve method]]) to decompose the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003010.png" /> into subsums of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003011.png" /></td> </tr></table>

and of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003012.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003013.png" />. 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 [[#References|[a6]]] found a much simpler approach to sums over prime numbers.

Vaughan's identity is most easily understood in the context of [[Dirichlet series|Dirichlet series]]. Suppose that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003014.png" /></td> </tr></table>

are both absolutely convergent in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003015.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003016.png" /></td> </tr></table>

in this same half-plane. One of the simplest and most useful Dirichlet series is the Riemann zeta-function (cf. also [[Zeta-function|Zeta-function]]), which is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003017.png" /> for complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003018.png" /> with real part exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003019.png" />. The Euler product formula states that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003020.png" /></td> </tr></table>

where the product is over all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003021.png" />. Taking the reciprocal of the Euler product, one sees that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003023.png" /> is the Möbius function defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003024.png" /></td> </tr></table>

By looking at the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003025.png" />, one obtains the useful identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003026.png" /></td> </tr></table>

By taking the logarithmic derivative of the Euler product formula, one sees that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003027.png" /></td> </tr></table>

where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003028.png" /> are defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003029.png" /></td> </tr></table>

This is the [[Mangoldt function|Mangoldt function]]. By computing the product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003030.png" /></td> </tr></table>

in two different ways, one sees that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003031.png" /></td> </tr></table>

For technical reasons, it is often simpler to work with sums of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003032.png" /> than with sums of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003033.png" />, and estimates for the latter sum can usually be easily derived from estimates for the former.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003035.png" /> be arbitrary real numbers, both exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003036.png" />, and define

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003037.png" /></td> </tr></table>

Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003039.png" /> are partial sums of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003041.png" /> respectively. In particular,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003042.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003043.png" /></td> </tr></table>

Now consider the Dirichlet series identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003044.png" /></td> </tr></table>

Comparing coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003045.png" /> on both sides of the equation, one sees that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003046.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003047.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003048.png" /></td> </tr></table>

If one multiplies this equation by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003049.png" /> and sums over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003050.png" />, one obtains the Vaughan identity:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003051.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003052.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003053.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003054.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003056.png" />, 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 [[#References|[a6]]]. There, he proves that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003057.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003058.png" /></td> </tr></table>

whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003059.png" />. Another self-contained account of this can be found in [[#References|[a1]]].

There are many applications of Vaughan's identity in the literature. Vaughan [[#References|[a7]]] used it to obtain new estimates on the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003060.png" />, and he also used it to give an elegant proof of the Bombieri–Vinogradov theorem on prime numbers in arithmetic progressions [[#References|[a8]]]. H.L. Montgomery and Vaughan [[#References|[a5]]] obtained a new estimate for the error term in the formula for the number of square-free integers up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003061.png" />, conditional on the Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]). This requires a slightly different form of Vaughan's identity. In this case, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003062.png" /> be as before, but take

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003063.png" /></td> </tr></table>

From the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003064.png" /></td> </tr></table>

one can obtain an identity for sums of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003065.png" />. D.R. Heath-Brown and S.J. Patterson [[#References|[a3]]] used Vaughan's identity to prove a long-standing conjecture of E. Kummer about distribution of cubic Gauss sums (cf. also [[Kummer hypothesis|Kummer hypothesis]]; [[Gauss sum|Gauss sum]]). Heath-Brown [[#References|[a2]]] developed a more general and more flexible version of Vaughan's identity, and G. Harman [[#References|[a4]]] has developed an alternative treatment that returns to Vinogradov's original idea of using the sieve of Eratosthenes (cf. also [[Eratosthenes, sieve of|Eratosthenes, sieve of]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Davenport,   "Multiplicative number theory" , Springer  (1980)  (Edition: Second)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R. Heath-Brown,   "Prime numbers in short intervals and a generalized Vaughan identity"  ''Canad. J. Math.'' , '''34'''  (1982)  pp. 1365–1377</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.R. Heath-Brown,   S.J. Patterson,   "The distribution of Kummer sums at prime arguments"  ''J. Reine Angew. Math.'' , '''310'''  (1979)  pp. 110–130</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.C. Vaughan,   "Sommes trigonométriques sur les nombres premiers"  ''C.R. Acad. Sci. Paris Sér. A'' , '''285'''  (1977)  pp. 981–983</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.C. Vaughan,   "On the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v130/v130030/v13003066.png" /> modulo one"  ''Mathematika'' , '''24'''  (1977)  pp. 135–141</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R.C. Vaughan,   "An elementary method in prime number theory"  ''Acta Arith.'' , '''37'''  (1980)  pp. 111–115</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  I.M. Vinogradov,   "A new estimation of a certain sum containing primes"  ''Mat. Sb.'' , '''44'''  (1937)  pp. 783–791  (In Russian)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I.M. Vinogradov,   "The method of trigonometric sums in the theory of numbers" , Wiley/Interscience  (1954)  (In Russian)</TD></TR></table>