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 $e(\alpha)$ denote $e^{2\pi\alpha}$ and let $S(\alpha, N) = \sum_{p\le N} e(p\alpha)$, where $p$ runs over the prime numbers. By simply observing that $|e(p\alpha)| \le 1$ for all $p$, $\alpha$ and using the prime number theorem (cf. [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]), one immediately sees that $S(\alpha, N) = O(N/\log N)$. Vinogradov was able to improve on this estimate on the  "minor arcs" ; in other words, he obtained a better estimate for those values of $\alpha$ 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 $S(\alpha, N)$ into subsums of the form

$$
\sum_{n\le u} \alpha(n) \sum_{m \le N/n} e(\alpha m)
$$

and of the form

$$
\sum_{m\le u}
\sum_{n\le v}
\sum_{r\le N/mn}
a(m) b(n) e(rmn \alpha).
$$

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 $\sum_{p\le N} f(p)$. 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

$$
F(s) = \sum_{n=1}^\infty f(n) n^{-s} 
\quad \text{and} \quad 
G(s) = \sum_{n=1}^\infty g(n) n^{-s} 
$$

are both absolutely convergent in the half-plane $\Re s > a$. Then

$$
F(s) G(s) = \sum_{n=1}^\infty \left(
\sum_{de = n} f(d) g(e)
\right) n^{-s}
$$

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 $\zeta(s) = \sum_{n=1}^\infty n^{-s}$ for complex numbers $s$ with real part exceeding $1$. The Euler product formula states that

$$
\zeta(s) = \prod_p (1-p^{-s})^{-1},
$$

where the product is over all prime numbers $p$. Taking the reciprocal of the Euler product, one sees that

$$
\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \mu(n) n^{-s},
$$

where $\mu(n)$ is the Möbius function defined by

$$
\mu(n) = \begin{cases}
(-1)^k & \text{if $n = p_1\dots p_k$ for distinct prime numbers $p_1, \ldots, p_k$}, \\ 
0 & \text{if $n$ is divisible by the square of some prime number}.
\end{cases}
$$

By looking at the coefficients of $\zeta(s) . \zeta(s)^{-1}$, one obtains the useful identity

$$
\sum_{d|n} \mu(n) = \begin{cases}
1 & \text{if $n=1$}, \\ 
0 & \text{otherwise}.
\end{cases}
$$

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

$$
- \frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \Lambda(n) n^{-s},
$$

where the coefficients $\Lambda(n)$ are defined as

$$
\Lambda(n) = \begin{cases}
\log p & \text{if $n=p^a$ for some prime number $p$}, \\ 
0 & \text{otherwise}.
\end{cases}
$$

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

$$
\zeta(s) . \frac{-\zeta'}{\zeta}(s)
$$

in two different ways, one sees that

$$
\sum_{d|n} \Lambda(d) = \log n.
$$

For technical reasons, it is often simpler to work with sums of the form $\sum_{n\le N}\Lambda(n) f(n)$ than with sums of the form $\sum_{p\le N} f(p)$, and estimates for the latter sum can usually be easily derived from estimates for the former.

Let $u,v$ be arbitrary real numbers, both exceeding $1$, and define

$$
M(s) = \sum_{d\le u} \mu(d) d^{-s}, \quad F(s) = \sum_{e \le v} \Lambda(e) e^{-s}.
$$

Thus, $M$ and $F$ are partial sums of $1/\zeta$ and $-\zeta'/\zeta$ respectively. In particular,

$$
\zeta(s) M(s) = \sum_{n=1} \left( \sum_{\substack{d|n \\ d\le u}} \mu(d)\right) n^{-s}
= 1 + \sum_{n > u} \left( \sum_{\substack{d|n \\ d\le u}} \mu(d) \right) n^{-s}.
$$

Now consider the Dirichlet series identity

$$
\frac{\zeta'}{\zeta} + F 
= \zeta' M + FM\zeta + \left( \frac{\zeta'}{\zeta} + F \right)(1-\zeta M).
$$

Comparing coefficients of $n^{-s}$ on both sides of the equation, one sees that if $n > v$, then

$$
\Lambda(n) = 
\sum_{\substack{dr = n \\ d \le u}} \mu(d) \log r - 
\sum_{\substack{kr = n \\ k \le uv}} a(k) -
\sum_{\substack{ek = n \\ e > v \\ k > u}} \Lambda(e) b(k),
$$

where

$$
a(k) = \sum_{\substack{d \le u \\ e \le v \\ de = k}} \Lambda(d) \mu(e)
\quad \text{and} \quad 
b(k) = \sum_{\substack{d | k \\ d \le u}} \mu(d)
$$


If one multiplies this equation by $f(n)$ and sums over $v < n \le N$, one obtains the Vaughan identity:

$$
\begin{aligned}
\sum_{v < n \le N} \Lambda(n) f(n) 
&= \sum_{d \le u} \mu(d) \sum_{v/d \le r \le N/d} (\log r) f(rd) 
- \sum_{k \le uv} a(k) \sum_{v/k \le r \le N/k} f(rk) \\
&\quad - \sum_{v < e \le N/u} \sum_{u < k \le N/e} \Lambda(e) b(k) f(dk).
\end{aligned}
$$

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 $k \le K$ and $K \le k \le uv$, 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

$$
\sum_{n \le N} \Lambda(n) e(\alpha n) = O\left( (N q^{-1/2} + N^{4/5} + N^{1/2} q^{1/2}) \log^4 N\right)
$$

whenever $|\alpha - a/q| \le 1/q^2$. 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 $\alpha p \pmod{1}$, 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 $x$, conditional on the Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]). This requires a slightly different form of Vaughan's identity. In this case, let $M(s)$ be as before, but take

$$
F(s) = \sum_{n\le v} \mu(n) n^{-s}.
$$

From the equation

$$
\left( \frac{1}{\zeta} + M \right) (1-\zeta M) = \frac{1}{\zeta} + F - M - FM\zeta
$$

one can obtain an identity for sums of the form $\sum_{n\le N} \mu(n) f(n)$. 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 $\alpha p$ 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>