In mathematics and
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
, Vaughan's identity is an
identity found by that can be used to simplify
Vinogradov's
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an ani ...
on
trigonometric sums. It can be used to estimate summatory functions of the form
:
where ''f'' is some
arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition th ...
of the natural integers ''n'', whose values in applications are often roots of unity, and Λ is the
von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Mang ...
.
Procedure for applying the method
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
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula
\frac
where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the in ...
of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
in terms of functions which are partial
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in anal ...
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, 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
* Vaughan's identity has been used to simplify the proof of the
Bombieri–Vinogradov theorem
In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over ...
and to study
Kummer sums (see the references and external links below).
* In Chapter 25 of Davenport, one application of Vaughan's identity is to estimate an important prime-related
exponential sum
In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function
:e(x) = \exp(2\pi ix).\,
Therefore, a typi ...
of
Vinogradov defined by
:
In particular, we obtain an asymptotic upper bound for these sums (typically evaluated at
irrational
Irrationality is cognition, thinking, talking, or acting without rationality.
Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
) 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
.
* Another application of Vaughan's identity is found in Chapter 26 of Davenport where the method is employed to derive estimates for sums (
exponential sum
In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function
:e(x) = \exp(2\pi ix).\,
Therefore, a typi ...
s) of
three primes.
* Examples of Vaughan's identity in practice are given as the following references / citations i
this informative post.
Generalizations
Vaughan's identity was generalized by .
Notes
References
*
*
*
*{{citation, first= R.C. , last=Vaughan, title=Sommes trigonométriques sur les nombres premiers, journal= Comptes Rendus de l'Académie des Sciences, Série A , volume=285 , year=1977, pages= 981–983, mr= 0498434
External links
Proof Wiki on Vaughan's Identity(very detailed exposition)
Encyclopedia of Mathematics Terry Tao's blog on the large sieve and the Bombieri-Vinogradov theorem
Theorems in analytic number theory
Mathematical identities