Prime Numbers

48 Chapter 1 PRIMES!
been beyond reach, the analogous error term ɛ ′′ , which includes yet noisier
components, being so very difficult to bound.)
In summary: The estimate (1.39) is used for major-arc “resonances,”
yielding the main-term sum of (1.40), while the estimate (1.41) is used to
bound the minor-arc “noise” and control the overall error ɛ ′′ . The relation
(1.40) leads finally to the ternary-Goldbach estimate (1.12). Though this
language has been qualitative, the reader may find the rigorous and compelling
details—on this and related additive problems—in the references [Hardy
1966], [Davenport 1980], [Vaughan 1977, 1997], [Ellison and Ellison 1985,
Theorem 9.4], [Nathanson 1996, Theorem 8.5], [Vinogradov 1985], [Estermann
1952].
Exponential-sum estimates can be, as we have just seen, incredibly
powerful. The techniques enjoy application beyond just the Goldbach problem,
even beyond the sphere of additive problems. Later, we shall witness the
groundwork of Gauss on quadratic sums; e.g., Definition 2.3.6 involves
variants of the form (1.33) with quadratic f. In Section 9.5.3 we take
up the issue of discrete convolutions (as opposed to continuous integrals)
and indicate through text and exercises how signal processing, especially
discrete spectral analysis, connects with analytic number theory. What is
more, exponential sums give rise to attractive and instructive computational
experiments and research problems. For reader convenience, we list here some
relevant Exercises: 1.35, 1.66, 1.68, 1.70, 2.27, 2.28, 9.41, 9.80.
1.4.5 Smooth numbers
Smooth numbers are extremely important for our computational interests,
notably in factoring tasks. And there are some fascinating theoretical
applications of smooth numbers, just one example being applications to a
celebrated problem upon which we just touched, namely the Waring problem
[Vaughan 1989]. We begin with a fundamental definition:
Definition 1.4.8. A positive integer is said to be y-smooth if it does not
have any prime factor exceeding y.
What is behind the usefulness of smooth numbers? Basically, it is that for y
not too large, the y-smooth numbers have a simple multiplicative structure,
yet they are surprisingly numerous. For example, though only a vanishingly
small fraction of the primes in [1,x]areintheinterval[1, √ x], nevertheless
more than 30% of the numbers in [1,x]are √ x-smooth (for x sufficiently
large). Another example illustrating this surprisingly high frequency of smooth
numbers: The number of (ln 2 x)-smooth numbers up to x exceeds √ x for all
sufficiently large numbers x.
These examples suggest that it is interesting to study the counting function
for smooth numbers. Let
ψ(x, y) =#{1 ≤ n ≤ x : n is y-smooth}. (1.42)
Part of the basic landscape is the Dickman theorem from 1930: