Prime Numbers

538 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
cubic case)? In such research, you would have to find upper bounds on general
U sums, and indeed these can be obtained; see [Vinogradov 1985], [Ellison and
Ellison 1985], [Nathanson 1996], [Vaughan 1997]. However, the hard part is
to establish explicit s, which means explicit bounding constants need to be
tracked; and many references, for theoretical and historical reasons, do not
bother with such detailed tracking.
One of the most fascinating aspects of this research area is the fusion of
theory and computation. That is, if you have bounding parameters ck, ɛk for
k-th power problems as above, then you will likely find yourself in a situation
where theory is handling the “sufficiently large” N, yet you need computation
to handle all the cases of N from the ground up to that theory threshold.
Computation looms especially important, in fact, when the constant ck is
large or, to a lesser extent, when ɛk is small. In this light, the great efforts
of 20th-century analysts to establish general bounds on exponential sums can
now be viewed from a computational perspective.
These studies are, of course, reminiscent of the literature on the celebrated
Waring conjecture, which conjecture claims representability by a fixed number
s of k-th powers, but among the nonnegative integers (e.g., the Lagrange
four-square theorem of Exercise 9.41 amounts to proof of the k =2, s =4
subcase of the general Waring conjecture). The issues in this full Waring
scenario are different, because for one thing the exponential sums are to
be taken not over all ring elements but only up to index x ≈ ⌊N 1/k ⌋ or
thereabouts, and the bounding procedures are accordingly more intricate.
In spite of such obstacles, a good research extension would be to establish
the classical Waring estimates on s for given k—which estimates historically
involve continuous integrals—using discrete convolution methods alone. (In
1909 D. Hilbert proved the Waring conjecture via an ingenious combinatorial
approach, while the incisive and powerful continuum methods appear in many
references, e.g., [Hardy 1966], [Nathanson 1996], [Vaughan 1997].) Incidentally,
many Waring-type questions for finite fields have been completely resolved;
see for example [Winterhof 1998].
9.81. Is there a way to handle large convolutions without DFT, by using
the kind of matrix idea that underlies Algorithm 9.5.7? That is, you would
be calculating a convolution in small pieces, with the usual idea in force: The
signals to be convolved can be stored on massive (say disk) media, while the
computations proceed in relatively small memory (i.e., about the size of some
matrix row/column).
Along these lines, design a standard three-FFT convolution for arbitrary
signals, except do it in matrix form reminiscent of Algorithm 9.5.7, yet do not
do unnecessary transposes. Hint: Arrange for the first FFT to leave the data
in such a state that after the usual dyadic (spectral) product, the inverse FFT
can start right off with row FFTs.
Incidentally, E. Mayer has worked out FFT schemes that do no transposes
of any kind; rather, his ideas involve columnwise FFTs that avoid common
memory problems. See [Crandall et al. 1999] for Mayer’s discussion.