One of the general methods in
analytic number theory.
Two problems in number theory required for their solution the creation
of the method of trigonometric sums: the problem of the
distribution of the fractional parts of a polynomial (cf.
Fractional part of a number),
and the problem of representing a positive integer as the sum of
terms of a specified type
(additive problems of number theory,
cf.
Additive number theory).
Let
be a real-valued function,
,
.
One says that the fractional parts of
are uniformly distributed if for any
and
,
,
the number
of fractional parts of
occurring in the interval
is proportional to the length of this interval, that is,
Now, let
be the characteristic function of the interval
,
that is,
Extending
periodically to the entire real axis, that is, setting
,
one obtains
Expanding
in a
Fourier series,
one obtains
Thus,
This last relation is not true in general, since there may be
such that
or
;
but
and
can be replaced by
and
that are close and are such that for all
,
and
;
the precision of the formula is practically unchanged by
this substitution, and the formula becomes true. In exactly the same way, the function
can be
"smoothed out"
( "corrected" ), so that the magnitude of
is practically unchanged and the coefficients
of the Fourier series rapidly decrease as
increases, that is, so that the series
rapidly converges.
The second term in equation
(1)
does not exceed
in absolute value, where
If it is known that
where
as
,
then one obtains for
:
that is, the fractional parts of
are uniformly distributed. Thus, one must provide an upper bound of
the modulus of a trigonometric sum. Since each term in
is equal to 1 in modulus, a trivial bound of
is
,
that is, the number of terms of the sum
.
An estimate of the form
(2)
is said to be non-trivial if
,
and
is called a
reducing factor.
In the problem of the fractional parts of
,
one can, by smoothing
if necessary, merely require that the estimate
(2)
be
obtained for
"a relatively small"
number of values of
,
for example, for
in the interval
,
where
is some constant.
A similar approach is applied in the derivation of an asymptotic formula for the sum
which occurs in problems on the number of integer
points in regions of the plane and in space.
In additive problems of number theory, trigonometric sums occur in the following way.
The following formula holds for an integer
:
Therefore, if
denotes the number of solutions of the equation
where the
are certain sets of natural numbers, then
where
In particular, by setting
one obtains the
Waring problem;
for
,
,
the ternary
Goldbach problem,
etc. As in the problem on the distribution of
fractional parts, the main question here is that of
finding an upper bound of the modulus of
,
that is, the question of an upper bound of
.
Thus, a variety of problems in number theory can
be formulated uniformly in the language of trigonometric sums.
The first non-trivial trigonometric sum appeared in the work of
C.F. Gauss
(1811)
in one of his proofs of the reciprocity law for quadratic residues (cf.
Quadratic reciprocity law):
Gauss calculated the precise value of
:
A whole series of independent articles applying trigonometric sums
appeared at the beginning of the
20th century.
H. Weyl
studied the distribution of the fractional parts of a polynomial
with real coefficients
,
and considered sums
with a function
of the form
,
where
is a non-zero integer.
I.M. Vinogradov
(
1917),
in the study of the
distribution of integer points in regions of the plane and in space, considered sums
with a function
of which it was required only that its second derivative satisfy the conditions
where
are absolute positive constants and
.
G.H. Hardy
and
J.E. Littlewood
(
1918),
having obtained
an approximate functional equation of the Riemann zeta-function, considered the sum
with a function
of the form
where
is a real parameter,
.
In all these papers it was required to find a
best possible bound of the modulus of the sum
.
The general scheme for studying these problems in number theory by the
method of trigonometric sums is as follows. One writes down the exact
formula expressing the number of solutions of the equation under study, or
the number of fractional parts of the function under study occurring in a
given interval, or the number of integer points in a given region, in the
form of an integral of trigonometric sums or in the form
of a series whose coefficients are trigonometric sums. The exact formula is
expressed as the sum of two terms, the principal and the secondary
term (e.g. if one is considering the Fourier series of the
characteristic function of an interval, then the principal term is
obtained from the zero coefficient of the Fourier series); the
principal term supplies the principal term of the asymptotic formula,
the secondary term supplies the remainder term. In additive problems
such as the Waring problem, the Goldbach problem, etc., the principal term
is studied by a method close to the circle
method of Hardy–Littlewood–Ramanujan (this method is called the
circle method
in the form of Vinogradov trigonometric sums). In the
majority of other problems (the distribution of fractional parts, integer
points in regions, etc.), the principal term is obtained trivially.
There now arises the problem of estimating the remainder term, and if
it can be proved that it is a quantity of smaller order
than the principal term, then the asymptotic formula has been proved.
The main problem in estimating the remainder term is the
problem of whether more precise estimates of trigonometric sums
are possible. Concerning methods of estimating trigonometric sums, see
Trigonometric sum,
and also
Vinogradov method;
Waring problem;
Goldbach problem;
Additive problems.
For references see
Trigonometric sum.