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One of the general methods in [[Analytic number theory|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|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|Additive number theory]]).

Let  $  f ( x) $
be a real-valued function,  $  x = 1 \dots P $, 
$  P \rightarrow + \infty $. 
One says that the fractional parts of  $  f ( x) $
are uniformly distributed if for any  $  \alpha $
and  $  \beta $, 
$  0 \leq  \alpha < \beta < 1 $, 
the number  $  \sigma $
of fractional parts of  $  f ( x) $
occurring in the interval  $  ( \alpha , \beta ) $
is proportional to the length of this interval, that is,

$$ 
\sigma  =  ( \beta - \alpha ) P + o ( P).
$$

Now, let  $  \psi ( x) $
be the characteristic function of the interval  $  ( \alpha , \beta ) $, 
that is,

$$ 
\psi ( x)  =  \left \{

Extending  $  \psi ( x) $
periodically to the entire real axis, that is, setting  $  \psi ( x + 1) = \psi ( x) $, 
one obtains

$$ 
\sigma  =  \sum _ {x = 1 } ^ { P }  \psi ( f ( x)).
$$

Expanding  $  \psi ( x) $
in a [[Fourier series|Fourier series]], one obtains

$$ 
\psi ( x)  =  \sum _ {m = - \infty } ^ { {+ }  \infty }
c ( m) e ^ {2 \pi imx } ,\ \ 
c ( 0)  =  \beta - \alpha .
$$

Thus,

$$ \tag{1 }
\sigma  =  ( \beta - \alpha )
P + \sum _ {\begin{array}{c}
m = - \infty \\
 m \neq 0 
\end{array}
 } ^ { {+ }  \infty } c ( m)
\sum _ {x = 1 } ^ { P }  e ^ {2 \pi imf ( x) } .
$$

This last relation is not true in general, since there may be  $  x $
such that  $  \{ f ( x) \} = \alpha $
or  $  \{ f ( x) \} = \beta $; 
but  $  \alpha $
and  $  \beta $
can be replaced by  $  \alpha  ^  \prime  $
and  $  \beta  ^  \prime  $
that are close and are such that for all  $  x = 1 \dots P $, 
$  \{ f ( x) \} \neq \alpha  ^  \prime  $
and  $  \{ f ( x) \} \neq \beta  ^  \prime  $; 
the precision of the formula is practically unchanged by this substitution, and the formula becomes true. In exactly the same way, the function  $  \psi ( x) $
can be  "smoothed out"  ( "corrected" ), so that the magnitude of  $  \sigma $
is practically unchanged and the coefficients  $  c ( m) $
of the Fourier series rapidly decrease as  $  m $
increases, that is, so that the series

$$ 
\sum _ {m = - \infty } ^ { {+ }  \infty } | c ( m) |
$$

rapidly converges.

The second term in equation (1) does not exceed  $  \kappa $
in absolute value, where

$$ 
\kappa  =  \sum _ {\begin{array}{c}
m = - \infty \\
 m \neq 0 
\end{array}
 } ^  \infty   | c ( m) |
\left | \sum _ {x = 1 } ^ { P }  e ^ {2 \pi imf ( x) } \right | .
$$

If it is known that

$$ \tag{2 }
| S |  =  \left | \sum _ {x = 1 } ^ { P } 
e ^ {2 \pi imf ( x) } \right |  \leq   P \Delta ,
$$

where  $  \Delta = \Delta ( P) \rightarrow 0 $
as  $  P \rightarrow + \infty $, 
then one obtains for  $  \sigma $:

$$ 
\sigma  =  ( \beta - \alpha ) P + o ( P),\ \ 
o ( P)  =  cP \Delta ,
$$

that is, the fractional parts of  $  f ( x) $
are uniformly distributed. Thus, one must provide an upper bound of the modulus of a trigonometric sum. Since each term in  $  S $
is equal to 1 in modulus, a trivial bound of  $  | S | $
is  $  P $, 
that is, the number of terms of the sum  $  S $. 
An estimate of the form (2) is said to be non-trivial if  $  \Delta < 1 $, 
and  $  \Delta $
is called a reducing factor.

In the problem of the fractional parts of  $  f ( x) $, 
one can, by smoothing  $  \psi ( x) $
if necessary, merely require that the estimate (2) be obtained for  "a relatively small"  number of values of  $  m $, 
for example, for  $  m $
in the interval  $  0 < | m | < (  \mathop{\rm ln}  P)  ^ {A} $, 
where  $  A > 0 $
is some constant.

A similar approach is applied in the derivation of an asymptotic formula for the sum

$$ 
\sum _ {x = 1 } ^ { P } 
\{ f ( x) \} ,
$$

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  $  m $:

$$ 
\int\limits _ { 0 } ^ { 1 }  e ^ {2 \pi i \alpha m }  d \alpha  =  \left \{

Therefore, if  $  I ( N) $
denotes the number of solutions of the equation

$$ 
N  =  u _ {1} + \dots + u _ {k} ,\ \ 
u _  \nu  \in U _ {N} ,\ \ 
\nu = 1 \dots k,
$$

where the  $  U _  \nu  $
are certain sets of natural numbers, then

$$ 
I ( N)  =  \int\limits _ { 0 } ^ { 1 } 
S _ {1} ( \alpha ) \dots S _ {k} ( \alpha )
e ^ {- 2 \pi i \alpha N }  d \alpha ,
$$

where

$$ 
S _  \nu  ( \alpha )  = \ 
\sum _ {u _  \nu  \in U _  \nu  }
e ^ {2 \pi i \alpha u _  \nu  } ,\ \ 
\nu = 1 \dots k.
$$

In particular, by setting  $  U _  \nu  = U = \{ 1  ^ {n} , 2  ^ {n} ,\dots \} $
one obtains the [[Waring problem|Waring problem]]; for  $  k = 3 $, 
$  U _  \nu  = U = \{ 2, 3, 5, 7, 11 \dots p ,\dots \} $, 
the ternary [[Goldbach problem|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  $  S _  \nu  ( \alpha ) $, 
that is, the question of an upper bound of  $  | S _  \nu  ( \alpha ) | $.

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|Quadratic reciprocity law]]):

$$ 
S  =  \sum _ {x = 1 } ^ { P }  e ^ {2 \pi iF ( x) } ,\ \ 
F ( x)  =  
\frac{ax  ^ {2} }{P }
 .
$$

Gauss calculated the precise value of  $  S $:

$$ 
S  =  \left \{

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

$$ 
f ( x)  =  \alpha _ {1} x + \dots + \alpha _ {n} x  ^ {n}
$$

with real coefficients  $  \alpha _ {1} \dots \alpha _ {n} $, 
and considered sums  $  S $
with a function  $  F ( x) $
of the form  $  F ( x) = mf ( x) $, 
where  $  m $
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  $  S $
with a function  $  F ( x) $
of which it was required only that its second derivative satisfy the conditions

$$ 
{
\frac{c _ {1} }{A}
 }  \leq  \ 
F ^ { \prime\prime } ( x)  \leq  \ 
{
\frac{c _ {2} }{A}
 } ,
$$

where  $  c _ {1} , c _ {2} $
are absolute positive constants and  $  A = A ( P) \rightarrow + \infty $. 
G.H. Hardy and J.E. Littlewood (1918), having obtained an approximate functional equation of the Riemann zeta-function, considered the sum  $  S $
with a function  $  F ( x) $
of the form

$$ 
F ( x)  =  t   \mathop{\rm ln}  x,
$$

where  $  t $
is a real parameter,  $  t = t ( P) $.

In all these papers it was required to find a best possible bound of the modulus of the sum  $  S $.

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|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|Trigonometric sum]], and also [[Vinogradov method|Vinogradov method]]; [[Waring problem|Waring problem]]; [[Goldbach problem|Goldbach problem]]; [[Additive problems|Additive problems]].

For references see [[Trigonometric sum|Trigonometric sum]].