An exponential sum is a finite sum of complex numbers of the form
 |
where 
is a finite index set, 
are complex
weights, and 
is a real-valued phase (McGehee et al. 1981). The unweighted case has 
. Although every summand then has
absolute value 1, cancellation can make 
much smaller than 
.
For a positive integer 
and an integer 
, the sum of 
over 
is zero when 
does not divide 
, and is 
when 
divides 
. Littlewood's
exponential sum problem concerns the average absolute
value of sums with distinct integer frequencies.
-Norm
of Exponential Sums." Ann. Math. 113, 613-618, 1981.