Vinogradov notation is a form of asymptotic notation, common in number theory, that expresses bounds up
to a constant factor. For functions 
and 
with 
, the notation 
means that 
for some constant 
independent of 
throughout the specified range. This is equivalent to 
in big-O
notation (Evertse and Győry 2022, p. xv).
When the bound is an asymptotic estimate as 
, it is required only for sufficiently large 
. For example, 
holds for 
with 
.
For positive functions 
and 
, the reversed notation 
means 
. Equivalently, 
for some constant 
.
A subscript lists parameters on which the implied constant may depend. Thus 
allows 
to depend on 
but not on 
.
In an asymptotic estimate, the threshold beyond which
the bound holds may also depend on 
. The same subscript convention applies to 
.
