The behavior of a number-theoretic function for large is often difficult to determine because the function values can fluctuate considerably as increases. It is more fruitful to study partial sums and seek asymptotic formulas of the form
where is a known function of , and represents the error, a function of smaller order than for all in some prescribed range. For example, Dirichlet (1849) proves that for all ,
Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. They are valid for all . The error terms given here are not necessarily the best known.
where again is Euler’s constant.
where is a constant.
where , , and is a constant depending on and .
where , .
for some positive constant ,