# §27.11 Asymptotic Formulas: Partial Sums

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 ,

27.11.2

where is Euler’s constant (§5.2(ii)). Dirichlet’s divisor problem (unsolved in 2009) is to determine the least number such that the error term in (27.11.2) is for all . Kolesnik (1969) proves that .

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 , .

Letting in (27.11.9) or in (27.11.11) we see that there are infinitely many primes if are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions.

for some positive constant ,

27.11.13
27.11.14
27.11.15

Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3). The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if , then the number of primes with is asymptotic to as .