10 1. Arithmetic Functions

1.4. The Mertens estimates

Sums

p≤x

f(p)

over primes occur frequently in analytic number theory. In the simpler cases,

f is a positive, continuous, monotone function of a real variable that does

not change rapidly. If the sum diverges as x → +∞, its asymptotic behavior

may be guessed from the heuristic

p≤x

f(p) ∼

2≤n≤x

f(n)

log(n)

∼

x

2

f(u) du

log(u)

,

which is inspired by the observation that the density of the primes near n

is close to 1/ log(n). The latter statement is a formulation of the Prime

Number Theorem, and indeed the PNT with a good estimate for the error

term is a natural tool with which to estimate such sums. As an example of

the heuristic in action, consider the sum of log(p)/p over primes p ≤ x. The

guess for the asymptotic behavior is

p≤x

log(p)

p

∼

2≤n≤x

log(n)/n

log(n)

∼

x

2

du

u

∼ log(x),

and this is actually correct. Indeed, F. C. J. Mertens proved in 1874 that

the absolute error in the asymptotic approximation is bounded. This may

be expressed as

p≤x

log(p)

p

= log(x) + O(1)

by means of the big-O notation for the error term.

The heuristic for guessing the asymptotic behavior of sums over primes

will not perform satisfactorily if f does not have the nice properties assumed.

If, for example, f changes sign, there will be cancellation in the sum, and

the underlying rationale for the heuristic does not take account of this.

Partial summation is perhaps the tool most frequently applied in analytic

number theory. The basic version is the identity

n

m=1

ambm = bn

n

m=1

am −

n−1

m=1

(bm+1 − bm)

m

k=1

ak.

This is an analogue, for sums, of integration by parts. The partial sum-

mation identity is easily proved by observing that bj(a1 + · · · + aj) =

bj(a1 + · · · + aj−1)+ ajbj and applying mathematical induction. The partial

summation identity yields a formula that is very convenient for estimating

weighted sums of arithmetic functions.