Mahler measures
Definition
The Mahler measure of a Laurent polynomial P ∈ C[x±1
1 , . . . , x±1
n ] is
m(P) =
Z 1
0
· · ·
Z 1
0
log |P(e2πit1
, . . . , e2πitn
)|dt1 · · · dtn.
• For P ∈ C[x] monic, Jensen’s formula gives m(P) =
X
P(α)=0
|α|≥1
log |α|.
• If P has coefficients in Q, then m(P) is a period in the sense of
Kontsevich and Zagier.
• In favorable situations, m(P) should be related to L-functions.
For example (Smyth, 1981):
m(1 + x + y) = L0
(χ−3, −1)
m(1 + x + y + z) = −14ζ0
(−2).
1