We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions
t
3
/
2
,
t
log
t
t^{3/2}, t\log t
and
e
log
t
e^{\sqrt {\log t}}
. We show that if all non-trivial linear combinations of the functions
a
1
a_1
, …,
a
k
a_k
stay logarithmically away from rational polynomials, then the
L
2
L^2
-limit of the ergodic averages
1
N
∑
n
=
1
N
f
1
(
T
⌊
a
1
(
n
)
⌋
x
)
⋅
⋯
⋅
f
k
(
T
⌊
a
k
(
n
)
⌋
x
)
\frac {1}{N} \sum _{n=1}^{N}f_1(T^{\lfloor {a_1(n)}\rfloor }x)\cdot \dots \cdot f_k(T^{\lfloor {a_k(n)}\rfloor }x)
exists and is equal to the product of the integrals of the functions
f
1
f_1
, …,
f
k
f_k
in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions
a
1
a_1
, …,
a
k
a_k
, we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.