For a positive integer
,
let
denote the number
of
-regular partitions of
a nonnegative integer
.
In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity
results for
modulo for
certain values of
.
Further, they proposed some conjectures on self-similarities of
modulo for
certain values of
.
For example, for a positive proportion of primes
, they
conjectured that
satisfies
where
,
. In this paper, we prove
their conjectures on
and
. We also prove a
self-similarity result for
modulo .
The proofs use a result of Serre which says that if
is an integer weight cusp form on the congruence subgroup
, then for any positive
integer
, a positive
proportion of the primes
has the property that
.
Keywords
$t$-regular partitions, eta-quotients, modular forms