"Hopefully I can come many more years here. As I say, I don't think it'll only be good ones but I hope it'll be more good times than bad ones, otherwise it won't work."
Medvedev cruised through in straight sets against van de Zandschulp, winning 6-4, 6-4, 6-2, while fourth-seeded Stefanos Tsitsipas needed an extra set to beat mercurial Frenchman Benoit Paire, triumphing 6-3, 7-5, 6-7 (2), 6-4.
He will next play No 20 seed Taylor Fritz, who tipped over 15th seed Roberto Bautista Agut in five sets.
Maxime Cressy, a 24-year-old American, is into the fourth round at a major for the first time.
The No 70-ranked Cressy beat Australian wildcard entry Chris O'Connell 6-2, 6-7 (6), 6-3, 6-2 and will next play Medvedev.
No 9 Felix Auger-Aliassime had a 6-4, 6-1, 6-1 win over No 24 Dan Evans, winning 14 of the last 16 games after it was 4-4 in the first set.
- news.com.au and AP