Serena Williams walked into her news conference at Wimbledon holding her phone, a cold bottle of water and a statistics sheet that reinforced what was clear from watching her third-round singles victory Saturday:

She is as close to being back to her best as she's been in a while.

COCO GAUFF, 15, ADVANCES TO ROUND 4 AT WIMBLEDON

Williams, hampered for much of this season by injuries or illness, took a step forward against 18th-seeded Julia Goerges, a powerful hitter in her own right who lost to the American in last year's semifinals at the All England Club. Sure enough, Williams hit serves at up to 120 mph, put in a tournament-best 71 percent of her first serves, never faced so much as one break point and won 6-3, 6-4.

"It's been an arduous year for me," said Williams, who had competed only 12 times in 2019 until this week, mostly because of a bothersome left knee that finally is pain-free. "So every match, I'm hoping to improve tons."

Maybe it was a good thing she played twice Saturday, then.

About 4½ hours after getting past Goerges at No. 1 Court, Williams headed out to Centre Court for her much-ballyhooed debut as Andy Murray's teammate in mixed doubles. Other than one slip near the net when she lost her footing in the first set — she was fine and laughed it off — Williams looked good during the 6-4, 6-1 win against Andreas Mies and Alexa Guarachi, including smacking one serve at 122 mph, equaling the fastest hit in singles by any woman (her, naturally) during the tournament.

"Andy and I both love the competition. I know we both want to do well," Williams said. "We're not here just for show."

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She rarely is.

But if Williams is going to win an eighth singles championship at Wimbledon, and a record-tying 24th Grand Slam singles title overall, she will want more performances like the one she gave against Goerges.

Forceful, yes, but nothing was forced.

"I play pretty good when I'm calm, but also super-intense, just finding the balance in between there," the 37-year-old Williams said. "So it's a hard balance to find, because sometimes when I'm too calm, I don't have enough energy. Still trying to find that balance."

Two more key stats on the paper she brought to her media session: She produced more winners than unforced errors, 19-15, while Goerges finished with 32 forced errors, a reflection of just how difficult Williams can make it for opponents to handle shots she sends their way.

Goerges credited Williams with causing havoc with her returns, as well.

After averaging 10 aces in the first two rounds, Goerges was limited to half that many.

Of more significance, perhaps, was that Williams' stinging replies to serves immediately put her in control of points.

"It's fair to say that she builds up enormous pressure with her returns," Goerges said. "That means I need to go to the limit in my service games."

After the traditional middle Sunday off, action resumes Monday with all fourth-round men's and women's singles matches.

Williams, who is seeded 11th, will face No. 30 Carla Suarez Navarro, while the other matchups on the top half of the women's field established Saturday are No. 1 Ash Barty, who has a 15-match winning streak, against unseeded Alison Riske of the U.S.; No. 21 Elise Mertens against Barbora Strycova; and two-time champion Petra Kvitova against No. 19 Johanna Konta of Britain.

On the bottom half, it will be the 15-year-old American sensation Coco Gauff vs. No. 7 Simona Halep; No. 3 Karolina Pliskova vs. Karolina Muchova; No. 8 Elina Svitolina vs. No. 24 Petra Martic; and Dayana Yastremska vs. Shuai Zhang.

In the men's draw, eight-time champion Roger Federer and two-time winner Rafael Nadal both won in straight sets Saturday to move closer to a semifinal showdown. Federer's record 17th visit to the fourth round at Wimbledon will come against No. 17 Matteo Berrettini, an Italian never before this far at the grass-court tournament.

"For me, I'm very happy how it's going so far," said Federer, a 7-5, 6-2, 7-6 (4) winner over No. 27 Lucas Pouille. "I hope it's going to take a special performance from somebody to stop me, not just a mediocre performance."

Nadal, who defeated Jo-Wilfried Tsonga 6-2, 6-3, 6-2, meets unseeded Joao Sousa next.

No. 8 Kei Nishikori meets Mikhail Kukushkin, and Sam Querrey plays Tennys Sandgren in the first Week 2 matchup at Wimbledon between two American men since Pete Sampras beat Jan-Michael Gambill in the 2000 quarterfinals.

Kukushkin's four-set victory over Jan-Lennard Struff at Court 12 was interrupted when a 60-year-old female spectator had to be resuscitated after collapsing.

Sandgren beat No. 12 Fabio Fognini 6-3, 7-6 (12), 6-3 at tiny Court 14, with its 318 seating capacity. Fognini unleashed a tirade in Italian at one moment, saying he wanted a bomb to explode at the All England Club. He later said his comments came in the heat of the moment because he was upset about not playing well and the condition of the court's grass.

"If I offended anyone, I apologize," said the volatile Fognini, who was fined $27,500 at Wimbledon in 2014 for unsportsmanlike conduct and is in a Grand Slam probationary period after getting kicked out of the 2017 U.S. Open. "That definitely wasn't my intention."