Miss Lawrence was 35 when she disappeared in March 2009.
Today North Yorkshire Police arrested three men in their 50s on suspicion of her murder.
Officers said a York man in his 50s who was arrested on suspicion of her murder in March remains on police bail.
Peter Lawrence, the chef's father and a committed Christian, was asked not to comment on the arrests by police but his official spokesman Martin Dales said it was "possitiveness" after years of negative news.
Detectives have been carrying out a number of searches, including a detailed re-examination of Miss Lawrence's home, in the Heworth area of York, over the last few months.
Miss Lawrence has still not been found despite the arrests, police said.
Last year a 60-year-old man was arrested in connection with her disappearance and suspected murder but was later released without charge.
Whilst last July a 47-year-old man was arrested on suspicion of perverting the course of justice but he was also released with no further action taken.
Detective Superintendent Dai Malyn, head of North Yorkshire Police's Major Crime Unit and the senior investigating officer in the Claudia Lawrence inquiry, said: "As with the previous arrest last month, I strongly advise the media and members of the public against identifying the men who have been arrested.
"This is to ensure the investigation and legal process are not compromised or potentially damaged in any way. This includes naming or publishing images of the men, their home addresses, places of work, their vehicles and properties being searched on traditional media platforms or social networking sites.
"I urge everyone to again show restraint and patience while we carry out these very important enquiries. Your co-operation and support is greatly appreciated by the investigation team."