Ricci was found dead on August 3 at his home in Utah, according to family members, reported Variety.
An exact cause of death has yet to be determined.
In the 1970s, Ricci collaborated with Carl Wilson of the Beach Boys to produce the album "Beached".
In the 1990s, he joined the rock band Dino, taking the place of his late brother Dean Paul Martin. Ricci performed with the band at the MGM Grand hotel in Las Vegas and other clubs with Desi Arnaz Jr and Billy Hinsche. Dean Paul was killed in a military jet crash in 1987.
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In the memoir he recalls his 21st birthday party with a guest list that included John Lennon, Ringo Starr, Elizabeth Taylor, David Bowie and Elton John.
In recent years, Ricci had been performing a touring tribute show to his father.
Ricci is survived by three daughters, Pepper, Montana and Rio; his mother, Jeanne Martin; his older brother, Craig; and four sisters, Gail, Deana, Gina and Sasha.