These are the songs that reached number one on the Top 100 Best Sellers chart in 1964 as published by Cash Box magazine.
Issue date Song ArtistJanuary 4 There! I've Said It Again Bobby VintonJanuary 11 Louie Louie The KingsmenJanuary 18 January 25 I Want to Hold Your Hand The Beatles February 1 February 8 February 15 February 22 February 29 March 7 March 14 March 21 She Loves You March 28 April 4 Twist and Shout April 11 Can't Buy Me Love April 18 April 25 May 2 May 9 May 16 Hello, Dolly! Louis Armstrong & All StarsMay 23 Love Me Do The Beatles May 30 My Guy Mary WellsJune 6 Chapel of Love The Dixie CupsJune 13 June 20 June 27 A World Without Love Peter and GordonJuly 4 I Get Around The Beach BoysJuly 11 Rag Doll The Four SeasonsJuly 18 July 25 A Hard Day's Night The Beatles August 1 August 8 August 15 Everybody Loves Somebody Dean MartinAugust 22 Where Did Our Love Go The SupremesAugust 29 September 5 The House of the Rising Sun The AnimalsSeptember 12 September 19 September 26 Oh, Pretty Woman Roy Orbison & Candy MenOctober 3 October 10 October 17 Do Wah Diddy Diddy Manfred Mann October 24 October 31 We'll Sing in the Sunshine Gale GarnettNovember 7 Last Kiss J. Frank Wilson and the CavaliersNovember 14 Baby Love The SupremesNovember 21 November 28 Leader of the Pack The Shangri-LasDecember 5 She's Not There The ZombiesDecember 12 Ringo Lorne GreeneDecember 19 I Feel Fine The Beatles December 26 Come See About Me The Supremes
See also 1964 in music List of Hot 100 number-one singles of 1964 (U.S.)
References https://web.archive.org/web/20101121003531/http://cashboxmagazine.com/archives/60s_files/1964.html