The custom is to recite the paragraph of “Hayom Haras Olam etc Areshes Sifaseinu etc” each time the Shofar is sounded during Musaf. [Some have the custom to slightly bow upon reciting this paragraph in order to show their subjugation and arouse mercy in their judgment.]
R”H that coincides with Shabbos: The paragraph of Hayom Haras Olam is recited even when R”H falls on Shabbos and the Shofar is hence not blown. However the paragraph of Areshes Sifaseinu is omitted.
Is Hayom Haras Olam recited in the silent prayer of Musaf?
The Ashkenazi custom is not to recite it while the Sefaradi custom is to recite it. The Chabad custom is not to recite Hayom Haras Olam in the silent prayer of Musaf even though Tekios are sounded. See the section on Rosh Hashanah Chapter 1 Halacha 16D for the full details of this subject!
 592/6; See Siddur Rashash and Kaf Hachaim 582/36 for the Kabalistic meaning of this paragraph
When was the world created? It is disputed in the Gemara between Rebbe Eliezer and Rebbe Yehoshua as to when the world was created, in Nissan or Tishrei. Hence we see that by us saying this paragraph we rule like Rebbe Eliezer that the world was created in Tishrei. [Abudarham 101; Elya Raba 592/6; Zohar Emor; Levush] However the Tosafus rules [R”H 27] that we follow Rebbe Yehoshua that the world was created in Nissan. Nonetheless this statement still stands true as even according to Rebbe Yehoshua the world was created in thought in the month of Tishrei. [M”A 592/1]
 Seder Hayom; Shaareiy Teshuvah 591/3; Kaf Hachaim 592/9
 Admur 592/6; Rama 592/1; Rashbatz 3/290 and 309; Birkeiy Yosef 591/5; Shaareiy Teshuvah 591/6; Machzor Chabad
 Admur ibid; Rama ibid; Machzor Chabad