We do not recite any vidui (confession) on Rosh Ha-shana – neither communal nor personal – because the theme of Rosh Ha-shana is to crown God king over us and the entire world, and to ask for the upcoming year to be a good one, enabling us to reveal the glory of His kingship. It is not appropriate to use this day for our personal self-improvement. That is what the rest of the Ten Days of Repentance are for. It is also not appropriate to mention sins, which diminish the glory of His kingship. There is even concern that mentioning them might strengthen the accusers’ case against us (Zohar II 186a; according to Arizal, in between the shofar blasts one may quietly confess; see below 4:7).
Some maintain that just as we do not recite vidui on Rosh Ha-shana, we should also leave out the lines of the Avinu Malkeinu prayer that mention sin, like, “Our Father, our King, we have sinned before You” (Beit Yosef; Arizal). Others maintain the custom of reciting the entire Avinu Malkeinu, arguing that the lines that mention sin are not confessional but simply note our general state (Rema 584:1; MB ad loc. 3).
Even though we normally recite Hallel on holidays, we do not do so on Rosh Ha-shana because it is a day of judgment (as explained above, section 4).
There are many changes made in the Amida during the Days of Awe. The most important one is in the conclusion of the third berakha where we substitute “ha-Melekh ha-kadosh” for “ha-Kel ha-kadosh.” One who forgot to incorporate this change has not fulfilled his prayer obligation, as explained below (5:2).
The laws and customs pertaining to the ḥazan are explained above (2:10). The custom to stand when the ark is open, and the admonition not to pray out loud nowadays, are explained above as well (2:11).
As on all the festivals, five people are called up to the Torah (Peninei Halakha: Festivals 2:8). On the first day, we read the story of Yitzḥak’s birth and the banishment of Hagar and Yishmael (Bereishit 21:1-34), and the haftara is the story of Ḥanna and the birth of Shmuel (SA 584:2). This is appropriate, as three righteous women conceived on Rosh Ha-shana: Our matriarch Sarah, mother of Yitzḥak; our matriarch Raḥel, mother of Yosef; and Ḥanna, mother of the prophet Shmuel (Rosh Ha-shana 10b). These women were so exceptionally righteous that they were destined to give birth to children whose souls were radically new. These souls could not be born naturally, which is why their mothers remained barren for so long. It was only through the renewal of Rosh Ha-shana that they could be helped.
Most of the Torah reading, however, tells of the banishment of Yishmael. We can learn two principles from this: 1) Despite the pain it caused, the banishment of Yishmael was not in any way immoral. Had it been, the Sages would not have ordained its reading on Rosh Ha-shana, as they would not have wanted to provide the accusers with material. 2) Especially on Rosh Ha-shana, the day of judgment, it is important to distinguish between the Jewish people and the rest of the nations, who are not willing to accept upon themselves the great and awe-inspiring mission of repairing the world under the kingship of God – just as it was necessary to separate Yishmael from Yisrael.
On the second day we read the story of the binding of Yitzḥak (Bereishit 22:1-24), in order to invoke the merit of our ancestors. We then read a comforting haftara from the Book of Yirmiyahu (31:1-19). On both days, the custom is to take out a second Torah scroll for the maftir reading, whose subject is the holiday offerings (Beit Yosef OḤ 488:2).