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Shacharis-The Morning Prayers:
The laws below apply to both the first and second day of Rosh Hashanah.
A. May one eat prior to Shacharis?
It is forbidden to eat a meal prior to hearing Shofar. [Even those which generally eat before prayer during the year so they have strength to pray, may not eat a meal prior to hearing Shofar on Rosh Hashanah. This applies equally to both days of Rosh Hashanah.]
Snacking: Some Poskim rule it is only forbidden to eat a meal, however one may eat up to a Kebeitza [55 grams] of Mezonos and have an unlimited amount of fruit and drink. [Based on this some communities are accustomed to eat after Kerias Hatorah, prior to Tekias Shofar.] Nevertheless many Poskim rule that one may not eat even a snack prior to hearing Shofar unless one is weak or sick. Practically many are accustomed to fast and not eat or drink anything until one hears Shofar. There is no prohibition in doing so even if one will only be able to eat after midday. However some say that even according to the Chabad custom one may drink prior to Davening.
It is accustomed not to eat prior to hearing Shofar although one may drink. Many are accustomed to fast until they hear Shofar.
Is one who is old or weak or sick and thus needs to eat to hear Shofar early in the morning?
As explained above one who is sick or weak may eat up to 55 grams of Mezonos, unlimited amount of fruit and vegetables, and unlimited amount of beverages prior to Shacharis. However they may not eat a full meal or over 55 grams of Mezonos until they hear Shofar. Thus if they feel a need to eat a meal they should hear the Shofar early before Davening and then eat. Furthermore, some Poskim rule that in all cases that one feels he needs to eat, if possible, he should hear Shofar beforehand and only then eat.
May women eat prior to hearing Shofar?
It is customary for women not to eat at all until they hear Shofar. However from the letter of the law women are allowed to eat up to 55 grams of Mezonos, unlimited amount of fruit and vegetables, and unlimited amount of beverages. This certainly applies to a woman who is pregnant, nursing or feels weak. However she should not eat a full meal or over 55 grams of Mezonos until she hears Shofar. If however they feel that they require this amount of food to eat then it is completely allowed. Nevertheless, some Poskim rule that in all cases that a woman feels she needs to eat, if possible, she is to hear Shofar beforehand and only then eat.
B. The length of the morning Prayer:
On Rosh Hashanah we lengthen the Piyutim and prayers until at least midday. When Rosh Hashanah coincides with Shabbos one should not lengthen the prayer past midday. When Rosh Hashanah falls on a weekday, one may lengthen the prayers past midday. However this only refers to the Piyutim and prayers however one should not lengthen the singing past midday, being that Rosh Hashanah is a holiday.
Customs related to Shacharis
It is customary to immerse in a Mikveh prior to Shacharis even if one already immersed on Erev Rosh Hashanah. This especially applies to the Baal Tokeia and Chazan.
Bringing children to Shul:
It is a custom of Jewry to place great effort that every child be present in Shul for some time on Rosh Hashanah. Based on his age he is to be present for the prayers, blessings, Tekias Shofar, answering Amen. The children perform this with joy until it affects even the adults.
The following guarantee was made by Rav Yehuda Hachassid; Rav Haiy Gaon and Rav Sherira Gaon: One who concentrates while reciting the opening paragraph of Adon Olam I guarantee that his prayers will be heard and the Satan will not prosecute against his prayers. He will not have any Satan [prosecute] during his prayers of Rosh Hashanah and Yom Kippur and his enemies will fall before him.
The custom is to have the Chazan of Shacharis switch the Chazan of Pesukei Dezimra starting from the word Hamelech. The word Hamelech is recited by the Chazan in the slow traditional tune, lengthening the verbalization of the word. Some have the custom to lengthen the letter “Hei” of the word Hamelech. Others have the custom that the Chazan recites the word Hamelech in his place and only afterwards comes to the Amud. Others have the custom to repeat the word Hamelech after the Chazan. The above customs are not the Chabad custom. [Nevertheless each person is to read all the words quietly to himself.]
Shir Hamaalos Mimamakim is recited between Yishtabach and Kaddish. See Halacha 3 for further details on this subject.
Many are accustomed to recite Piyutim within Birchas Shema. The Chabad custom is not to recite any Piyutim other than those that are recited during the Chazan’s repetition.
Leaving the Tallis over one’s head for the duration of prayer:
The Rebbe did not remove his Tallis from over his head for the entire duration of the prayers.
C. Chazaras Hashatz:
Being Yotzie Shemoneh Esrei with another: Those communities who are accustomed to recite long Piyutim during the Chazan’s repetition are unable to have anyone fulfill their obligation with the Chazan’s repetition, even if they listen to the entire prayer from the Chazan, from beginning to end. One is never able to fulfill the obligation of Shemoneh Esrei through hearing it from another person unless the person is saying it in the presence of another nine people that are listening and answering Amen to his blessings. This applies even if the person reciting the Shemoneh Esrei is Davening in order to fulfill his own personal obligation.
Standing when the ark is open:
From the letter of the law one is not required to stand when the ark is open, however the custom of the world is to stand out of respect. Nevertheless if it is difficult for one to stand he may be seated.
Standing at the end of each blessing:
In certain years the Rebbe was accustomed to slightly lift himself towards the conclusion of each blessing of Chazaras Hashatz and by the recital of Ukesov and Ubisefer. However in the later years the Rebbe remained sitting throughout the repetition, including the concluding blessings.
Reading along with the Chazan by Chazaras Hashatz:
Although one is required to listen to each word recited by the Chazan in his repetition of the Amidah, nevertheless on Rosh Hashanah the custom in some communities is to read along together with the Chazan. Nevertheless one may not read along louder than the actual Chazan. In however many communities this custom is not followed and they listen silently to the Chazan’s repetition, with exception to the verses of Malchiyos, Zichronos, Shofros of Musaf which some read along silently together with the Chazan. Those that desire to add in their concentration to the words of the Chazan may however read along with him. There is no prohibition in reciting Hashem’s name in these verses as it is like reading verses from the Torah. Those that follow the previous custom may not recite any of Hashem’s names that are part of the prayer. Thus he may not say Elokeinu Velokei Avoseinu, and is rather to start from the word Yimloch.
What is the law if the Chazan made an invalidating mistake in his private Shemoneh Esrei on Rosh Hashanah, may he rely on his repetition as is the law during the regular year?
See Halacha 17A Q&A regarding the law by the Musaf repetition and the same applies for the Shacharis repetition, if the Chazan recites long Piyutim in the Chazarah.
Niggun by Kaddish Tiskabel:
It is customary to sing a hymn when reciting the Kaddish Tiskabel which follows the Chazan’s repetition. This shows that we trust that our prayers have been accepted before Hashem and we will be blessed with a good year. Some however have the custom to end the hymn upon reciting the words Tiskabel.
The Halel prayer is not recited on Rosh Hashanah or Yom Kippur.
Saying Halel in Tehillim: Those that say Tehillim on a daily basis and have reached the psalms of Halel on Rosh Hashanah and Yom Kippur [as part of their regular cycle] are permitted to recite these psalms as usual. [However those that are not saying the Tehillim as part of any cycle may not specifically recite the psalms of Halel in Tehillim.]
On Rosh Hashanah one may not specifically recite the psalms of Halel in Tehillim. If however one is saying a regular cycle of Tehillim he is not required to skip the psalms of Halel.
E. Avinu Malkeinu:
Avinu Malkeinu is recited on Rosh Hashanah [after each Shacharis and Mincha Shemoneh Esrei]. [It is recited even if one is not Davening with a Minyan. The custom is to open the ark prior to its recital. The custom is to stand during its recital.]
Omitting part of the Stanzas: [Some opinions say that] Avinu Malkeinu is recited as usual. However some have the custom to omit the stanza of “Avinu Malkeinu Chatanu Lefanecha”. [Practically we omit on Rosh Hashanah all stanzas that mention sin such as Chet or Avon and the like.]
Mincha Erev Shabbos: [In addition to the above] Avinu Malkeinu is also omitted by Mincha of Erev Shabbos. [Thus Avinu Malkeinu is omitted by Mincha of Erev Shabbos Shuva and by Mincha of the second day of Rosh Hashanah that coincides with Erev Shabbos.]
Sparks Of Chassidus:
Why do we not confess on Rosh Hashanah?
On Rosh Hashanah we avoid mentioning the confession prayer. This is because the main focus of Rosh Hashanah is the coronation of G-d as our King, as it is on Rosh Hashanah that we accept the yolk of Heaven for the entire coming year. It is for this reason that no confession is made, as confession brings one to a state of remorse and broken heartedness which is a most unwanting atmosphere for the day of the Kings coronation. Rather we must rejoice over the acceptance of G-d to be our King. Now, although certainly one must also repent and have remorse on Rosh Hashanah, as how can there be a coronation if the nation is still in the midst of rebellion, nevertheless this is only a general remorse of one’s lack of discipline in following the Kings commands throughout the past year and a general re-acceptance of the Kings commands. It however is not remorse on each particular sin. A particular confession relates to Yom Kippur which is designated as the day of Teshuvah.
F. Davening Shacharis without a Minyan:
Although one who is Davening alone is not to Daven Musaf of Rosh Hashanah until three hours into the day have passed nevertheless one who is Davening Shacharis of Rosh Hashanah without a Minyan may Daven even within the first three hours of the day. [Furthermore it is a Mitzvah Min Hamuvchar to pray at the same time as the congregation is praying in Shul.] This is opposed to Musaf which one is not to Daven within the first three hours of the day.
 Shulchan Aruch 584
 See Piskeiy Teshuvos 585/2; Hearos Hatemimim Veanash 770 vol. 322 p. 64 for a general overview of this subject; See “Noam” 19 p. 181-188 for an article of Rav Yosef Simcha Ginzberg on this topic.
 Mateh Efraim 588/2; Alef Hamagen 581/81; Ashel Avraham Butchach 589/3; Takanos Rav Akiva Eiger that even during the epidemic he ruled to only eat after Tekios Meyushav; See Kaf Hachaim 585/26 for a discussion why this was not explicitly mentioned in Poskim by Shofar; See Piskeiy Teshuvos ibid footnote 10; This prohibition applies before all time dependent Mitzvos: 1) Regarding Mincha and Maariv: See 232/2; 235/2; 2) Regarding Bedikas Chameitz see: 431/5; 3) Regarding Lulav see: 652/2; 4) Regarding Megillah see: Rama 692/4; M”B 692/15; Kaf Hachaim 692/37.
The reason: As is the case whenever one has a Mitzvah that must be fulfilled by a certain time, that one may not eat a meal until he fulfills the Mitzvah due to fear he may come to forget to perform the Mitzvah.
The law of a Shomer: The general rule regarding eating before Mitzvos is that a Shomer validates one to be able to eat. [see Rama 232/2]; However this allowance seemingly only applies by Davening and not before Mitzvos. This explains why the allowance of a Shomer is not mentioned in Poskim by the Mitzvah of Megillah, Lulav, Shofar etc. [See Piskeiy Teshuvos ibid footnote 11; Admur 431/11 Kuntrus Achron 2]
 The reason: See previous footnote. This is similar to the custom that even those that eat before Davening do not eat prior to reciting Shema.
 See Alef Hamagen 581/81 [in footnote that all snacking is permitted before a Mitzvah]; Darkei Chaim Veshalom 716 [regarding tea and coffee]; Hisorerus Teshuvah 147; Olas Shmuel 82; Tzitz Eliezer 6/7; 7/32; 8/21; Az Nidbaru 1/10; Kaf Hachaim 585/26 [brings opinions in this matter]; See “Noam” ibid.
 Regarding eating between Kerias Hatorah and Tekias Shofar and whether Kiddush is required-see Halacha 13A!
 Mateh Efraim 588/2; Kaf Hachaim 585/26; 588/11; Chasam Sofer 7; Takanas Rav Akiva Eiger ibid; Mahrsham 1/1; Heishiv Moshe 19; Shaareiy Teshuvah 584/3; Minchas Yitzchak 5/111; Betzeil Hachochmah 4/147; Sheivet Halevy 4/54; Lehoros Nasan 1/29; Piskeiy Teshuvos ibid footnote 15; Regarding the prohibition to snack prior to Megillah reading unless one is weak or sick see: M”A 692/7; Elya Raba 692/11; M”B 692/14; Kaf Hachaim 692/36
 Rokeach Hagadol 353, brought in Alef Hamagen 581/81 and 588/2; Beshamayim Rosh 74; Pischeiy Olam 74; see Sdei Chemed Daled Minim 3/22 and so writes Hiskashrus, and Otzer Minhagei Chabad 176, and so is custom of many of Anash.
 The following Poskim rule that on R”H one may fast even past midday: Ateres Zekeinim 597; M”B 597/2; Kaf Hachaim 597/6; Implication of Rokeach and Alef Hamagen, and so is the practical custom of many of Anash.
 Rav Zalmin Shimon Dworkin, and so is the custom in Tomchei Temimim in 770. [brought in Hiskashrus], and so ruled to me Rav Avraham Osdaba, as well as Rav Tuvia Bloy
 Mateh Efraim 585/9; Sdei Chemed R”H 2/40; Piskeiy Teshuvos ibid footnote 14; See Ashel Avraham Butchach 689 and Mahadurah Tinyana 592; Ateres Zikeinim 689
 Ateres Zikeinim 589; Ashel Avraham Tinyana 592; See Kaf Hachaim 585/25; Olas Shmuel 82, 83, 87
 Ashel Avraham Butchach 589 and Mahadurah Tinyana 592; so is implied also from Kitzur SHU”A 129/19; Ateres Zikeinim 589
The reason: Although women are not obligated to hear Shofar, nevertheless since they accepted it upon themselves as an obligation [see Admur 17/3], they are therefore also to follow the ruling of not eating prior to doing the Mitzvah. [Ashel Avraham ibid ]
 As according to some Poskim even a man is allowed from the letter of the law to eat these foods, and if he is weak according to all it is allowed. Thus certainly women who are not even obligated in the Mitzvah are allowed. [see Piskeiy Teshuvos 585/2 footnote 26 who uses this logic regarding even slightly weak women]
 Piskeiy Teshuvos 585/2 footnote 26
 This complies with the ruling of the previously mentioned Poskim that women do not eat prior to Shofar. Thus if one is able to satiate themselves with other foods which are allowed from the letter of the law, it is better to do so. Furthermore, some Poskim [Ashel Avraham; Ateres Zikeinim ibid] rule that they should hear Shofar early in all cases rather than eat beforehand. Thus it is certainly best to avoid eating a meal or over 55 grams of Mezonos when avoidable.
 Chayeh Adam 141/7; Kitzur SH”A ibid; Ashel Avraham ibid [They do not limit the amount of food that they may eat]; Piskeiy Teshuvos 585/2.
 Ateres Zekeinim ibid; Ashel Avraham Tinyana ibid
 584/6; Rama 584/1; Maharil; See Otzer Minhagei Chabad p. 91 for the custom of Chabad Chassidim in the length of the morning prayer and when it would start. In Lubavitch during the times of the Rebbe Rashab the prayers began at 8:00 and concluded at 4:00. [Reshimos Devarim p. 328]
 Ibid; Rama 584/1; Mahril; Mateh Moshe 812
 Ibid; Rashal Beitza; M”A 584/4; M”B 584/5
 Ibid; M”A 584/4; M”B 584/5
Other Opinions: Some Poskim write that even when Rosh Hashanah falls during the week one is not to lengthen the prayers past midday. [Chayeh Adam 139/8; Maaseh Rav 200; brought in Kaf Hachaim 584/12]
 Mateh Efraim 584/2; Otzer Minhagei Chabad p. 90
 Likkutei Sichos 29 p. 536
 Brought in Hiskashrus 947
 M”E 584/9; Otzer Minhagei Chabad 183
 Otzer ibid
 Minhag Yisrael Torah 584
 Aruch Hashulchan 584/1; Menachem Meishiv 91
 Piskeiy Teshuvos 584/1
 See Otzer ibid
 See Otzer Minhagei Chabad 185; Piskeiy Teshuvos 584/1
 See Admur 68/1 “Whoever has the power to abolish this custom of saying Piyutim during Birchas Shema without causing discord should do so.”; Hakdama of Shaar Hakolel 7 “Admur was stringent in his Siddur not to say the Piyutim”; This was also the custom of the Baal Shem Tov. [Shulchan Hatahor 68/1-2]
 See Otzer Minhagei Chabad 87
 See 591/3; Halacha 17; Thus if one made an invalidating mistake he may not rely on the Chazan’s repetition to be Yotzei.
 594/1; Michaber 594/1
Other Opinions: Some Poskim rule that if one does not know how to pray he can fulfill his obligation with hearing the prayer of another even if there isn’t a Minyan. [Rama 594/1]
 The reason: Although in such a case one’s prayer is not considered in vain even if there are not nine people answering Amen, nevertheless the person cannot fulfill his obligation with hearing the prayer of another unless he hears the prayer of a Tzibur, which means that the prayer is being said in front of nine people answering Amen. This is due to the fact that the Sages originally instituted that the repetition of the Chazan be within a congregation for the sake of fulfilling the obligation of those that don’t know how to pray. However when a congregation is not present then one does not fulfill his obligation, as since prayer is a request of mercy, therefore it is not befitting to say it through an emissary unless he is the emissary of a congregation. [Admur ibid]
 Taz 242/13; Siddur Yaavetz; P”M 141 M”Z 3; Beir Moshe 1/23; Custom of Rebbe in the early years. In the later years the Rebbe sat. [Otzer Minhagei Chabad 190]
Other opinions: See Panim Meiros 1/74 that one is to stand if he is able to see the Sefer Torah.
 Beir Moshe ibid as there is no requirement from the letter of the law to stand while the ark is open.
 Otzer Minhagei Chabad 190; Hiskashrus 947
 Shaareiy Teshuvah 582/8; Kaf Hachaim 582/59
 Beis David 371 and 378 brought in Shalmei Tzibur 309; Shaareiy Teshuvah ibid; Kaf Hachaim ibid
 Shaareiy Teshuvah ibid; Kaf Hachaim ibid
 Mateh Efraim 584/15; See also 596/1
 M”E ibid; Others however sing the hymn also by Tiskabel. [Piskeiy Teshuvos 584 footnote 10]
 584/1; Michaber 584/1; R”H 32a
 The reason: As Hashem is sitting in judgment with the books of life and death open before him, and hence how can one say Shira. [ibid] This is because it is improper for one to be in a state of joy when he is being judged and should have a face of awe and trepidation. [Taz 583/1; M”B 584/1] Likewise it is improper to be in joy when others are being written in the book of opposite of life.
 Admur ibid; M”A 584/1; Elya Raba 581/1; Chayeh Adam 139/11; M”E 584/19; M”B 584/2; Kaf Hachaim 581/2
 As they are not saying the psalms of Halel as hymns of praise but rather as a request and supplication. [ibid]
 So is implied from Admur which specifically mentions the allowance of saying Tehillim for one who says it on a daily basis and just happens to reach these psalms on Rosh Hashanah. This implies that to pick out the Halel section in Tehillim is forbidden even if one plans on saying it as part of Tehillim.
 584/2; Rama 584/1
Background of Avinu Malkeinu:
The Gemara in Tanis 25b states that one time a fast was decreed and they were not answered until Rebbe Akiva Davened for the Amud and recited the prayer of Avinu Malkeinu [the stanza of “Ein Lanu Melech Ela Ata” and of “Lemancha Rachem Aleinu”], upon which he was immediately answered. After the affect of this prayer was witnessed more stanzas were added and it was established to recite it during Aseres Yimei Teshuvah. [Beis Yosef in name of Kol Bo; Kaf Hachaim 584/3; See Mateh Moshe 801; Shaar Hakolel 42/12] Some say the reason we recite Avinu Malkeinu is in correspondence to the middle blessings of Shemoneh Esrei that are omitted on Rosh Hashanah. [Levush brought in Kaf Hachaim 584/7]
Repeating the middle stanzas after the Chazan: It is not our custom to repeat the middle stanzas aloud after the Chazan. [Otzer Minhagei Chabad 203; See Shaar Yissachar p. 219 which protests against those which do so being that they only emphasize the stanzas dealing with physical benefit while those dealing with spiritual matters are said in regular tone.] However there are Poskim which write that the custom is to say it aloud. [Mateh Efraim 602/13; Yalkut Avraham 584; Piskeiy Teshuvos 602/2]
 Kitzur Shlah; Shvus Yaakov 3/42; Elya Raba 584/5; Shaareiy Teshuvah 584/2; M”E 584/13; Kaf Hachaim 584/11; Daas Torah 584/1
Other Opinions: Those that are not accustomed to recite Avinu Malkeinu unless they are with a Minyan have upon what to rely as the source of Avinu Malkeinu is from Rebbe Akiva who said it with a Minyan. [Elya Raba ibid brought in Kaf Hachaim ibid; Daas Torah 584/1]
 Mateh Efraim 619/46; Otzer Minhagei Chabad 199 in name of Sefer Haminhagim
 Makor Chaim of Chavos Yair 602; Piskeiy Teshuvos 602/2
The reason: As it was established to be recited corresponding to Shemoneh Esrei. [ibid]
 Darkei Moshe 584/1; Rama 584/1 [custom of Ashkenazim]
The reason: The M”A 584/2 explains that the reason they do not omit Chatanu Lefanecha is because it refers to our forefathers and not to us.
 Beis Yosef 584; M”A 584/2 in name of Zohar; Arizal in Shaar Hakavanos 90a; Peri Eitz Chaim Shaar Rosh Hashanah 7; brought in Kaf Hachaim 582/16 and Kaf Hachaim 584/4
The reason: This is because we do not recite confessions on Rosh Hashanah in order not to arouse the prosecutor. However we are not particular against reciting verses which mention sin being that it is not said in a way of confession [and the heavenly prosecutor will hence not be aroused]. [Admur ibid; M”A 584/2; M”B 584/3]
During Tekios: Furthermore even actual confession is not to be refrained from being said between the Tekios, as the heavenly prosecutor is in a state of confusion during that time. It is [thus] proper for one to silently confess between the Tekios while the congregation is reciting the prayer of Yehi Ratzon. [Admur ibid; M”A 584/2; Shlah 214b; Arizal in Shaar Hakavanos 90a; Peri Eitz Chaim Shaar 26/5; Kaf Hachaim 584/6] See Halacha 13H for further details on this matter.
 Siddur Admur; Arizal in Shaar Hakavanos 90a; Peri Eitz Chaim Shaar Rosh Hashanah 7; Kaf Hachaim 582/16; Shaareiy Teshuvah 584/2
Ruling of Admur in 584: In 584/2-3 it seems that according to Admur only the stanza of Chatanu Lefanecha is omitted while Selach Umechal is recited. [In 584/2 Admur rules: “One needs to say Selach; Umechal; Mechei; Vihaver Chatoseinu Upeshaeinu as Mechila is greater than Selicha, and Pesha is greater than Chatas, and one must first ask for a minute matter and then add more.” Thus it seems that even the second opinion in Admur held that sin can be mentioned in the stanzas on Rosh Hashanah. This applies also according to the M”A 584/3. Thus in total there are three customs in this matter: a) Recite everything [Rama] b) Only omit Chatanu. [Beis Yosef] c) Omit every mention of sin. [Arizal] See however Shaareiy Teshuvah 584/2, based on Kneses Hagedola, that those which omit the first stanza but recite the stanzas that contain the word sin are mistaken.
The reason: The Arizal ibid states based on the Zohar that on Rosh Hashanah one is not to mention any sin and hence one is to omit all the stanzas of sin, such as Mechol and Selach Lanu. [Kaf Hachaim ibid]
List of stanzas written by Admur in the Siddur that are to be omitted: “Avinu Malkeinu Chatanu Lefanecha”; “Avinu Malkeinu Selach Umechol Lechol Avonoseinu” “Avinu Malkeinu Michei Vehaver Peshaeinu Mineged Einecha” “Avinu Malkeinu Michok Birachamecha Harabim Kol Shitrei Chovoseinu”; “Avinu Malkeinu Kasveinu Besefer Selicha Umechila”.
 584/4; Taz 622/6
 The reason: This implies that only the evil should be torn from the decree, while the mercy that is in the decree is to remain. [ibid]
 584/5; 602/2; Rama 584/1; 602/1; Rivash 512; Kneses Hagedola 584/2; Peri Chadash; Kisei Eliyahu 584/3
Other opinions: Many Poskim rule that Avinu Malkeinu is to be recited even on Shabbos. [Rashbatz 3/186 brought in Beis Yosef; Hatanya; Mateh Yehuda 584; implication of Arizal in Shaar Hakavanos; see Kaf Hachaim 584/8] Based on the Arizal ibid it appears that one is to recite all the stanzas of Avinu Malkeinu, with exception to the one’s that mention sin, even on Shabbos Shuva, and so is the custom of the Beis Keil community in Jerusalem. [Kaf Hachaim 582/16] The Kaf Hachaim 584/8 concludes: It seems that the Ashkenazi custom is to omit it while the Sefardi custom is to say it and for this reason the Michaber omitted this ruling from his Shulchan Aruch. Each community is to follow their custom.
 The reason: As it is forbidden to request one’s needs on Shabbos. [ibid; Ran] Alternatively the reason we omit it on Shabbos is because the entire reason that we recite Avinu Malkeinu is in correspondence to the middle blessings of Shemoneh Esrei that are omitted on Rosh Hashanah during the week. [Levush brought in Kaf Hachaim 584/7]
Does this also apply to private requests? Some suggest that only a set prayer of request was negated on Shabbos Rosh Hashanah while a private request is not only allowed but is motivated to be expressed. [Piskeiy Teshuvos 582 footnote 21]
 602/2; Rama 602/1
 584/5; 602/2; Meil Tzedaka; Rivash 512; M”A 584/3; M”E 584/11; M”B 584/4; P”M 584 A”A 6; Kaf Hachaim 584/7
Other opinions: From the Levush it is implied that Avinu Malkeinu is recited on Mincha Erev Shabbos. [Kaf Hachaim ibid]
 The reason: From Admur it is implied that this is for the same reason mentioned above, that it is forbidden to make personal requests on Shabbos. Vetzaruch Iyun as it is not yet Shabbos and hence should not carry this prohibition? [See Kaf Hachaim 584/7]
 602/2; P”M 584 A”A 6; Shaareiy Tziyon 584/7; Kaf Hachaim 584/7
 Likkutei Sichos 9 p. 434; Igros Kodesh 22 p. 510
 Similarly we do not mention Tzidkascha when Rosh Hashanah falls on Shabbos in order not to arouse Divine judgment on a day of judgment. [598/1]
 591/14 based on Rashi as is understood by the M”A 591/9
Other Opinions: Some Poskim rule that even Shacharis of Rosh Hashanah is to be initially prayed within the first hour of the day, as that is when most communities are Davening. If he did not Daven in the first hour he is to do so in the second hour. He is however to avoid doing so in the third hour as that is a time that G-d’s wrath is revealed. [M”A 591/9; Elya Raba 591/13; Yeshuos Yaakov; Kaf Hachaim 591/35]
The reason: The reason for this is because during the Musaf prayer one recites the blessings of Malchiyos, Zichronos and Shofros, and he is thus to worry that perhaps the supernal courts will investigate his actions and his prayer will be denied. The reason for this worry specifically by the first three hours of the day is because within the first three hours of the day G-D does not judge beyond the letter of the law, as during this time He learns His Torah which is called Truth [and thus judges people in accordance to Truth]. However after three hours into the day He judges the entire world and once He sees that the world cannot up stand to His judgment He therefore judges them in a manner of above the letter of the law, and the individuals prayer will not be differed. [ibid]
 The reason: As it is merely a praise of Hashem [and not requests from G-d] and hence even if he Davens alone [within the first three hours] they will not investigate his actions. [591/14]
 This is not an obligation but rather a mere extra Mitzvah. Hence if one desires to pray prior to the congregation’s prayer he may do so. [90/11]
 The reason: As this is considered an auspicious time above which as result of his prayers are not repelled [90/11], as also when praying simultaneously to a Minyan, even if not in the same building, on it G-d says “I will not repel their prayer”. [591/14] Nevertheless this prayer is not accepted above to the same extent as is prayer of a Minyan within a Shul. [90/11]
 90/11; Kaf Hachaim 591/35