HaRav Shmuel Eliahu is Chief Rabbi of Tzfat.
In the book, “Pirkei deRabbi Eliezer (46) it is written: "’And on the first day of the month of Elul, the Holy One said to him (Exodus 24:12), ‘Come up to me on the mountain,’ and they blew the shofar in the camp, for Moses ascended the mountain so that they would no longer err in worshipping the stars and the constellations, and the Holy One ascended that day with that shofar, as it is said (Psalms 44:6), ‘God ascended with a call of the shofar.’
"Therefore the Sages established that the shofar should be blown on the first day of the month of Elul every year."
The Ashkenazi Custom
The Tur on the Laws of Rosh Hashanah wrote: "Therefore the Sages established that the shofar should be blown on the first day of Elul every year and all through the month in order to warn Israel that they should repent, as it is said (Amos 3:11) ‘If a Shofar is blown in a city,’ etc., and in order to confuse Satan, and so it is customary in Ashkenaz to blow every morning and evening after prayer."
And the Shulchan Aruch did not bring the custom of blowing the shofar (during selichot). And the Rema wrote about the selichot throughout the month of Elul, "And the custom of the Ashkenazim is not like that, but from the new moon onwards they begin to blow after the morning prayer, and there are places where it is even done in the evening."
And today, the Ashkenazim do not practice blowing the shofar during the mincha service. However, if they did not have time to blow the shofar during the morning prayer, they will blow the it during mincha. And the “Kaf HaHayim" wrote that the Sephardim also practice blowing during the selichot, "And among those who practice according to the words of the Shulchan Aruch, some people also practice blowing during the selichot while reciting the Thirteen Attributes, and also before “Titkabel" in order to fulfill the words of all."
Why are there no blessings for blowing the selichot?
And the “Avne Nezer" asks why they do not bless over the blowing of the shofar that is sounded every day in the month of Elul, since it is a “Takanah" (correcting decree) of the Sages, as stated in the “Pirkei deRabbi Eliezer" and in the Tur? And they reasoned that the blowing of the shofar in the month of Elul is only a preparation for Rosh Hashanah. And they said in the name of the “Chiddushei HaRim" (Rabbi Yitzhak Meir of Gur) that the blowing of the month of Elul is to strike the heart, that is, to prepare the heart so that we can receive the blowing of Rosh Hashanah. And because it is only a preparation for Rosh Hashanah, they do not bless and they rely on the blessing of the main mitzvah on Rosh Hashanah.
Aliyah to the Holy Land
The principle message in the blowing of the shofar is to gather the exiled Jews home to Israel. Thus says the prophet Isaiah, "And it shall be in that day that the Lord will beat out His harvest from the strongly flowing river as far as the wadi of Mitzrayim, and you shall be gathered up one by one, O Children of Israel" (27:12). The Radak explains that this is how olives are gathered and how wheat is separated from chaff. This is how Hashem will separate Israel from the nations. And then "it shall come to pass in that day that a great shofar shall be blown, and the lost in the land of Assyria and the outcasts in the land of Egypt shall come and worship the Lord on the holy mountain in Jerusalem."
Gathering of the Exiles - God is King
This gathering of the exiles is the great sanctification of the Name, that all the nations see that God is gathering the people of Israel to their Land just as He spoke through His servants the prophets. And although the nations at first agreed to this in a wondrous way, they soon after resisted. Yet despite the opposition of leading world powers, the Children of Israel gathered and ascended to their Land against all odds. And this gathering is a miraculous historic event that has no equal in the annuls of mankind.
This is none other than the ultimate declaration that God is King over all nations and all events in the world.
Therefore, after “The Lord is King, the Lord is King, the Lord will reign forever and ever," there are congregations who say in the morning prayer before Baruch Hu, “And the Lord will be king over all the earth. On that day the Lord will be one and His Name one."
How does this come about? Through the ingathering of the exiles, as is said: "Save us, O Lord our God, and gather us from the nations to give thanks to Your holy Name and to glory in Your praise" and then "To triumph with instruments of music from a psalm of song" which speaks of the illumination of the faces of the immigrants who ascend to the Land of Israel, "A Land which grants us its harvest. May God bless us, our God." And when the nations of the world see how Israel is blessed, they too long for Hashem’s blessing, “And all the ends of the earth shall fear Him." Amen and amen.