The gematria of Rosh Hashanah ראשׁ השׁנה is 861, the same value as בּית המקדשׁ, Bais Hamikdosh (the Temple). And considering that the atbash, or reverse gematria of Rosh Hashanah is 596, the same value as ירושׁלים, Jerusalem, there appears to be an important connection between Rosh Hashanah and our Holy Temple, soon to be rebuilt by G-d Himself in Jerusalem, and thus, last forever.
The main focus of Rosh Hashanah is the sovereignty of Hashem. For two days our service is filled with references to G-d as King. Without the subjects of a kingdom voluntarily proclaiming their king, as their king, there is no real recognition of his sovereignty. Before the creation of man, G-d had no one with free will to proclaim Him as the King of the Universe. The angels who were created before man, have no evil inclination and are pure agents of G-d's will. They see the truth and only the truth of G-d's existence and ways, and are not capable of any doubt. But because of this lack of choice, the angels see G-d not as their King, but as their Ruler. G-d wants the people of His Kingdom to proclaim Him their King not because He needs the recognition. By accepting Him as the King of the Universe we would be benefiting G-d by showing Him how we have elevated ourselves. With free will G-d is our King. With no free will G-d is our ruler. Obviously, because there are forces attempting to pull us off the path of the truth, it is much greater for us to proclaim G-d as our King rather than as our Ruler. But time is running short. When Mashiach comes and chases the Soton away, we too will resemble the angels and only see the ways of truth, and will also proclaim G-d as our Ruler. In fact the atbash of the word for Ruler מושׁל has the same atbash as Mashiach.
On Rosh Hashanah the shofar is sounded both before and during Mussaf. We have been taught that it is done this way in order to confuse and trick the Soton. But the confusion caused by all our different customs (how many times we blow, when do we blow during Mussaf and at the end of Mussaf) actually gives the Soton ammunition. This provides evidence against us as to the total lack of unity that we have. We trick the Soton not because of the number of blows or when we blow. We trick the Soton because of what the prophet Amos states in 3(6), "Is the shofar ever sounded in a city and the people not tremble?" When the Soton sees the intensity of the Jewish People as they prepare to listen, when he hears the total silence during the blowing, when he feels our souls trembling, the Soton temporarily loses his power and goes into hiding. Our trembling neutralizes the Soton. He is tricked into believing that this time it's for real. This time the Jewish People have totally repented. This time it will last and we won't revert back. The Soton knows that the gematria of Amos' words, עם לא יחרדו, "the people not tremble," is 369, the same as ונשׁובה, "And we shall return."
On Yom Kippur, we resemble an angel. We don't eat, we don't drink, we dress in white, we stand with our feet together (this resembles angels which appear to have one foot), and we say בּרוך שׁם aloud instead of in a whisper. But we are greater than angels for we have free will to choose. We can grow to greater heights. We can turn our lives completely around. The connection between Rosh Hashanah and our redemption period, when we will all be in Jerusalem together and witness the Temple with our own eyes, is that at that time, Hashem will be our ruler. We will finally be free of the Soton and will only see the path of truth. We won't need the shofar blasts to cause our souls to tremble for we will all follow Hashem's ways as the knowledge of His words fill the air. And then we will learn to Love Hashem, and when we do, He will finally be our King, not as before for just a few days, but forever and ever.