The day after Rosh HaShana, the third day of Tishrei, is the Fast of Gedalya. The Gemora in Rosh HaShana explains that we first see this fast mentioned in Zecharia 8:19, when the verse speaks about "...the fast of the seventh...." As the seventh month (starting from Nissan) is the month of Tishrei, the fast mentioned in that verse is referring to Tzom Gedalya, the third day of Tishrei. ..On this day, the Gemora says, Gedalya ben Achikam was killed by Yishmael ben Nesania. The Gemora concludes that this fast teaches us that the death of a righteous person is on par with the burning and destruction of the Holy Temple.
...We see that after the two days on which the whole nation of Israel prayed for life and a good year, we suffered a great downfall. On this day, we should truly feel troubled and worried about our devotion to Hashem. We should focus our prayers on requesting mercy from Hashem. We should not be so confident that the prayers we just completed on Rosh HaShana sufficed. We should ask from Hashem that not only should He raise us from the depths to which we have sunk after our downfalls, but He should decree a good and long life for the whole nation.