The Rebbe: “Beginning from the Chamisha Asar BeAv it’s customary to greet people with the blessing “May you be inscribed and sealed for a good year” (ספר-השיחות תנש”א ח”ב עמ’ 770).
The Talmud says that from the Fifteenth of Av onward, the power of the sun decreases and “whoever increases his Torah study will receive an increase.”
From this point forward, when the nights lengthen, one who adds [demosif ] to his nightly Torah study will add [yosif ] to his life (Bava Batra, 121B).
The Rebbe explains:
– This reflects the true purpose of the night as our Sages declared, “the night was created solely for the purpose of study.” Therefore, the Shulchan Aruch mentions the importance of increasing Torah study at night from the fifteenth of Av onward. Since the Torah is called “our life and the length of our days,” an increase in Torah study will lead to an increase in our lifespans…
See more in the Rebbe’s talk on Shabbat “Vaetchanan”, 5750 (1990).