The new moon of Ramadhan is special to Muslims. It announces the start of the holy month and the beginning of the blessings and mercy associated with Ramadhan. According to the Holy Prophet (peace be upon him and his holy progeny) whoever sees the new moon of Ramadhan should recite the following Du'a.
Du'a on sighting the new moon of Ramadhan :
In the name of Allah the Beneficent the Merciful.
My Lord and your Lord is Allah, Lord of the worlds
O Allah let the new moon bring us, peace and faith, safety and submission and a hastening towards what you like and are pleased with.
O Allah blesses us in this month of ours, and gives us of its good, and its assistance Keep us away from its harm, and its evil, its difficulties, and its trials.
This Du'a is highly beneficial, and may be read at the beginning of every lunar month. (Translator)
Allah has decreed that Muslims use the movement of the moon to determine the passing of months and years. When the early Muslims were asked about the new moon, Allah revealed a verse saying:
They ask you concerning the new moons. Say; they are times appointed for (the benefit of) people, and (for) pilgrimage.
source : sibtayn