Whenever the Prophet—peace and blessings be upon him—sighted the new crescent, whether for Ramadan or for any other month, he would say: “Allāhumma ahillhu ʿalaynā bil-yumni wal-īmāni was-salāmati wal-islām, rabbī wa rabbuka Allāh” (O Allah, let it appear over us with prosperity, faith, safety, and submission; my Lord and your Lord is Allah). In another wording he ﷺ said: “Allāhu akbar, Allāhu akbar. Allāhumma ahillhu ʿalaynā bil-amni wal-īmāni was-salāmati wal-islām, rabbī wa rabbuka Allāh” (Allah is Greatest, Allah is Greatest. O Allah, let it appear over us with security, faith, safety, and submission; my Lord and your Lord is Allah). This is the supplication reported for sighting the crescent, whether for Ramadan or any other month. As for a special invocation reserved exclusively for the beginning of Ramadan, I know of nothing authentic in that regard.
Still, if a Muslim prays that Allah assist him in fasting the month and accept it from him, there is no harm in doing so. Such a plea is not restricted to a set wording; he may simply ask Allah to help him, to accept his deeds, and to praise Allah—Exalted and Majestic—for allowing him to reach Ramadan.
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Sighting the Crescent
Question: If the first day of Ramadan is Saturday in the Kingdom of Saudi Arabia, yet in Algeria the first day is Sunday, what is the ruling on someone living in Algeria who began fasting with the schedule of Saudi Arabia? Is that permissible or not? And with whom should he break the fast? If he ends the fast with Saudi Arabia, that day will still be a fasting day in his own country, but if he fasts that day it will coincide with the day of ʿEid in the country with which he originally fasted.