No specific adhkaar have been narrated for the month of Ramadan
Question

I noticed that the imam in our area repeats after every prayer the words “Ashhadu an laa ilaaha ill-Allah, astaghfir-Allah, nas’aluka al-jannah wa na‘oodhu bika min an-naar (I bear witness that there is no god but Allah; I ask Allah for forgiveness; we ask You for Paradise and we seek refuge in You from the Fire)” three times. Is this Sunnah, and can we repeat it? What are the other du‘aa’s (supplications)? I want to know what they are so that I can repeat them from here on in.
Praise be to Allah.There are no reports in the Prophet’s Sunnah to suggest that there are specific du‘aa’s or adhkaar for the blessed month of Ramadan, apart from what is narrated about seeking Laylat al-Qadr in the last ten nights of Ramadan. It was narrated that ‘Aa’ishah (may Allah be pleased with her) said: I said: O Messenger of Allah, what you think, if I know that a particular night is Laylat al-Qadr, what should I say on that night? He said: “Say: ‘Allaahumma innaka ‘afuwwun tuhibb al-‘afwa fa‘fu ‘anni (O Allah, You are forgiving and You love forgiveness, so forgive me).’”
Narrated by at-Tirmidhi (3513); he said: it is hasan saheeh. See also question no. 36832 
Apart from that, there are no reports in the Prophet’s Sunnah to suggest that there are specific adhkaar for the month of Ramadan, specifying a particular number of times to recite them, or a specific reward for saying them. Rather it is mustahabb for the Muslim to remember Allah at all times and in all circumstances, as the Prophet (blessings and peace of Allah be upon him) used to combine dhikr and du‘aa’ in order to make the most of the days and nights of this month, especially the times at which supplication is more likely to be answered, such as the time before dawn, after ‘Asr on Fridays, and so on. So he would ask Allah for Paradise and seek refuge with Him from Hell. 
Imam ash-Shaatibi (may Allah have mercy on him) said: 
The word bid‘ah (innovation) refers to invented practices in religion that are similar to prescribed practices, the purpose of which is to go to extremes in worshipping Allah, may He be glorified… That includes committing oneself to specific manners and wordings, such as reciting dhikr in unison, taking the day of the birth of the Prophet (blessings and peace of Allah be upon him) as a festival, and so on. It also includes committing oneself to doing specific acts of worship at specific times when that particular time is not specific for that act of worship. 
End quote from al-I‘tisaam (1/37-39) 
We would like to take this opportunity to point out that what is widespread in many chat rooms and social media websites, of specific adhkaar and du‘aa’s for each day of the month of Ramadan is all something that has been fabricated and made up by people. These are nothing more than some adhkaar that have been selected and spread by some people, and many people think that this is a prescribed act of worship that is to be done in this blessed month, but in fact this is not part of the Sunnah and is not narrated in any hadith. 
So the Muslim should be keen to recite the adhkaar of the morning and the evening, and the adhkaar that are to be recited following the prayers, and on other occasions as prescribed in Islam, and he should try hard to read the Holy Qur’an, study it, and reflect on its meanings, and that will suffice him, by Allah’s leave, and bring the reward that he seeks. 
And Allah knows best.