Is There a Specific Du`a for Each Day and Night of Ramadan?
Question

I heard that Allah, may He be Exalted, divided Ramadan into three parts: the first ten days of it are mercy, the second ten days are forgiveness and the third ten days are ransom from the fire. It is said that there are specific aDu`asfor each part, so in the first part we say: Allahumm arhamni ya arham ar-rahimeen (O Allah, have mercy on me, O most merciful of those who show mercy); in the second part we say Allahumm ighfir li ya Rabb Al-`Alameen (O Allah forgive me, O Lord of the Worlds); and in the third part we say Allahumm a`tiqni min an-nar wa adkhilni Al-Jannah (O Allah, ransom me from the Fire and admit me to Paradise). Is this correct and is there any evidence for it? What are the Du`a`s that should be recited a great deal in Ramadan? To the best of my knowledge, Allahumma innaka `afuwwun tuhibb Al-`afwa fa`affu `anni (O Allah, You are forgiving and You love forgiveness, so forgive me) is one of the Du`a`s that should be repeated often in the last ten days, when seeking Laylat Al-Qadr. What about the other days of Ramadan? Are there any specific Du`a`s for them?
Praise be to Allah.Is Ramadan divided into three parts?
Ibn Khuzaymah (may Allah have mercy on him) narrated in his Sahih (1887) that Salman (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) addressed us on the last day of Sha`ban and said: “O people, a great month has come upon you, a blessed month… It is a month the beginning of which is mercy, its middle is forgiveness and its end is ransom from the Fire.” 
In the answer to question no. 21364 , we explained that this Hadith isinauthentic.  
The entire month of Ramadan is mercy from Allah; the entire month is also forgiveness and ransom from the Fire. None of these blessings is restricted to any one part of the month to the exclusion of any other part, and this is a reflection of the vastness of Allah`s mercy. 
Muslim (1079) narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “When Ramadan comes, the gates of Paradise are opened and the gates of Hell are closed, and the devils are fettered.” 
At-Tirmidhi (682) narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “When the first night of Ramadan comes, the devils and mischievous jinn are chained up, and the gates of Hell are closed, and none of its gates are opened. The gates of Paradise are opened and none of its gates are closed. And a caller cries out: `O seeker of good, proceed; O seeker of evil, desist.` And Allah has people whom He frees (from the Fire), and that happens every day.” (Classed as authentic by Al-Albani in Sahih At-Tirmidhi) 
Based on that, singling out the first third of Ramadan to pray for mercy , the second third to pray for forgiveness and the last third to pray for ransom from Hell is an innovation for which there is no basis in Islamic teachings. There is also no justification for singling out these times for these supplications, because all the days of Ramadan are equal in that regard. Rather the Muslim may pray for whatever he wants of goodness in this world and the hereafter throughout Ramadan; that includes asking Allah for mercy, forgiveness, ransom from the Fire and admission to Paradise. 
The recommendation to make a lot of Du`a in Ramadan
The Muslim should offer a great deal of Du`a, asking for goodness and mercy, especially in this month, making the most of this time of goodness and blessing (Barakah), seeking the mercy and forgiveness of their Lord, may He be Exalted. Allah, may He be Exalted, says (interpretation of the meaning):
“And when My slaves ask you (O Muhammad blessings and peace of Allah be upon him) concerning Me, then (answer them), I am indeed near (to them by My Knowledge). I respond to the invocations of the supplicant when he calls on Me (without any mediator or intercessor). So let them obey Me and believe in Me, so that they may be led aright.” [Al-Baqarah 2:186] 
Ibn Kathir (may Allah have mercy on him) said: 
“The fact that Allah, may He be Exalted, mentions this verse, which encourages one to offer Du`a, in the midst of the verses that speak of the rulings on fasting, indicates that one should strive hard in offering Du`a at the end of the fast and, indeed, every time one breaks the fast.” (Tafsir Ibn Kathir, 1/509) 
Examples of Du`as in Ramadan
It is good for the one who offers Du`a to follow proper etiquette and to recite many of the Du`as that have been narrated from the Prophet (blessings and peace of Allah be upon him), and not to overstep the mark in offering Du`a. He should observe the etiquette of Du`a and recite often the Du`as that it is recommended to recite a great deal in Ramadan and also at times other than Ramadan. These include the following: 

 “Rabbana atina fi`d-dunya hasanah wa fi`l-akhirah hasanah wa qina `adhab An-nar (Our Lord, give us that which is good in this world and that which is good in the Hereafter, and protect us from the torment of the Fire),” [Al-Baqarah 2:201]
“Rabbana hab lana min azwajina wa dhurriyyatina qurrata a`yunin waj`alna lil-muttaqina imama (Our Lord! Bestow on us from our wives and our offspring who will be the comfort of our eyes, and make us leaders for the Muttaqun (the pious))” [Al-Furqan 25:74].
“Rabb ij`alni muqima as-salati wa min dhurriyyati Rabbana wa taqabbal Du`a. Rabbana ighfir li wa li walidayya wa lil-mu`minina yawma yaqum ul-hisab (O my Lord! Make me one who performs As-Salat (Iqamat-as-Salat), and (also) from my offspring, our Lord! And accept my invocation. Our Lord! Forgive me and my parents, and (all) the believers on the Day when the reckoning will be established).” [Ibrahim 14:40-41]
Allahumma innaka `afuwwun tuhibb Al-`afwa fa`affu `anni (O Allah, You are forgiving and You love forgiveness, so forgive me).
Allahumma inni as`aluka min Al-khayri kullihi `ajilihi wa ajilihi, ma `alimtu minhu wa ma lam a`lam, wa a`udhu bika min ash-sharri kullihi `ajilihi wa ajilihi, ma `alimtu minhu wa ma lam a`lam. Allahumma inni as`aluka min khayri ma sa`alaka `abduka wa nabiyyuka, wa a`udhu bika min sharri ma `adha minhu `abduka wa nabiyyuka. Allahumma inni as`aluka Al-jannata wa ma qarraba ilayha min qawlin aw `amal, wa a`udhu bika min an-nari wa ma qarraba ilayha min qawlin aw `amal, wa as`aluka an taj`ala kulla qada`in qadaytahu li khayran (O Allah, verily I ask you for all that is good in this world and the hereafter, what I know of it and what I do not know, and I seek refuge with You from all that is evil in this world and the hereafter, what I know of it and what I do not know. O Allah, verily I ask You for the good of that which Your slave and Prophet asked of You, and I seek refuge with You from the evil of that from which Your slave and Prophet sought refuge with You. O Allah, verily I ask you for Paradise and that which will bring me nearer to it of words and deeds, and I seek refuge with You from Hell and that which would bring me nearer to it of words and deeds. I ask You to make everything You decree for me good).
Allahumma inni as`aluka Al-`afiyata fi`d-dunya wa`l-akhirah. Allahumma inni as`aluka Al-`afwa wa`l-`afiyata fi deeni wa dunyaya wa ahli wa mali. Allahumma astur `awrati wa amin raw`ati. Allahumma ihfazni min bayna yadayya wa min khalfi wa `an yameeni wa `an shimali, wa min fawqi, wa a`udhu bi `adhamatika an ughtala min tahti (O Allah, I ask You for pardon and well-being in my religious and worldly affairs, my family and my wealth. O Allah, conceal my faults and protect me from that which causes me to worry. O Allah, protect me from before me and from behind me, from my right and from my left, and from above me, and I seek refuge in Your greatness lest I be destroyed from beneath me).

Similarly, it is recommended to recite general Du`as from the Quran and Sunnah, any Du`a is good in which the individual strives hard, beseeching his Lord. None of that is limited to Ramadan only. 
It is also recommended to say after breaking the fast: “Dhahaba Adh-dhama` wa abtalat Al-`uruq wa thabata Al-ajr in sha Allah (Thirst is gone, the veins are moistened and the reward is certain if Allah wills).”
Please see also the answer to question no. 14103  and 26879 . 
One should strive hard in Du`a in the last third of the night in particular, every night. 
During the last ten days, one should recite a great deal the words: “Allahumma innaka `afuwwan tuhibb ul-`afwa fa`fu `anni (O Allah, You are forgiving and love forgiveness, so forgive me).”
Please see the answer to question no. 36832 . 
Please see the answer to question no. 36902  for more information on the etiquette of Du`a. 
And Allah knows best.