Many of the followers of the Sanathan Dharma do the Nama Japa of Lord Rama extensively.
Is Rama Nama Japa capable to give moksha?
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Yes, reciting the name of Rama not only gives moksha but is perhaps the easiest and the most effective way in Kali Yuga. The Adhyatma Ramayana of the Brahmanda Purana states:
रामनाम्नैव मुक्तिः स्यात्कलौ नान्येन केनचित् ॥
rāmanāmnaiva muktiḥ syātkalau nānyēna kēnacit ॥
By the name ‘Rama’ only, there is liberation in the age of Kali, not by means of any other mantra. (Adhyatma Ramayana 2.5.27)
In the Skanda Purana, Sanatkumara says:
रामनामैव नामैव नामैव मम् जीवनम् । कलौ नास्त्येव नास्त्येव नास्त्येव गतिरन्यथा ॥
rāma-nāmaiva nāmaiva nāmaiva mam jīvanam । kalau nāstyēva nāstyēva nāstyēva gatiranyathā ॥
The name of Sri Rama, one and only ‘Rama’ Nama is my life. There is no other means, no other means and no other means except Rama Nama in this age of Kali for Jivas to attain liberation. (Skanda Purana, Uttarakhanda 5.51)
Tulsidas's Ramcharitmanas also extols the glory of chanting the name of Rama:
नहिं कलि करम न भगति बिबेकू। राम नाम अवलंबन एकू॥
nahiṃ kali karama na bhagati bibēkū । rāma nāma avalaṃbana ēkū॥
In the age of Kali neither Karma nor Bhakti nor Jnana avails; the name of Rama is the only resort. (Ramcharitmanas 1.27.4)
The great Hindu saint Swami Ramdas (https://en.wikipedia.org/wiki/Swami_Ramdas) wrote:
Sri Samarth Ramdas assures the aspirant that if he takes "Sri Ram Jai Ram Jai Jai Ram" 13 crores of times, he will have the vision of Sri Rama. The mantra mentioned by Samarth Ramdas is without 'OM'. The Mantra Ramdas gives you is "Om Sri Ram Jai Jai Ram Jai Jai Ram". 'OM' has untold spiritual power. Hence Ramdas, from his own experience, tells you that by repeating this Mantra with 'OM' six crores of times, you will attain salvation. Repeat the Mantra at all times, until the target is reached. You need not keep count of the Mantra. When it reaches six crores you will automatically realize Ram--the Supreme Self. Chanting His name is the way to make Him manifest Himself in you.