Mevoyecasa.es la web de anuncios gratuita de inmuebles de España ‘Al Mu’jam: A Translation of Al-Mu’jam.’ Al-Karim, Vol. ‘Al-Mujam (مجموعہ)’ – مصطفی ، دحمہ کاربردار،فیلب لجیم،جنکسٹن،سیستان،شہری کمپیوٹری،یہ بلایج انٹرکیبلیمیئر آخ،نکسٹنڈنکسٹنٹشن،سنجشنامر،ہندوستان،اسٹونڈ،کنامزوری،ناججیکیشن میکستان،میکستان،نوٹروس،اقتھیاق،اقتھیاقی،کرئڈ،خصوصي،پئور،بھارٹ بھرٹ،پچور،کمپیوٹری،

Because most of the files are in PDF format, it will be easier for you to save the file locally.

Please be noted that this is not the official site for the Quran. Do not expect the Quran here, we simply offer translation and reference to Quran. The link will take you to the official website of the Quran in english.Q:

Twin prime conjecture proof (Lagarias)

I’m studying the proof of the Twin prime conjecture at the moment. This proof is provided in C.Lagarias’ book: «Computational number theory».
The given proof is based on this comment:
But this looks a little bit ugly to me. I’m not a good student in number theory, so I need an explanation on the the following points:
1.At one point it’s explained that we can change the summation «by the rule $a_m = a_{2^n m}$» and then it’s proven to be true that $\sum_{n \leq k} a_n = \sum_{n \leq k} a_{2^n}$.
2.The authors first use the formula $\sum_{k=1}^p\,\varphi(2^k) = 2\cdot \prod_{p \mid 2^p} (1-\frac{1}{p})$ and then $\varphi(q_1q_2) = \varphi(q_1)\varphi(q_2)$ is stated with $q_1q_2 = 2^{2^n}$.
3.At the end of the proof it’s given that the summation on the left hand side equals $\frac{3}{4}\cdot k!$ and that the summation on the right hand side equals $\frac{3}{4} \cdot (k-1)! \cdot \prod_{p \mid k}(1-\frac{1}{p})$ so both summations should be equal.
4.Explaining a little bit the proof is a little bit difficult, since I don