I'm trying to understand On Arithmetical Properties of Lambert Series by Erdös, but am stuck on the first page.
He states:
Put $k=\left[(\log n)^{1/10}\right]$ and let $p_1,p_2,\ldots$ be the sequence of consecutive primes greater than $(\log n)^2$. (...) From elementary results about the distribution of primes, it follows that $p_i < 2(\log n)^2$ for $i \leq \frac{k(k+1}{2}$.
I think the "elementary results" refers to Bertrand's Postulate. But, from that, I can only say $(\log n)^2 < p_1 < 2((\log n)^2)$ -- not the statement about all $p_i$ for the given $i$. How is this result obtained?
He further states:
Put $$A = \left\{\prod_{1\leq i \leq \frac{k(k+1)}{2}} p_i\right\}^t$$ (...) by a simple computation, we obtain $$A < \left\{2(\log n)\right\}^{tk^2} < e^{(\log n)^{1/4}}$$
How does this follow?
Finally, the page ends with a listing of congruences. Let's say the first congruence is true. Is there a guarantee that all other congruences will be true as well?