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It is not hard prove the bounds you want by purely real variable techniques. First note that the $a_n$ are non-negative for all $n$. For a general non-negative sequence $a_n$, and real numbers $N>0$, …

If one argues carefully using the saddle point method, one could obtain asymptotics for $P(k,y)$; as the argument above indicates, the likely values of $y$ are sharply concentrated around $\sqrt{2nk} … For larger $k$ one can use the previous argument of bounding the probability away from the peak (or work harder and get asymptotics using the saddle point method). …

This is a binary additive divisor problem (also known as the shifted convolution problem) and related questions are to understand asymptotics for $\sum_{n\le x} d(n) d(n+k)$, or $\sum_{n\le x} \lambda_f …

The sequence grows too rapidly and I doubt that one can get meaningful asymptotics for the largest prime factor. …

But we can continue with actual inequalities rather than asymptotics. Consider the probability that the size of the intersection is at most $J$ (and assume that $J\le p^2/(n-2p)$). …

It may be possible to refine this to get asymptotics but I have not thought about that. …

For general $\alpha >1$, a similar argument would show very good asymptotics for large $n$. …

This is not true. In fact $$ x(\log x)^{-1+1/\pi} \gg \sup_n \Big| \sum_{\substack{ d|n \\ d\le x}} \mu(d) \Big| \gg x (\log x)^{-1+1/\pi}. $$ The upper bound is due to Montgomery and Vaughan (see …

point method, and typically one would get that the contribution to the integral around $z= r$ is dominant, and that there is an arc (probably of length about $1/\sqrt{n}$ around $r$) which will give fine asymptotics …

We can attack this problem by using Fourier transforms (i.e. characteristic functions). I'll consider the example in the problem where $X$ is a random variable taking the value $0$ with probabilit …

I'll address Joel's edited question of getting good asymptotics on RH. … The usual way in which such results are carried out in the literature is to start with asymptotics for coefficients of $\zeta(s)^z$ for complex $z$, and then to use the saddle point method to identify …

Assuming that $|x+y|<1$ and $4|xy| \le 1$, here's a proof of the decay. First suppose that $|x|> |y|$. The desired sum is $$ \le \binom{2n}{n} |xy|^n \sum_{j=0}^{n} |y/x|^j \le \binom{2n}{n} |x …

One can obtain more precise asymptotics by working harder with the argument below. …