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Theorem 3.2 in Maynard shows that there are many values $x$ for which the interval $[x,x+\log x]$ contains $\gg \log \log x$ primes. This is a quantification of his earlier breakthrough work where he …

I'll try to unify all the previous answers (of GH from MO, Will Sawin and Hurkyl) and also indicate unconditional results on this problem. It turns out that one can get a surprisingly decent uncon …

"Those oft are stratagems which errors seem, Nor is it Homer nods, but we that Dream." Heath-Brown's proof is fine. It needs just one more line of explanation. Note that $$ \sum_{\substack{\rho \n …

GH from MO and Anonymous have commented above on (modest) lower bounds for the first problem. Let me mention here that a version of problem 2 (of producing many non-residues) appeared in work of Bour …

Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see http://renyi.hu/~p_erdos/1951-13.pdf (page 107; in fact he gives an expli …

Note that $$ s_{1(3)}(n)-s_{2(3)}(n) = \sum_{p\le n} \frac{\chi_{-3}(p)}{p} $$ where $\chi_{-3}$ is the real Dirichlet character $\pmod 3$ (ie the Legendre symbol). This sum converges (as in the p …

Sure. The generating function for the sum you want is the Dirichlet series $$ \sum_{\substack{ n=1\\p|n \implies p\equiv a\pmod q}}^{\infty} \frac{d(n)}{n^s} = \prod_{p\equiv a\pmod q} \Big(1- \fra …

If $m=p_1\cdots p_{\ell}$ is square-free, then the $k$-free integers $n$ that have $m$ as a radical are given by $$ \prod_{j=1}^{\ell} p_j^{a_j} $$ with $1\le a_j \le (k-1)$. Clearly there are $(k …

This is not true for the all $k<r$ problem. Consider random $n$ below $x$, and put $z=\log x$. How many prime factors would a random number have in $[z,z^e]$? This is approximately Poisson with par …

Let $R(n)$ denote the number of ways of writing $n$ as a sum of $4$ squares, and $r(n)$ the number of ways where gcd of $(a,b,c,d) =1$. Then grouping representations of $n$ as a sum of $4$ squares ac …

Let $f$ be any multiplicative function with $|f(n)| \le 1$ and such that $\sum_{d|n} f(d)$ is non-negative for all $n$. It is easy to check that $\mu$ and $\lambda$ satisfy this constraint. Then $$ …

In fact one can prove a stronger result -- namely the probability is bounded away from zero, so long as $B \ge (\log N)^\delta$ for some $\delta >0$. This is best possible, by taking $N$ to be the pr …

Yes, this was proved by Ford, Luca and Pomerance in 2010 (paper in Bulletin of the London Math. Soc.).

Nice question, with an amusing answer $$ R(x) \sim 2 x\log x, $$ so that the trivial upper bound that you get from the triangle inequality is tight. The point is that the average of the divisor fun …

Balog and Sarkozy (Stud. Sci. Math. Hungarica 1984) showed that large $N$ may be written as $x+y+z$ where $x$, $y$, and $z$ are all $\exp(3\sqrt{\log N \log \log N})$ smooth. An analogous result appl …