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To answer the main question "Is this close enough to be of use in any practical application?" I do not believe so. However, one cannot say "no" with certainty, so it seems unlikely that you will get a …

Up to a factor of logarithm, $E(x)$'s oscillation has an amplitude which is of the same magnitude as that of $\frac{1}{x}\left(\psi(x)-x\right)$, that is the error in the prime number theorem. Specifi …

We can take $f(n)=\alpha n$ for any $\alpha<0.7375$. In particular, the set of primes with more than twice as many ones that zeros in their binary expansion is infinite. I posted a short article on …

This answer is a heuristic along the lines of Joro's. We use $p,q,r$ to denote primes. Let $S(p,a)$ denote the number of pairs of primes $(q,r)$ with $q,r\leq p$ and $p|(qr+a)$. We are interested i …

We'll prove that the maximal cardinality of such a set for $M^2\leq N$ has size equal to $$\pi(N)+\sum_{1<n\leq M} \pi(p(n))$$ where $p(n)$ is the smallest prime factor of $n$. Since $$\sum_{1<n\le …

Note: This question was answered in the comments, and I am posting to remove the question from the unanswered list. The result of Maynard and Tao applies to all admissible $k$-tuples of linear forms. …

As mentioned by Aaron Meyerowitz, a set of the form $$S=\left\{\left[k^{1/\gamma}\right]:\ k\in\mathbb{N}\right\}$$ should work for any $1/2<\gamma<1$. Notice that for such a value of $\gamma$, $$\lef …

Let $g(n)$ denote the indicator function for the fourth powers. Then your sum equals $$\sum_{n\leq x}\left(h_{r}*g\right)(n),$$ where $*$ denotes Dirichlet convolution. We may rewrite the given as …

A standard approach to think about is partial summation. Suppose that $A(x)=\sum_{n\leq x} a(n)$ and $S(x)=\sum_{n\leq x} a(n)e^{-n/x}$. Then we can relate these two sums in the following way: $$ …

This is a comment rather than an answer, but it is too long. Let $g$ be some generator of the multiplicative group. Then $$\sum_{a=1}^{p-1}e\left(\frac{a}{p}\right)\frac{p-1}{ord_{p}a}=\sum_{k=1}^{ …

I just wanted to add, we also have a similar upper bound which combined with unknown(google)'s answer shows that $$E_{r}(x)\asymp\frac{x\left(\log\log x\right)^{r}}{\left(\log x\right)^{\frac{1}{2}} …

Your above identity stems from $$\text{li(x)}-\Pi(x)+\text{small}=\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty} \log \left((s-1)\zeta(s)\right)\frac{x^s}{s}ds,$$ and a Taylor expansion of the logarithm …

This has been answered in the comments by Lucia. The identity $$P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(4s)}-\sqrt{\frac{2}{\zeta(8s)}-\cdots}}}}$$ is false. By …

Question 1: The set is dense. Suppose that we are given a fixed $x\in\mathbb{R}$. Then let $p$ be a large prime. If $p$ is sufficiently large, then there will be a prime $$q\in\left[px,\ px+\left …

Edit (20/11/2013) : Yesterday James Maynard posted the paper Small gaps between primes on the arxiv in which he shows that for any $m$ there exists a constant $C_m$ such that $$ p_{n+m}-p_n\leq C_m$$ …