Number of the primes of the form $\pm2\pmod 5$ up to given large $x$

Question

I need to know the estimate on number of primes of the form $\pm2\pmod5$ less than given large $x$ for :

1. Known under prime number theorem

2. known under improved (best possible) error term of PNT

3. Under Riemann hypothesis .

Answer 1

You're asking (essentially) about the prime number theorem for arithmetic progressions, a generalization of the prime number theorem. So your categories aren't quite relevant. However, here's the summary you hopefully want:

Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a\pmod q$; we'll always assume that $\gcd(a,q)=1$ to avoid trivial situations. You are asking about $\pi(x;5,2)+\pi(x;5,3)$, but the results are typically stated for individual $\pi(x;q,a)$, so that's what I'll do here.

Unconditionally, it is known that for any $A>1$, $$ \pi(x;q,a) = \frac{\mathop{\rm li}(x)}{\phi(q)} + O_A\bigg( \frac x{(\log x)^A} \bigg), $$ where $\mathop{\rm li}(x) = \int_2^x \frac{du}{\log u}$, and where the last term indicates that the difference between the first two terms is bounded in absolute value by some constant (depending on $A$) times $\frac x{(\log x)^A}$. Indeed, that error term can be replaced by something more complicated, for example $O\big( x\exp(-c\sqrt{\log x}) \big)$ for some absolute constant $c>0$.

If one assumes the generalized Riemann hypothesis for Dirichlet $L$-functions (mod $q$) (which includes the usual Riemann hypothesis), then the error term can be improved to $\sqrt x\log qx$.

In any case, for your question the main term is $2\frac{\mathop{\rm li}(x)}{\phi(5)} = \frac{\mathop{\rm li}(x)}2$; in other words, asymptotically half the primes are of the desired form. (The field of "comparative prime number theory" attempts to explain the emperical phenomenon that for finite values of $x$, it's usually the case that slightly more than half the primes are of the desired form.)