Some version of the Prime Number Theorem provides the asymptotic behavior of the number of primes in arithmetic progression $qn+a$ with $(q,a)=1$, $n \ge 1$. I was wondering there are Chebyshev-type arguments (using the binomial coefficients or variants thereof) that establish the existence of at least $$c x/\log x$$ primes in such arithmetic progressions up to $x$ for some even small $c>0$ and some values of $a, q>1$. For instance, is there such an argument for primes of the form $ 4k \pm 1$? This is mostly a curiosity triggered by thinking about a number theory course I will teach soon. 



Elementary lower bounds for the number of primes in arithmetic progressions
In Section 9 of


*Diamond, Harold G.*, [**Elementary methods in the study of the distribution of prime numbers**](http://dx.doi.org/10.1090/S0273-0979-1982-15057-1), Bull. Am. Math. Soc., New Ser. 7, 553-589 (1982). [ZBL0505.10021](https://zbmath.org/?q=an:0505.10021).


an argument from


*Diamond, Harold G.; Erdős, Paul*, On sharp elementary prime number estimates, Enseign. Math., II. Sér. 26, 313-321 (1980). [ZBL0453.10007](https://zbmath.org/?q=an:0453.10007).


is sketched, where it is shown that for every $\varepsilon > 0$ there is a Chebyshev style argument that shows that


$$ (1-\varepsilon) x \leq \psi(x) \leq (1+\varepsilon) x$$


for all sufficiently large $x$. The idea is to compare $\Lambda(n) = \mu \* \log(n)$ with a truncated version $\Lambda\_T(n) = \mu\_T \* \log(n)$ where $\mu\_T$ is $\mu$ truncated to a large interval $[1,T]$ (and modified at $T$ to ensure the normalisation $\sum\_n \mu\_T(n)/n=0$). Elementary methods can show that $|\sum\_{n \leq x} \Lambda\_T(n) - m\_1(T) x| \leq o(x)$ for an explicitly computable quantity $m\_1(T)$ (in fact $m\_1(T) = \sum\_{n<T} \log \frac{T}{n} \frac{\mu(n)}{n}$) that can be easily demonstrated to converge to one as $T \to \infty$ in a suitably averaged sense. In particular $|\sum\_{n \leq x} \Lambda\_T(n) - x| \leq |m\_1(T)-1| x + o(x)$. The Dirichlet hyperbola method can also be used to produce an upper bound $|\sum\_{n \leq x} \Lambda\_T(n) - \psi(x)| \leq \delta\_T x + o(x)$ for some explicitly computable quantity $\delta\_T$ depending on $T$; with the aid of the PNT (in the form $\sum\_{n \leq x} \mu(n) = o(x)$) one can show that $\delta\_T \to 0$ as $T \to \infty$, and this is enough to conclude the claim. Amusingly, this gives a circular proof of the PNT assuming the PNT; however if one only wishes to get the PNT up to a given error $\varepsilon$ then this argument does not require the PNT, one simply has to locate a concrete value of $T$ for which the upper bounds $|m\_1(T)-1|, \delta\_T$ in the above inequalities (which can be explicitly computed by Chebyshev type methods) are small enough.


It looks very likely that one can twist this argument by characters and conclude that for every $\varepsilon > 0$ and primitive residue class $a \hbox{ mod } q$ there is a Chebyshev style argument that shows that


$$ (1-\varepsilon) \frac{x}{\phi(q)} \leq \psi(x,q,a) \leq (1+\varepsilon) \frac{x}{\phi(q)}$$


for all sufficiently large $x$, where one again needs the prime number theorem in arithmetic progressions (presumably this time in the form $\sum\_{n \leq x} \mu(n) \chi(n) = o(x)$) to guarantee that the elementary upper bounds for the error are indeed small. It seems plausible though that for specific residue classes like $1 \hbox{ mod } 4$ or $-1 \hbox{ mod } 4$ one could actually produce explicit numerical choices of the $T$ parameter that obey all the required properties and in particular produce non-trivial lower bounds of Chebyshev type.


