# Primes in arithmetic progressions: weak version of Linnik's theorem with prime power modulus?

Looking at a problem in representation theory I ran into a question on small primes in arithmetic progressions.

Let me begin with a short summary of results on small primes in arithmetic progressions. By Linnik's theorem there are constants $$c,L$$ such that for every $$d \geq 2$$ and $$1\leq a with $$(d,a) = 1$$ the least prime $$p_{\text{min}}(d,a)$$ congruent $$a$$ modulo $$d$$ satisfies $$p_{\text{min}}(d,a) \leq c d^L.$$ Currently the best known value for the exponent is $$L=5$$ (Xyloris). On the extended Riemann hypothesis or the generalized Riemann hypothesis, we have $$L = 2+\epsilon$$ for every $$\epsilon > 0$$. A folklore conjecture (sometimes attributed to Chowla, sometimes to Heath-Brown) states that $$L = 1+\epsilon$$ for all $$\epsilon$$.

Fix a prime number $$p$$ and fix $$a$$, say $$a=1$$. For my purposes the only relevant case is when $$d$$ a power of the fixed prime number $$p$$. In this case stronger results are known. Let $$L(p)$$ be defined as $$L(p) = \limsup_{j \to \infty} \frac{\log(p_{\text{min}}(p^j,1))}{j \log(p)}.$$ In other words $$L(p)$$ is the infimum over all real numbers $$L> 0$$ such that $$p_{\text{min}}(p^j) \leq c_L p^{jL}$$ for some $$c_L > 0$$ and all $$j \geq 1$$. Barban, Linnik and Tshudakov proved $$L(p) \leq \frac{8}{3}$$. Gallagher established $$L(p) < 2.5$$ and Huxley improved this to $$L(p) \leq 2.4$$. The best bound I am aware of can be found in a paper of Banks-Shparlinski: $$L(p) < 2.1115$$.

My question is: what can be said if $$\limsup$$ is replaced by $$\liminf$$? Let's define $$K(p) = \liminf_{j \to \infty} \frac{\log(p_{\text{min}}(p^j,1))}{j \log(p)}.$$ Clearly, $$K(p) \leq L(p)$$. So according to the strongest conjectures on $$L(p)$$ one would have $$K(p) =1$$. However, it seems possible that one can approach $$K(p)$$ with different methods. Put differently: one "only" needs to show that for infinitely many $$j$$ there is a small prime in the arithmetic progression $$\equiv 1 \bmod p^j$$

Question: Are there upper bounds for $$K(p)$$ which are better than the known bounds for $$L(p)$$?

(For me the case $$a = 1$$ is sufficient, but I don't see how this might be useful.)

With $$p_{\min}(d,a)$$ as the OP defines it, let us take

$$p_{\min}(d)=\max_{(a,d)=1}p_{\min}(d,a).$$

Li, Pratt, and Shakan proved (see their Theorem 1.1) that for all $$0<\varepsilon<\frac{1}{2}$$, there exists $$d(\varepsilon)>0$$ such that if $$d>d(\varepsilon)$$ and $$d$$ has no more than

$$\exp\Big(\Big(\frac{1}{2}-\varepsilon\Big)\frac{(\log\log d)(\log\log\log\log d)}{\log\log\log d}\Big)$$

distinct prime factors, then

$$p_{\min}(d)\gg \varphi(d)\frac{(\log d)(\log\log d)(\log\log\log\log d)}{\log\log\log d}.$$

Take $$d=p^j$$. Then $$d$$ has one distinct prime factor, and $$\varphi(d)=p^j-p^{j-1}$$. From this, it follows that $$K(p)\geq 1$$.

• Hi! I don't see the connection. A lower bound for $p_{\text{min}}(d)$ should only give a lower bound for $K(p)$ (here: $K(p) \geq 1$). An upper bound should involve proving the existence of small primes in some arithmetic progressions. Apr 9 at 7:44
• @SteffenKionke Right, I updated so that $K(p)\geq 1$. You said "My question is: what can be said if limsup is replaced by liminf?" I tried to answer that. Apr 9 at 8:22