This post is related to questions asked here and here. However, I include the relevant background on least prime in arithmetic progressions presented here for benefit of the reader.

By Linnik's theorem there are constants $c,L$ such that for every $d \geq 2$ and $1\leq a<d$ 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 known bound can be found in a paper of Banks-Shparlinski: $L(p) < 2.1115$.

I wanted to ask if we could possibly strengthen the upper bound on $L(p)$ to the conjectural bound $1-o(1)$ under Generalised Riemann Hypothesis. One can achieve the bound $p(a,d)\ll d^{1-o(1)}$ if one assumes the Montgomery's conjecture as answered in this post but as sharper upper bounds have been proven for $L(p)$ than the general case, is there any possible way to get the conjectured upper bound on GRH? Thanks in advance for any assistance.

This post is related to questions asked here and here. However, I include the relevant background on least prime in arithmetic progressions presented here for benefit of the reader.

By Linnik's theorem there are constants $c,L$ such that for every $d \geq 2$ and $1\leq a<d$ 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 known bound can be found in a paper of Banks-Shparlinski: $L(p) < 2.1115$.

I wanted to ask if we could possibly strengthen the upper bound on $L(p)$ to the conjectural bound $1+o(1)$ under Generalised Riemann Hypothesis. One can achieve the bound $p(a,d)\ll d^{1+o(1)}$ if one assumes the Montgomery's conjecture as answered in this post but as sharper upper bounds have been proven for $L(p)$ than the general case, is there any possible way to get the conjectured upper bound on GRH? Thanks in advance for any assistance.

This post is related to questions asked here and here. However, I include the relevant background on least prime in arithmetic progressions presented here for benefit of the reader.

By Linnik's theorem there are constants $c,L$ such that for every $d \geq 2$ and $1\leq a<d$ 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 known bound can be found in a paper of Banks-Shparlinski: $L(p) < 2.1115$.

I wanted to ask if we could possibly strengthen the upper bound on $L(p)$ to the conjectural bound $1-o(1)$ under Generalised Riemann Hypothesis. One can achieve the bound $p(a,d)\ll d^{1-o(1)}$ if one assumes the Montgomery's conjecture as answered in this post but as sharper upper bounds have been proven for $L(p)$ than the general case, is there any possible way to get the conjectured upper bound on GRH? Thanks in advance for any assistance.

This post is related to questions asked here and here. However, I include the relevant background on least prime in arithmetic progressions presented here for benefit of the reader.

By Linnik's theorem there are constants $c,L$ such that for every $d \geq 2$ and $1\leq a<d$ 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 known bound can be found in a paper of Banks-Shparlinski: $L(p) < 2.1115$.

I wanted to ask if we could possibly strengthen the upper bound on $L(p)$ to the conjectural bound $1-o(1)$ under Generalised Riemann Hypothesis. One can achieve the bound $p(a,d)\ll d^{1-o(1)}$ if one assumes the Montgomery's conjecture as answered in this post but as sharper upper bounds have been proven for $L(p)$ than the general case, is there any possible way to get the conjectured upper bound on GRH? Thanks in advance for any assistance.