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REFINEMENTS TO THE PRIME NUMBER THEOREM FOR ARITHMETIC PROGRESSIONS

• $q\geq 2,\quad x\to \infty ,\quad p\leq x,\quad a{\pmod {q}},\quad \gcd(a,q)=1,\quad {\frac {h}{\varphi (q)}}\geq {\sqrt {x}}(\log x)^{2}$

• $\sum _{\begin{aligned}x-h<p\leq x\\p\equiv a{\pmod {q}}\end{aligned}}\log p={\frac {h}{\varphi (q)}}+O\left({\sqrt {x}}(\log X)^{2}\right)\quad x\geq q$

• ${\begin{aligned}&h=x,\quad h=x^{1-\delta },\quad q\geq (\log x)^{100},\quad q\geq x^{\delta },\quad s=1,\quad \prod _{\chi {\pmod {q}}}L(s,\chi )\neq 0,\\&\mathrm {Re} (s)\geq 1-{\frac {1}{13\log \left(q\left(\left|\mathrm {Im} (s)\right|+3\right)\right)}},\quad \chi _{1}{\pmod {q}},\quad L(\beta _{1},\chi _{1})=0\end{aligned}}$

• $q\geq 2,\quad 4\geq h\geq x$

• $\lambda =\lambda (x,q,a,h):={\begin{cases}{\frac {1}{h}}\int _{x-h}^{x}(1-\chi _{1}(a)t^{\beta _{1}-1})dt&{\text{if }}\beta _{1}{\text{ exists,}}\\1&{\text{otherwise,}}\end{cases}}\qquad \theta ={\begin{cases}{\frac {32}{37}}&{\text{if }}\beta _{1}{\text{ exists,}}\\{\frac {7}{12}}&{\text{otherwise.}}\end{cases}}$

• $0<\varepsilon <1-\theta ,\quad c_{1}=c_{1}(\varepsilon )>0,\quad {\frac {\lambda h}{\varphi (q)}}\geq x^{\theta -\varepsilon }$

• $\sum _{\begin{aligned}x-h<p\leq x\\p\equiv a{\pmod {q}}\end{aligned}}\log p={\frac {\lambda h}{\varphi (q)}}\left[1+O_{\varepsilon }\left(\exp \left({\frac {-c_{1}\log x}{\log q+\left(\log {\frac {x}{h}}\right)^{\frac {2}{3}}\left(\log ^{+}\log {\frac {x}{h}}\right)^{\frac {1}{3}}+\left(\log x\right)^{\frac {2}{5}}\left(\log \log x\right)^{\frac {1}{5}}}}\right)\right)\right]$

• $\log ^{+}(u)=\max\{0,\log u\}$

• $A\geq 1,\quad q\geq (\log x)^{A},\quad \theta ={\frac {7}{12}},\quad \varepsilon >0,\quad b_{\varepsilon }>0$

• $1-\beta _{1}\geq b_{\varepsilon }q^{-\varepsilon }$

• $\varepsilon \geq {\frac {1}{2}}$

• $c_{2}>0,\quad \beta _{1}\leq 1-{\frac {c_{2}}{\log q}},\quad \lambda =1,\quad \theta ={\frac {7}{12}}$

• $\lambda =1,\quad {\frac {h}{\varphi (q)}}$

• ${\frac {\lambda h}{\varphi (q)}}={\frac {h}{\varphi (q)}}-{\frac {\chi (a)}{\varphi (q)}}{\frac {x^{\beta _{1}}-(x-h)^{\beta _{1}}}{\beta _{1}}}$

• ${\frac {\lambda h}{\varphi (q)}}\exp \left({\frac {-c_{1}\log x}{\log q+\left(\log {\frac {x}{h}}\right)^{\frac {2}{3}}\left(\log ^{+}\log {\frac {x}{h}}\right)^{\frac {1}{3}}+\left(\log x\right)^{\frac {2}{5}}\left(\log \log x\right)^{\frac {1}{5}}}}\right)$

• $p\equiv a{\pmod {q}},\quad \xi \in (x-h,x)$

• $\lambda =1-\chi (a)\xi ^{\beta _{1}-1}$