Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < cm^{L}$. You can read more on the history of Linnik's Theorem on Wikipedia. Various sources, the Wikipedia link included, claim that in certain cases both these constants $c$ and $L$ can be made explicit. I have however not been able to find any paper where this is actually done. Can someone provide me with a reference where explicit values of both $c$ and $L$ are given?
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$\begingroup$ Where does the wikipedia article claim that "in certain cases both these constants c and L can be made explicit" without a reference? $\endgroup$– efsCommented Jul 11, 2021 at 18:35
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$\begingroup$ The Wikipedia article says 'Linnik's proof showed c and L to be effectively computable' and 'in Heath-Brown's result the constant c is effectively computable.' And, for example, the most recent paper by Matti Jutila that is mentioned and referenced on the Wikipedia page also states 'everything can be made explicit. In fact it would not be too difficult to calculate'. But I have not been able to actually find such a calculation yet. $\endgroup$– WoettCommented Jul 11, 2021 at 19:02
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1$\begingroup$ If I'm not wrong, Heath-Brown's article gives numerical values for the $c$'s involved. See Theorems 1, 2 and 3. $\endgroup$– efsCommented Jul 11, 2021 at 19:17
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$\begingroup$ I really don't see how it does. Those 3 theorems don't mention Linnik's Theorem and specifically talk about 'sufficiently large' moduli. $\endgroup$– WoettCommented Jul 11, 2021 at 21:23
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1$\begingroup$ @efs Writing Linnik's bound as $cm^L$, it is clear from a careful read of the introduction to Heath-Brown's paper that he only computes $L$ for all $m\geq 2$. On the other hand, it follows from Heath-Brown's paper (dig into the details) that there exists an absolute and effectively computable constant $m_0\geq 2$ such that if $m\geq m_0$, then one can take $c=1$. That being said, in order to use Heath-Brown's value (or Xylouris's improved value) of $L$, one must take $m_0$ to be epically gigantic. $\endgroup$– 2734364041Commented Jul 21, 2022 at 20:14
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1 Answer
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As far as I have investigated in literature, the problem of providing a full explicit form of Linnik's theorem is still open. Although, the best known unconditional value for $L$ is equal to $5$, due to Triantafyllos Xylouris [https://bonndoc.ulb.uni-bonn.de/xmlui/handle/20.500.11811/5074], but providing an explicit value for both of $c$ and $L$, even with larger value for $L$, is welcome.
