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Significantly rewritten, yet the main message stays the same.

It is quite likely that this conjecture is false yet no counter example was found so far.

The reason why this seems likely is that there seems to be no supporting evidence for this conjecture beyond numerics.

And, there are investigations base on quite natural random models of the primes that contradict it. What is commonly known as Cramér's conjecture, that is that maximal gaps between consecutive primes are of sizes at most $(\log p_n)^2$ (up to lower order terms) does not contradict this conjecture, and one might even think it supports it. However, on the one hand it is not quite clear Cramér even conjectured this in precisely this form; he conjectured gaps are $O((\log p_n)^2)$ and somehow implied that about $(\log p_n)^2$ might be true. On the other hand, and more importantly, Granville note that a finer investigations of Cramér's reasoning rather suggests maximal gaps of size $2 e^{-\gamma} (\log p_n)^2$ (up to lower order terms). And if this were true it would contradict the conjecture mentioned in OP (this is what is referred to in OP as Cramér-Granville conjecture).

It should however be noted that Granville did not conjecture that the gaps are of this size, yet pointed out what taking an additional aspect into account would mean for Cramér's reasoning. For Granville on this matter on MO see Consequences of Legendre's conjectureConsequences of Legendre's conjecture

For details on Granville's arguments see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

Significantly rewritten, yet the main message stays the same.

It is quite likely that this conjecture is false yet no counter example was found so far.

The reason why this seems likely is that there seems to be no supporting evidence for this conjecture beyond numerics.

And, there are investigations base on quite natural random models of the primes that contradict it. What is commonly known as Cramér's conjecture, that is that maximal gaps between consecutive primes are of sizes at most $(\log p_n)^2$ (up to lower order terms) does not contradict this conjecture, and one might even think it supports it. However, on the one hand it is not quite clear Cramér even conjectured this in precisely this form; he conjectured gaps are $O((\log p_n)^2)$ and somehow implied that about $(\log p_n)^2$ might be true. On the other hand, and more importantly, Granville note that a finer investigations of Cramér's reasoning rather suggests maximal gaps of size $2 e^{-\gamma} (\log p_n)^2$ (up to lower order terms). And if this were true it would contradict the conjecture mentioned in OP (this is what is referred to in OP as Cramér-Granville conjecture).

It should however be noted that Granville did not conjecture that the gaps are of this size, yet pointed out what taking an additional aspect into account would mean for Cramér's reasoning. For Granville on this matter on MO see Consequences of Legendre's conjecture

For details on Granville's arguments see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

Significantly rewritten, yet the main message stays the same.

It is quite likely that this conjecture is false yet no counter example was found so far.

The reason why this seems likely is that there seems to be no supporting evidence for this conjecture beyond numerics.

And, there are investigations base on quite natural random models of the primes that contradict it. What is commonly known as Cramér's conjecture, that is that maximal gaps between consecutive primes are of sizes at most $(\log p_n)^2$ (up to lower order terms) does not contradict this conjecture, and one might even think it supports it. However, on the one hand it is not quite clear Cramér even conjectured this in precisely this form; he conjectured gaps are $O((\log p_n)^2)$ and somehow implied that about $(\log p_n)^2$ might be true. On the other hand, and more importantly, Granville note that a finer investigations of Cramér's reasoning rather suggests maximal gaps of size $2 e^{-\gamma} (\log p_n)^2$ (up to lower order terms). And if this were true it would contradict the conjecture mentioned in OP (this is what is referred to in OP as Cramér-Granville conjecture).

It should however be noted that Granville did not conjecture that the gaps are of this size, yet pointed out what taking an additional aspect into account would mean for Cramér's reasoning. For Granville on this matter on MO see Consequences of Legendre's conjecture

For details on Granville's arguments see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

significant rewrite
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user9072

Perhaps in view of the comments the following clarfication is helpful and since it also sort of an answer and a bit long I give it as answer.Significantly rewritten, yet the main message stays the same.

It is quite likely that this conjecture is false yet no counter example was found so far.

The pointreason why this seems likely is that there seems to be no supporting evidence for this conjecture beyond numerics.

And, which is mentioned by OP but got a bit blurred viathere are investigations base on quite natural random models of the commentsprimes that contradict it. What is commonly known as Cramér's conjecture, that Granville arguesis that largemaximal gaps between successiveconsecutive primes in fact will not be bounded byare of sizes at most $(\log p_n)^2$ (contraryup to what Cramér conjecturedlower order terms) but thatdoes not contradict this conjecture, and one actually needs a constant factor larger thanmight even think it supports it. However, on the one hand it is not quite clear Cramér even conjectured this in precisely this form; he conjectured gaps are $O((\log p_n)^2)$ and somehow implied that about $(\log p_n)^2$ might be true. On the other hand, and more specificallyimportantly, Granville note that there will be infinitely manya finer investigations of Cramér's reasoning rather suggests maximal gaps of size $2 e^{-\gamma} (\log p_n)^2$ (up to lower order terms) $2 e^{-\gamma} (\log p_n)^2$.` And if this were true it would contradict the conjecture mentioned in OP (this is what is referred to in OP as Cramér-Granville conjecture).

It should however be noted that Granville did not conjecture that the gaps are of this size, yet pointed out what taking an additional aspect into account would mean for Cramér's reasoning. For Granville on this matter on MO see Consequences of Legendre's conjecture

For details on Granville's arguments, based eg on results by Maier, see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

Perhaps in view of the comments the following clarfication is helpful and since it also sort of an answer and a bit long I give it as answer.

It is quite likely that this conjecture is false yet no counter example was found so far.

The point is, which is mentioned by OP but got a bit blurred via the comments, that Granville argues that large gaps between successive primes in fact will not be bounded by $(\log p_n)^2$ (contrary to what Cramér conjectured) but that one actually needs a constant factor larger than one, and more specifically that there will be infinitely many gaps of size (up to lower order terms) $2 e^{-\gamma} (\log p_n)^2$.`

For details on Granville's arguments, based eg on results by Maier, see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

Significantly rewritten, yet the main message stays the same.

It is quite likely that this conjecture is false yet no counter example was found so far.

The reason why this seems likely is that there seems to be no supporting evidence for this conjecture beyond numerics.

And, there are investigations base on quite natural random models of the primes that contradict it. What is commonly known as Cramér's conjecture, that is that maximal gaps between consecutive primes are of sizes at most $(\log p_n)^2$ (up to lower order terms) does not contradict this conjecture, and one might even think it supports it. However, on the one hand it is not quite clear Cramér even conjectured this in precisely this form; he conjectured gaps are $O((\log p_n)^2)$ and somehow implied that about $(\log p_n)^2$ might be true. On the other hand, and more importantly, Granville note that a finer investigations of Cramér's reasoning rather suggests maximal gaps of size $2 e^{-\gamma} (\log p_n)^2$ (up to lower order terms). And if this were true it would contradict the conjecture mentioned in OP (this is what is referred to in OP as Cramér-Granville conjecture).

It should however be noted that Granville did not conjecture that the gaps are of this size, yet pointed out what taking an additional aspect into account would mean for Cramér's reasoning. For Granville on this matter on MO see Consequences of Legendre's conjecture

For details on Granville's arguments see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

corrected confusing choice of word;
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user9072
user9072

Perhaps in view of the comments the following clarfication is helpful and since it also sort of an answer and a bit long I give it as answer.

It is quite likely that this conjecture is false yet no counter example was found so far.

The point is, which is mentioned by OP but got a bit blurred via the comments, that Granville argues that large gaps between successive primes in fact will not be bounded by $(\log p_n)^2$ (contrary to what Cramér conjectured) but that one actually needs a constant factor larger than one, and more specifically that there will be infinitely many gaps of size (up to lower order factorsterms) $2 e^{-\gamma} (\log p_n)^2$.`

For details on Granville's arguments, based eg on results by Maier, see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

Perhaps in view of the comments the following clarfication is helpful and since it also sort of an answer and a bit long I give it as answer.

It is quite likely that this conjecture is false yet no counter example was found so far.

The point is, which is mentioned by OP but got a bit blurred via the comments, that Granville argues that large gaps between successive primes in fact will not be bounded by $(\log p_n)^2$ (contrary to what Cramér conjectured) but that one actually needs a constant factor larger than one, and more specifically that there will be infinitely many gaps of size (up to lower order factors) $2 e^{-\gamma} (\log p_n)^2$.`

For details on Granville's arguments, based eg on results by Maier, see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

Perhaps in view of the comments the following clarfication is helpful and since it also sort of an answer and a bit long I give it as answer.

It is quite likely that this conjecture is false yet no counter example was found so far.

The point is, which is mentioned by OP but got a bit blurred via the comments, that Granville argues that large gaps between successive primes in fact will not be bounded by $(\log p_n)^2$ (contrary to what Cramér conjectured) but that one actually needs a constant factor larger than one, and more specifically that there will be infinitely many gaps of size (up to lower order terms) $2 e^{-\gamma} (\log p_n)^2$.`

For details on Granville's arguments, based eg on results by Maier, see for example http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf

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user9072
user9072