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In this post we denote the $k-th$ prime number as $p_k$. I present two conjectures, the first about the asymptotic behaviour of a product and the other about an alternating series. I don't know if these are in the literature. I think that these are interesting because involve the sequence of prime numbers and were an attempt to capture some aspect of the so-called Firoozbakht's conjecture and a conjecture from section E7 of [1].

The interesting thing (and I assume that surely my creations/inventions failed in this, I believe that my sequences don't have enough precision to do it) should be try to create sequences, recurrences, products or series capturing the mathematical content of the mentioned Firoozbakht's conjecture, I say that the convergence or divergence of such objects is closely related to the veracity of Firoozbakht's conjecture.

Conjecture 1. One has that $$\prod_{n=1}^N\left(1+\frac{p_{n+1}^n}{p_{n}^{n+1}}\right)=O\left(N\log N\right)$$ as $N\to \infty$.

Conjecture 2. The series $$\sum_{n=2}^\infty(-1)^n n\cdot \left(p_{n-1}\right)^{-\frac{n}{n-1}}$$ is a convergent series.

Question. Can we refute any of previous conjectures? In other case, can be proved? In this last case provide hints or the proof to get the conjectures as propositions. Many thanks.

If you can/are able to set more precise conjectures, in the spirit of previous paragraphs, feel free add your comments or as remarks in your answer.

References:

[1] Richard K. Guy, Unsolved Problems in Number Theory, Problem Books in Mathematics, Unsolved Problems in Intuitive Mathematics Volume I, Springer-Verlag (1994).

[2] See if you want the statement of Firoozbakht's conjecture from this article of Wikipedia.

Conjecture 30. The Firoozbakht Conjecture, from Carlos Rivera's web The Prime Puzzles & Problems Connection (August, 2012).

EDIT: Firoozbakht's conjecture is that if $p_n$ is the $n$-th prime, then for all $n\ge1$, $\root{n+1}\of{p_{n+1}}<\root n\of{p_n}$.

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    $\begingroup$ How hard would it have been to include a statement of Firoozbakht, instead of (or in addition to) a link to Wikipedia? $\endgroup$
    – Gerry Myerson
    Commented Aug 13, 2019 at 13:25
  • $\begingroup$ (1/2) Yes you are right, next time I should include it @GerryMyerson . I wrote a conjecture as a present for your patience and attention. I am inspired in Robin's equivalence to the Riemann hypothesis and the last identity of the linked Wikipedia in my post, if I am right it is the conjecture due to John W. Nicholson. For each integer $n>5040$ one has that $e^{\gamma}\sqrt{n}\log\left(n\left(\frac{p_{n}}{p_{n+1}}\right)^{n}\right)<\sigma(n)$. I had few computational evidence, thus I don't know if it is the best conjecture or if it is compatible with the theory. $\endgroup$
    – user142929
    Commented Aug 13, 2019 at 21:00
  • $\begingroup$ (2/2) See also, if you want this line of code using Wolfram Alpha online calculator) plot sigma(n)-n^(5/8) e^(EulerGamma) log(n (prime(n)/prime(n+1))^n) , from n=5040 to 20000 As it was evoked I don't know if this last inequality has a good mathematical content. This is all, I hope you enjoy with the mathematics posted in the web MO and with this summer time. $\endgroup$
    – user142929
    Commented Aug 13, 2019 at 21:09
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    $\begingroup$ I edited the conjecture in, since you couldn't be bothered to do it yourself. Also, it's Winter here. $\endgroup$
    – Gerry Myerson
    Commented Aug 13, 2019 at 23:02

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