Replacing the n-th prime by the sum of logarithms of the first n primes makes analogues of Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures provable theorems.
Bornes effectives pour certaines fonctions concernant les nombres premiers
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Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$
Replacing the n-th prime by the sum of logarithms of the first n primes makes analogues of Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures provable theorems.