Geodesic
On prime numbers of special kind on short intervals
Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 908-912 
N. N. Mot'kina. On prime numbers of special kind on short intervals. Matematičeskie zametki, Tome 79 (2006) no. 6, pp. 908-912. http://geodesic.mathdoc.fr/item/MZM_2006_79_6_a7/
@article{MZM_2006_79_6_a7,
     author = {N. N. Mot'kina},
     title = {On prime numbers of special kind on short intervals},
     journal = {Matemati\v{c}eskie zametki},
     pages = {908--912},
     year = {2006},
     volume = {79},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_6_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Suppose that the Riemann hypothesis holds. Suppose that $$ \psi_1(x)=\sum_{\substack{n\le x\\ \{(1/2)n^{1/c}\}<1/2}}\Lambda(n), $$ where $c$ is a real number, $1. We prove that, for $H>N^{1/2+10\varepsilon}$, $\varepsilon>0$, the following asymptotic formula is valid: $$ \psi_1(N+H)-\psi_1(N)=\frac H2\biggl(1+O\biggl(\frac1{N^\varepsilon}\biggr)\biggr). $$

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