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GH from MO
GH from MO
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It is not know that RH implies EH, or that EH implies RH. Let us denote $$S(x):=\sum_{p_n < x} (p_n -p_{n-1})^2.$$ Assuming the Lindelöf hypothesis, Yu (1996)Yu (1996) proved that $S(x)\ll_\varepsilon x^{1+\varepsilon}$. Unconditionally, Stadlmann (2022) proved that $S(x)\ll_\varepsilon x^{1.23+\varepsilon}$.

It is not know that RH implies EH, or that EH implies RH. Let us denote $$S(x):=\sum_{p_n < x} (p_n -p_{n-1})^2.$$ Assuming the Lindelöf hypothesis, Yu (1996) proved that $S(x)\ll_\varepsilon x^{1+\varepsilon}$. Unconditionally, Stadlmann (2022) proved that $S(x)\ll_\varepsilon x^{1.23+\varepsilon}$.

It is not know that RH implies EH, or that EH implies RH. Let us denote $$S(x):=\sum_{p_n < x} (p_n -p_{n-1})^2.$$ Assuming the Lindelöf hypothesis, Yu (1996) proved that $S(x)\ll_\varepsilon x^{1+\varepsilon}$. Unconditionally, Stadlmann (2022) proved that $S(x)\ll_\varepsilon x^{1.23+\varepsilon}$.

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GH from MO
GH from MO
  • 105.2k
  • 8
  • 292
  • 398

It is not know that RH implies EH, or that EH implies RH. Let us denote $$S(x):=\sum_{p_n < x} (p_n -p_{n-1})^2.$$ Assuming the Lindelöf hypothesis, Yu (1996) proved that $S(x)\ll_\varepsilon x^{1+\varepsilon}$. Unconditionally, Stadlmann (2022) proved that $S(x)\ll_\varepsilon x^{1.23+\varepsilon}$.