L - ??????? — ??? ??????????????? ???????, ??????????? ?-??????? , ???????????? ??? ${\displaystyle L_{n}[\alpha ,c]}$ ??? ${\displaystyle n}$ ??????????? ? ????????????? . ??????? O-???????? , L-??????? ?????? ???????????? ??? ???????????? ?????? ?????????????? ????????? ??????????? ????????? . ??? ???? ${\displaystyle n}$ ???????????? ????? ????????? ???????? ??????? ?????? ?????????, ???????????????? ?? ???????: ????????, ????? ?????? ? ????? ?? ??????? ????? ? ?????????? ?????? ? ??? ??????????? ????, ??? ??????????? ????? ? ?????????? ?????????? ??? ?? ??????? ??????????? .

${\displaystyle L_{n}[\alpha ,c]}$ ???????????? ????????

${\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}$ ,

??? ${\displaystyle c}$ — ????????????? ?????????, ? ${\displaystyle \alpha }$ — ????????? ${\displaystyle 0\leq \alpha \leq 1}$ .

L -??????? ???????????? ? ???????? ? ?????????????? ?????? ????? ??? ????????? ????????? ?????????? ??? ??????? ??????? ?????? ????? , ????????, ?????????? ?????? ?????????? ??????????? ????? ?? ??????? ??????????? ? ??????? ?????????? ?????????? ?????????? . ???????????? ????? ??????? ??????????? ? ????????? ??????? ??????????.

??????????? ${\displaystyle e^{c(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}$ ? ${\displaystyle L_{n}[\alpha ,c]}$ ???????? ???????????? ????????????, ? ??????????? ${\displaystyle e^{o(1)(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}$ ????????? ?? ????? ????? ?????????????. ??? ????, ????? ${\displaystyle \alpha }$ ????? 0,

${\displaystyle L_{n}[0,c]=e^{(c+o(1))\ln \ln n}=(\ln n)^{c+o(1)}}$

???????? ??????????? ?? ln n , ? ?? ????? ??? ??? ${\displaystyle \alpha }$ ?????? 1,

${\displaystyle L_{n}[1,c]=e^{(c+o(1))\ln n}=n^{c+o(1)}}$

???????? ??????????? ?? ln n (?, ???????, ????????? ?? n ). ???? ?? ${\displaystyle \alpha }$ ????????? ???-?? ????? 0 ? 1, ?? ??????? ${\displaystyle L_{n}[\alpha ,c]}$ ???????????????????, ?. ?. ?????? ?????????, ??? ???????????????? ??????? ? ?????????? ?????? 1 (??? ?????-??????????????).

???????

?????? ????????? ?????????? ????? ?? ??????? ??????????? ????? ??????????????????? ?????????? ?????????. ?????? ??????? ? ????? ?????? ???????? ?????????????? ???????? ???????? ????? ????? ?????? ????????? ???? , ??????? ????? ??????:

${\displaystyle L_{n}[1/3,c]=e^{(c+o(1))(\ln n)^{1/3}(\ln \ln n)^{2/3}}}$

??? ${\displaystyle c=(64/9)^{(1/3)}}$ .

?????? ??????????, ?? ?????????? ?????? ????????? ????, ??? ????? ????????????? ?????? , ??????? ????? ?????? ?????????:

${\displaystyle L_{n}[1/2,1]=e^{(1+o(1))(\ln n)^{1/2}(\ln \ln n)^{1/2}}}$

??? ?????? ??????????? ???????????????? ????????????? ?????? , ????? ??????? ?????????????? ?????????? ???????? ???????? ??????? ? ????? ????? - ???????? ?????? , ??????? ????????????????? ?????? ??????? ?????? ?????? ??????????? ????? ?? ??????? ?????? n . ? L -??????? ??? ????????????:

${\displaystyle L_{n}[1,1/2]=n^{1/2+o(1)}}$

????????????? ????? ???????? AKS , ??????? ???????? ? ?????????????? ????? , ????????, ??? ????????? ????? ???????? ?????? ???? ?? ?????

${\displaystyle L_{n}[0,c]=(\ln n)^{c+o(1)}}$

? ????????, ??? c ?? ?????? ????????? 6.

???????

${\displaystyle L}$ -??????? ???? ?????????? ? ?????????? ? ????????? ????. ?????? ???????? ${\displaystyle L}$ -??????? ???? ???????? ? ??? ?????? «?????? ? ????????? ????????? ?????????? ???????????? ????? ?????» .

??? ????? ????? ?????? ???? ???????? ${\displaystyle c}$ , ${\displaystyle \alpha }$ ? ??????? ??? ?????????? ${\displaystyle 1/2}$ . ???????? ??????????? ????? ${\displaystyle L}$ (??? ????????? ${\displaystyle l}$ ) ? ???? ? ?????????? ?????? ??? ??????, ?????????? ????? ??????????.

??????????????? ???????, ?????????? ??? ?????????, ???? ??????? ??????? ???????? ? ????????? ???????? ? ?? ?????? «????????? ? ?????? ?????» , ??? ??????? ???? ???????????? ??? ??????? ??????????? ???????????????? ????????? ??????????? . ? ????????? ????? ??????? ???????? ???????? ??????????? ? ??????????.

??????????? ?? ?????????? ???????????? ?????????? L -??????? ??? :

${\displaystyle L_{n}[\alpha ,c]=O(e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}).}$

??? ?? ???????? ??????????? ????????????. ${\displaystyle O}$ ????????????, ??? ????? ?????? ??????, ???????????? ????????, ?????????? ??????. ??????, ??? ?????????? ?????? ????? ? ??????????? ???????????????? ${\displaystyle L}$ -???????, ???????????? ??? ??????, ?? ???????? ??????? ????????, ??? ??? ????? ??????????? ?? ?????? ?????????.

??????????