a a a a ^an and and and and and and and and applying as as as $\mathit{a}\mathit{s}$ auxiliarybe below bounded by by

^can complete ^consequence Consequently Consider counterpart

direct domain domain

every express

first following: following: follows: For For for for for for FUNCTIONS

$\mathrm{\supset }{\mathrm{0}}^{\mathrm{ċ}}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}$

HARMONIC here

if if In INCREASING integral integral is is is is

large. left-hand left-hand LEMMA Lemma Lemma Lemma

make mapping

$\mathit{\zeta }\mathrm{\prime }\mathrm{=}{\mathrm{2}}^{\mathrm{-}\mathrm{1}}{\mathit{e}}^{\mathrm{-}\mathit{\pi }\mathrm{/}\mathrm{2}}\mathrm{\left|}\mathit{\eta }\mathrm{\right|}\mathrm{0}\mathrm{<}{\mathit{\xi }}_{\mathrm{2}}\mathrm{<}{\mathit{\xi }}_{\mathrm{1}}\mathrm{\le }\mathit{\sigma }{\mathit{e}}^{\mathrm{-}\mathit{\pi }\mathrm{/}\mathrm{2}}\mathrm{;}\mathit{\zeta }\mathrm{\to }\log \mathrm{\left(}\mathrm{\right(}\mathit{\zeta }\mathrm{-}\mathit{i}{\mathit{\eta }}_{\mathrm{0}}{\mathrm{)}}^{\mathrm{-}\mathrm{1}}\mathrm{)}$

$\mathrm{\Delta }\mathit{R}\mathit{\sigma }\mathit{\xi }{\mathit{\eta }}_{\mathrm{0}}\mathit{\eta }$. R. ${\mathit{\xi }}^{\mathrm{*}}{\mathit{\delta }}^{\mathrm{*}}\mathrm{,}$ $\mathrm{\left|}\mathit{\zeta }\mathrm{\right|}\mathit{\zeta }\mathrm{\to }\mathrm{0}$. $\mathrm{\left|}\mathit{\eta }\mathrm{\right|}\mathrm{/}\mathit{\xi }{\mathit{\xi }}_{\mathrm{1}}\mathrm{,}$ ${\mathit{\xi }}_{\mathrm{2}}\mathrm{0}\mathrm{<}\mathit{\xi }\mathrm{<}{\mathit{\xi }}^{\mathrm{*}}\mathrm{\left|}{\mathit{\eta }}_{\mathrm{0}}\mathrm{\right|}\mathrm{<}\mathit{\delta }\mathit{\zeta }\mathrm{=}\mathit{\xi }\mathrm{+}\mathit{i}\mathit{\eta }\mathrm{\left|}\mathit{\eta }\mathrm{\right|}\mathrm{<}{\mathit{\delta }}^{\mathrm{*}}$.$\mathrm{=}\mathit{\xi }\mathrm{+}\mathit{i}\mathit{\eta }\mathrm{\notin }\mathrm{\Delta }\mathit{\zeta }\mathrm{=}\mathit{\xi }\mathrm{+}\mathit{i}\mathit{\eta }\mathrm{\notin }\mathrm{\Delta }\mathrm{\prime }\mathit{\zeta }\mathrm{\to }\mathrm{0}\mathrm{,}$ $\mathit{\zeta }\mathrm{\notin }\mathrm{\Delta }\mathrm{\prime }\mathrm{,}$ $\mathit{\zeta }\mathrm{\to }\mathrm{0}\mathrm{,}$ $\mathit{\zeta }\mathrm{\notin }\mathrm{\Delta }\mathrm{\prime }$.

${{\displaystyle \int }}_{\mathit{u}\mathrm{(}\mathit{\zeta }\mathrm{⟩}}^{\mathit{u}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}}\frac{\mathit{d}\mathit{c}}{{\mathrm{\Theta }}_{\mathit{R}}\mathrm{\left(}\mathit{c}\mathrm{\right)}}\mathrm{>}\mathrm{0}\mathrm{,}$ $\mathrm{\Delta }\mathrm{\prime }\mathrm{=}\mathrm{\{}\mathit{\zeta }\mathrm{=}\mathit{\xi }\mathrm{+}\mathit{i}\mathit{\eta }\mathrm{|}\mathrm{\left|}\mathit{\eta }\mathrm{\right|}\mathrm{<}\mathrm{2}{\mathit{e}}^{\mathit{\pi }\mathrm{/}}\mathrm{2}\mathit{\xi }\mathrm{\}}$. $\mathrm{0}\mathrm{<}\mathit{\sigma }\mathrm{<}\min \mathrm{(}\mathit{\delta }\mathrm{-}\mathrm{|}{\mathit{\eta }}_{\mathrm{0}}\mathrm{|}\mathrm{,}\mathrm{}\log \mathrm{(}{\mathit{r}}_{{\mathrm{0}}^{\mathrm{1}}}\mathrm{\left)}\mathrm{\right)}$,

$\mathit{D}\mathrm{=}\mathit{D}\mathrm{(}{\mathit{\eta }}_{\mathrm{0}}\mathrm{,}\mathrm{}\mathit{\sigma }\mathrm{)}\mathrm{=}\mathrm{\left\{}\mathit{\zeta }\mathrm{\right|}\mathrm{Re}\mathit{\zeta }\mathrm{>}\mathrm{0}\mathrm{,}\mathrm{}\mathrm{|}\mathit{\zeta }\mathrm{-}\mathit{i}{\mathit{\eta }}_{\mathrm{0}}\mathrm{|}\mathrm{<}\mathit{\sigma }\mathrm{\}}$. $\mathrm{\left|}\mathit{\eta }\mathrm{\right|}\mathrm{<}{\mathit{\delta }}_{\mathrm{0}}\mathrm{=}\min \mathrm{(}\mathrm{2}\mathit{\delta }\mathrm{/}\mathrm{3}\mathrm{,}\mathrm{}\frac{\mathrm{1}}{\mathrm{2}}\log \mathrm{(}{\mathit{r}}_{\mathrm{0}}^{\mathrm{-}\mathrm{1}}\mathrm{\left)}\mathrm{\right)}$,

$\frac{\mathrm{1}}{\mathit{\pi }}\log \frac{\mathrm{\left|}\mathit{\eta }\mathrm{\right|}}{\mathit{\xi }}\mathrm{+}\mathit{O}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{-}\mathrm{(}\log \frac{\mathrm{\left|}\mathit{\eta }\mathrm{\right|}}{\mathit{\xi }}\mathrm{+}\mathit{O}\mathrm{(}\mathrm{1}\mathrm{)}{\mathrm{)}}^{\mathrm{*}}{{\displaystyle \int }}_{\mathit{u}\mathrm{(}\mathit{\xi }\mathrm{\prime }\mathrm{)}}^{\mathit{u}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}}\frac{\mathit{d}\mathit{c}}{{\mathrm{\Theta }}_{\mathit{R}}\mathrm{\left(}\mathit{c}\mathrm{\right)}}\mathrm{=}\frac{\mathrm{1}}{\mathit{\pi }}\log \frac{\mathrm{\left|}\mathit{\eta }\mathrm{\right|}}{\mathit{\xi }}\mathrm{+}\mathit{O}\mathrm{(}\mathrm{1}\mathrm{)}{{\displaystyle \int }}_{\mathit{u}\mathrm{\left(}\mathit{\zeta }\mathrm{\right)}}^{\mathit{u}\mathrm{(}\mathit{\xi }\mathit{\gamma }}\frac{\mathit{d}\mathit{c}}{{\mathrm{\Theta }}_{\mathit{R}}\mathrm{\left(}\mathit{c}\mathrm{\right)}}\mathrm{+}{{\displaystyle \int }}_{\mathit{u}\mathrm{⟨}\mathit{\xi }\mathrm{\prime }\mathrm{)}}^{\mathit{u}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}}\frac{\mathit{d}\mathit{c}}{{\mathrm{\Theta }}_{\mathit{R}}\mathrm{\left(}\mathit{c}\mathrm{\right)}}$${{\displaystyle \int }}_{\mathit{u}\mathrm{(}{\mathit{\xi }}_{\mathrm{1}}\mathrm{+}\mathit{i}{\mathit{\eta }}_{\mathrm{0}}\mathrm{)}}^{\mathit{u}\mathrm{(}\mathit{\xi }\mathrm{.}\mathrm{+}\mathit{i}{\mathit{\eta }}_{\mathrm{0}}\mathrm{)}}\frac{\mathit{d}\mathit{c}}{{\mathrm{\Theta }}_{\mathit{D}}\mathrm{\left(}\mathit{c}\mathrm{\right)}}\mathrm{|}\mathrm{\le }\frac{\mathrm{1}}{\mathit{\pi }}\log \frac{{\mathit{\xi }}_{\mathrm{1}}}{{\mathit{\xi }}_{\mathrm{2}}}\mathrm{+}\frac{\mathit{\pi }}{\mathrm{4}}{{\displaystyle \int }}_{\mathit{u}\mathrm{(}\mathit{\xi }\mathrm{\prime }\mathrm{)}}^{\mathit{u}\mathrm{\left(}\mathit{\zeta }\mathrm{\right)}}\frac{\mathit{d}\mathit{c}}{{\mathrm{\Theta }}_{\mathit{R}}\mathrm{\left(}\mathit{c}\mathrm{\right)}}\mathrm{\le }\mathrm{(}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{+}\mathit{o}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{\left)}\mathrm{\right(}\log \frac{\mathrm{\left|}\mathit{\eta }\mathrm{\right|}}{\mathit{\xi }}\mathrm{+}\mathit{O}\mathrm{\left(}\mathrm{1}\mathrm{\right)}{\mathrm{)}}^{\mathrm{*}}$

need need

obtain of of of of of of On ^on other

1, 1' 24. 5 55 6 6. 6. (34). (36) (36) (36) (36) (37)

positive proof Proof

oatisfied satisfies second set side side SLOWLY small small Stolz Stolz such such sufficientlyoufficiently sufficiently sufficiently

Cake taking that that The The the the the the the then Theorem Theorem these this

UNBOUNDED

verified.

$\mathit{W}\mathit{e}$ We we we ^we which wide will with with with words,