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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

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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

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

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

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

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

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

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positive proof Proof

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

UNBOUNDED

verified.

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