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! Undefined control sequence.
l.9 \chapter
            [Building $\pi(X)$ knowing the spectrum]{How to build $\pi(X)$ k...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <7> on input line 9.
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(Font)              <5> on input line 9.

LaTeX Warning: Reference `part3' on page 1 undefined on input line 12.

! Undefined control sequence.
l.22   assuming the \RH
                       {} as formulated in Chapter~\ref{sec:tinkering}---can
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.


LaTeX Warning: Reference `sec:tinkering' on page 1 undefined on input line 22.

! Undefined control sequence.
l.30   The answer to this is yes (given the \RH
                                               {}), and is discussed very
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<recently read> \Li 
                    
l.38 ...ex{Gauss, Carl Friedrich} offered his $\Li
                                                  (X)$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.


LaTeX Warning: Reference `sec:rh1' on page 1 undefined on input line 40.

! Undefined control sequence.
<recently read> \Li 
                    
l.41   $\Li
           (X)= \int_2^{X}dt/{\rm log}(t).$ Riemann's guess for a better
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Misplaced alignment tab character &.
l.46     1 &
            
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Misplaced alignment tab character &.
l.50     -1&
             \mbox{\begin{minipage}{0.6\textwidth}if $n$ is a square-free
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Misplaced alignment tab character &.
l.53     0 &
             \mbox{if $n$ is not square-free.}
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.


Overfull \hbox (230.9011pt too wide) detected at line 55
\OML/cmm/m/it/10 ^^V\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 ) = [] [
][]  \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 1[][] 0[]
 []

! Undefined control sequence.
l.58 \ill
         {moebius}{1}{The blue dots plot the values of the Moebius function\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.63 R(X) = \sum_{n=1}^{\infty}{\mu(n)\over n} \Li
                                                  (X^{1\over n}),
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.67 \ill
         {riemann_RX}{0.8}{Riemann\index{Riemann, Bernhard} defining $R(X)$ ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

[1

{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
! Missing $ inserted.
<inserted text> 
                $
l.67 \ill{riemann_
                  RX}{0.8}{Riemann\index{Riemann, Bernhard} defining $R(X)$ ...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Extra }, or forgotten $.
l.67 \ill{riemann_RX}
                     {0.8}{Riemann\index{Riemann, Bernhard} defining $R(X)$ ...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.67 ...iemann\index{Riemann, Bernhard} defining $
                                                  R(X)$ in his manuscript}
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Extra }, or forgotten $.
l.67 ...rnhard} defining $R(X)$ in his manuscript}
                                                  
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing $ inserted.
<inserted text> 
                $
l.68 
     
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


LaTeX Warning: Reference `sec:pnt' on page 2 undefined on input line 69.

! Undefined control sequence.
<recently read> \Li 
                    
l.70 asserts that $X/\log(X)$ and $\Li
                                      (X)$ are both approximations for
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.72 Our first formulation of the \RH
                                     {} (see page~\pageref{rh:first}) was
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.


LaTeX Warning: Reference `rh:first' on page 2 undefined on input line 72.

! Undefined control sequence.
<recently read> \Li 
                    
l.73 that $\Li
              (X)$ is an essentially square root accurate approximation of
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of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.78 \illtwo
            {pi_riemann_gauss_100}{pi_riemann_gauss_1000}{0.47}{Comparisons ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.78 \illtwo{pi_
                riemann_gauss_100}{pi_riemann_gauss_1000}{0.47}{Comparisons ...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Extra }, or forgotten $.
l.78 \illtwo{pi_riemann_gauss_100}
                                  {pi_riemann_gauss_1000}{0.47}{Comparisons ...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.78 ...riemann_gauss_1000}{0.47}{Comparisons of $
                                                  \Li(X)$ (top), $\pi(X)$ (m...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Undefined control sequence.
l.78 ...mann_gauss_1000}{0.47}{Comparisons of $\Li
                                                  (X)$ (top), $\pi(X)$ (midd...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.78 ...{0.47}{Comparisons of $\Li(X)$ (top), $\pi
                                                  (X)$ (middle), and $R(X)$ ...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


Overfull \hbox (25.26248pt too wide) in paragraph at lines 78--79
[]\OT1/cmr/m/n/10 pi$[]\OML/cmm/m/it/10 iemann[]auss[]\OT1/cmr/m/n/10 00[][][]$
(X)$(\OML/cmm/m/it/10 top\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ;$$^^Y\OT1/cmr/m/n/1
0 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 )$ (mid-
 []

! Undefined control sequence.
l.81 \ill
         {pi_riemann_gauss_10000-11000}{0.5}{Closeup comparison of  $\Li(X)$...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.81 \ill{pi_
             riemann_gauss_10000-11000}{0.5}{Closeup comparison of  $\Li(X)$...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Extra }, or forgotten $.
l.81 \ill{pi_riemann_gauss_10000-11000}
                                       {0.5}{Closeup comparison of  $\Li(X)$...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.81 ...10000-11000}{0.5}{Closeup comparison of  $
                                                  \Li(X)$ (top), $\pi(X)$ (m...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Undefined control sequence.
l.81 ...00-11000}{0.5}{Closeup comparison of  $\Li
                                                  (X)$ (top), $\pi(X)$ (midd...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.81 ...loseup comparison of  $\Li(X)$ (top), $\pi
                                                  (X)$ (middle), and $R(X)$ ...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.111 \ill
          {riemann_Rk}{0.8}{Riemann\index{Riemann, Bernhard} analytic formul...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.111 \ill{riemann_
                   Rk}{0.8}{Riemann\index{Riemann, Bernhard} analytic formul...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Extra }, or forgotten $.
l.111 \ill{riemann_Rk}
                      {0.8}{Riemann\index{Riemann, Bernhard} analytic formul...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.111 ...Riemann, Bernhard} analytic formula for $
                                                  \pi(X)$.}
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Missing $ inserted.
<inserted text> 
                $
l.111 ...mann, Bernhard} analytic formula for $\pi
                                                  (X)$.}
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.113 The \RH
             {}, if true, would tell us that these correction
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of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.


Overfull \hbox (9.49744pt too wide) in paragraph at lines 113--115
[]\OT1/cmr/m/n/10 The , if true, would tell us that these cor-rec-tion terms $\
OML/cmm/m/it/10 C[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 )\OML/cmm
/m/it/10 ; C[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 )\OML/cmm/m/it
/10 ; C[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ;
 []$
 []

[2]
! Undefined control sequence.
<recently read> \Li 
                    
l.130 the definition of the function $\Li
                                         (X)$ to make sense when given
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.131 complex numbers $X=a+bi$.  Assuming the \RH
                                                 {}, the Riemann\index{Riema...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.137 ...rime numbers\index{prime number} \bibnote
                                                  {You may well ask how we p...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.156 \ill
          {Rk_1_2_100}{.9}{The function $R_{1}$ approximating the staircase ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.156 \ill{Rk_
              1_2_100}{.9}{The function $R_{1}$ approximating the staircase ...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Double subscript.
l.156 \ill{Rk_1_
                2_100}{.9}{The function $R_{1}$ approximating the staircase ...
I treat `x_1_2' essentially like `x_1{}_2'.

! Double subscript.
l.156 \ill{Rk_1_2_
                  100}{.9}{The function $R_{1}$ approximating the staircase ...
I treat `x_1_2' essentially like `x_1{}_2'.

! Extra }, or forgotten $.
l.156 \ill{Rk_1_2_100}
                      {.9}{The function $R_{1}$ approximating the staircase ...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.156 \ill{Rk_1_2_100}{.9}{The function $
                                         R_{1}$ approximating the staircase ...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Missing $ inserted.
<inserted text> 
                $
l.156 \ill{Rk_1_2_100}{.9}{The function $R_
                                           {1}$ approximating the staircase ...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.158 \ill
          {Rk_10_2_100}{.9}{The function $R_{10}$ approximating the staircas...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.158 \ill{Rk_
              10_2_100}{.9}{The function $R_{10}$ approximating the staircas...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Double subscript.
l.158 \ill{Rk_10_2_
                   100}{.9}{The function $R_{10}$ approximating the staircas...
I treat `x_1_2' essentially like `x_1{}_2'.

! Extra }, or forgotten $.
l.158 \ill{Rk_10_2_100}
                       {.9}{The function $R_{10}$ approximating the staircas...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.158 \ill{Rk_10_2_100}{.9}{The function $
                                          R_{10}$ approximating the staircas...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Missing $ inserted.
<inserted text> 
                $
l.158 \ill{Rk_10_2_100}{.9}{The function $R_
                                            {10}$ approximating the staircas...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.160 \ill
          {Rk_25_2_100}{.9}{The function $R_{25}$ approximating the staircas...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.160 \ill{Rk_
              25_2_100}{.9}{The function $R_{25}$ approximating the staircas...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Double subscript.
l.160 \ill{Rk_25_2_
                   100}{.9}{The function $R_{25}$ approximating the staircas...
I treat `x_1_2' essentially like `x_1{}_2'.

! Extra }, or forgotten $.
l.160 \ill{Rk_25_2_100}
                       {.9}{The function $R_{25}$ approximating the staircas...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.160 \ill{Rk_25_2_100}{.9}{The function $
                                          R_{25}$ approximating the staircas...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Missing $ inserted.
<inserted text> 
                $
l.160 \ill{Rk_25_2_100}{.9}{The function $R_
                                            {25}$ approximating the staircas...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.162 \ill
          {Rk_50_2_100}{.9}{The function $R_{50}$ approximating the staircas...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.162 \ill{Rk_
              50_2_100}{.9}{The function $R_{50}$ approximating the staircas...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Double subscript.
l.162 \ill{Rk_50_2_
                   100}{.9}{The function $R_{50}$ approximating the staircas...
I treat `x_1_2' essentially like `x_1{}_2'.

! Extra }, or forgotten $.
l.162 \ill{Rk_50_2_100}
                       {.9}{The function $R_{50}$ approximating the staircas...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.162 \ill{Rk_50_2_100}{.9}{The function $
                                          R_{50}$ approximating the staircas...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Missing $ inserted.
<inserted text> 
                $
l.162 \ill{Rk_50_2_100}{.9}{The function $R_
                                            {50}$ approximating the staircas...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.165 \ill
          {Rk_50_2_500}{.9}{The function $R_{50}$ approximating the staircas...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.165 \ill{Rk_
              50_2_500}{.9}{The function $R_{50}$ approximating the staircas...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Double subscript.
l.165 \ill{Rk_50_2_
                   500}{.9}{The function $R_{50}$ approximating the staircas...
I treat `x_1_2' essentially like `x_1{}_2'.

! Extra }, or forgotten $.
l.165 \ill{Rk_50_2_500}
                       {.9}{The function $R_{50}$ approximating the staircas...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.165 \ill{Rk_50_2_500}{.9}{The function $
                                          R_{50}$ approximating the staircas...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Missing $ inserted.
<inserted text> 
                $
l.165 \ill{Rk_50_2_500}{.9}{The function $R_
                                            {50}$ approximating the staircas...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.167 \ill
          {Rk_50_350_400}{.9}{The function $\Li(X)$ (top, green), the functi...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.167 \ill{Rk_
              50_350_400}{.9}{The function $\Li(X)$ (top, green), the functi...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Extra }, or forgotten $.
l.167 \ill{Rk_50_350_400}
                         {.9}{The function $\Li(X)$ (top, green), the functi...
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Missing } inserted.
<inserted text> 
                }
l.167 \ill{Rk_50_350_400}{.9}{The function $
                                            \Li(X)$ (top, green), the functi...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Undefined control sequence.
l.167 \ill{Rk_50_350_400}{.9}{The function $\Li
                                               (X)$ (top, green), the functi...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.167 ...n $\Li(X)$ (top, green), the function $R_
                                                  {50}(X)$ (in blue), and th...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


Overfull \hbox (3.98949pt too wide) in paragraph at lines 167--168
[]\OT1/cmr/m/n/10 Rk$[]0[]50[]00[][]$(X)$(\OML/cmm/m/it/10 top; green\OT1/cmr/m
/n/10 )\OML/cmm/m/it/10 ; thefunction$\OT1/cmr/m/n/10 R$[](\OML/cmm/m/it/10 X\O
T1/cmr/m/n/10 )$ (in blue), and
 []

[3] (./Rk.aux)

LaTeX Warning: There were undefined references.

 )
(\end occurred inside a group at level 1)

### simple group (level 1) entered at line 67 ({)
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