is consistent with Granville's claims and all of the conjectures above and is enough to yield  both  (\ref{cramer})
$$\dege G  = \dege  g_n = (0,2,0,0,0,\ldots).$$
However, Pintz suggests in \cite[p.\ 286]{pintz} that
\begin{align}\label{pintzcon}
G(x) = O ((\log x)^t) \ (x \to \infty), \quad \forall t > 2,
\end{align}
and  speculates that one might have $G(x) \neq O ((\log x)^2) \ (x \to \infty)$.
At the very least, $\deg g_n = \deg G = 0$ and $\dege_1 g_n  =  \dege_1 G  \leq 2$ are implied by all related conjectures we can find in the literature.
Thus, the following conjecture is an extension of (Piltz's) Conjecture \ref{gapconj} that is implied by each of the conjectures discussed above.

\begin{conjecture}\label{gapconj2}
Pintz's suggestion (\ref{pintzcon}) holds.
\end{conjecture}

The following result follows from Theorem \ref{GTM}.

\begin{proposition}
Each of the following statements is equivalent to (\ref{pintzcon}), i.e., to Conjecture \ref{gapconj2}.
\begin{enumerate}
\item $\deg g_n = 0$ and $\dege_1 g_n \leq 2$.
\item $\deg G = 0$ and $\dege_1 G \leq 2$.
\item $g_n = O((\log n)^t) \ (n \to \infty)$ for all $t > 2$.
\item For all $t > 2$, one has $\pi(x+(\log x)^t))-\pi(x) \geq 1$ for all $x \gg 0$.
\item $(0,1,1,-1,1,0,0,0,\ldots) \leq \dege g_n \leq (0,2,\infty,1,0,0,0,\ldots).$
\item $(0,1,1,-1,1,0,0,0,\ldots) \leq \dege G \leq (0,2,\infty,1,0,0,0,\ldots).$
\end{enumerate}
\end{proposition}

One might also suspect that $\dege_1 g_n = 2$, which is an analogue of the conjecture $\dege_1(\gamma_{n+1}-\gamma_n) = -\frac{1}{2}$ discussed briefly in the previous section.   Indeed,  in 2018, Maynard conjectured the following.

\begin{conjecture}[Maynard {\cite[Conjecture 2]{mayn}}]\label{mayncram}
One has
$$G(x) = (\log x)^{2+o(1)} \ (x \to \infty).$$
\end{conjecture}

Note the following.

\begin{proposition}
Each of the following statements is equivalent to Maynard's conjecture  $G(x) = (\log x)^{2+o(1)} \ (x \to \infty)$.
\begin{enumerate}
\item $G(e^x)$ has exact degree $2$.
\item $\lim_{x \to \infty} \frac{ \log G(x)}{\log \log x} = 2$.
\item  $G(x)$ has (exact) degree $0$ and exact logexponential degree $2$ of order $1$.
\item $(0,2,-\infty,-1,0,0,0,\ldots) \leq \underline{\dege} \, G \leq\dege G \leq (0,2,\infty,1,0,0,0,\ldots)$.
\item  $G(x) \ll (\log x)^t$ for all $t > 2$ and $G(x) \gg (\log x)^t$ for all $t < 2$.
\item $(\log x)^{2-\varepsilon} \ll G(x) \ll (\log x)^{2+\varepsilon}$ for all $\varepsilon > 0$.
\item For any  eventually nonnegative nondecreasing function  $h(x)$ defined on a neighborhood of $\infty$,  if $h(x) \gg (\log x)^t$ for some $t > 2$, then
$$\pi(x+ h(x))-\pi(x)) \geq 1, \quad \forall x \gg 0,$$
and, conversely, if
$$\pi(x+ h(x))-\pi(x)) \geq 1, \quad \forall x \gg 0,$$
then $h(x) \gg (\log x)^t$ for some $t < 2$.
\end{enumerate}
\end{proposition}

To test Maynard's conjecture,  one most naturally plots and samples the statistic $\frac{ \log G(e^x)}{\log x}$ and checks to see if it appears to have a limit of $2$ as $x \to \infty$.  Such an analysis is a bit more straightforward than that of the function $\frac{G(x)}{(\log x)^2}$.    In particular, Maynard's conjecture  sidesteps  the various subtleties involved in conjecturing appropriate bounds on $\frac{G(x)}{(\log x)^2}$.    See Figures \ref{Gx},  \ref{LGx}, and \ref{SGx} for plots of $G(e^x)$, $\frac{ \log G(e^x)}{\log x}-2$, and $\frac{G(e^x)}{x^2}$,  respectively,  in black, and their respective limits from the left in green, at each (prime integer) value $n$ of $e^x$ where $G(e^x)$ has a discontinuity,  i.e.,  where $g_{\pi(n)}$  is a  (record) maximal  prime gap, for the first $80$ maximal prime gaps.  Note that the step function $G(e^x)$ is constant in between each such point.  The 80 $x$-coordinates in these plots are just $\log a(n)$, where $a(n)$ is OEIS Sequence A002386, while the $y$-coordinates for the points in black, respectively, are $b(n)$,  $\frac{\log b(n)}{\log \log a(n)}-2$,  and  $\frac{b(n)}{(\log a(n))^2}$, where $b(n)$ is OEIS Sequence A005250, for $n = 1,2,3,\ldots,80$.  The largest (i.e., the $80$th) of these maximal prime gaps occurs at the $20$-digit prime $$p_{423731791997205041} =  18361375334787046697,$$
for which the gap between the next prime is $g_{423731791997205041} = 1550$.  To date this is the largest known maximal prime gap, which was found by B.\ Nyman in 2018 and verified  by analyzing the prime gaps for all primes less than $2^{64} = 18446744073709551616$.
The clear outlier in these three plots occurs at $\log p = 35.065386\ldots$, where $p$ is the prime number $$p_{49749629143526} =  1693182318746371,$$ for which the value of $$\frac{g_{\pi(p)}}{(\log p)^2}  = \frac{1132}{(\log( 1693182318746371))^2} = 0.920638\ldots$$
is the largest known, for $p > 7$.  The merit $\widehat{g}_{\pi(p)} = \frac{g_{\pi(p)}}{\log p}$, or normalized prime gap, of this prime gap of $1132$ is $32.282547\ldots$.  As of September 2022, the largest known merit of a prime gap, and the first found with merit over 40 (discovered by the Gapcoin network), is $41.938783\ldots$,  for a particular $87$-digit prime $p$ with $\log p = 199. 099717\ldots$ and with a prime gap of $8350$.  See Figure \ref{MGx} for a corresponding  plot of $\frac{G(e^x)}{x}$, i.e., a lin-log plot of the merits of the first $80$ maximal prime gaps.

\begin{figure}[h!]
\includegraphics[width=70mm]{G_ex_.png}
\caption{\centering Plot of $G(e^x)$ (in black) and $G((e^x)^-)$ (in green) at each (prime integer) value $n$ of $e^x$ where $G(e^x)$ has a discontinuity,  i.e., where $g_{\pi(n)}$ is a maximal prime gap, for the first $80$ maximal prime gaps}
\label{Gx}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=70mm]{G_ex_L.png}
\caption{\centering Plot of $\frac{\log G(e^x)}{\log x}-2$ (in black) and $\frac{\log G((e^x)^-)}{\log x}-2$ (in green) at each  value $n$ of $e^x$ where $G(e^x)$ has a discontinuity,  for the first $80$ maximal prime gaps $g_{\pi(n)}$, excluding $n = 2,3$}
\label{LGx}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=70mm]{G_ex_S.png}
\caption{\centering Plot of $\frac{G(e^x)}{x^2}$ (in black) and $\frac{G((e^x)^-)}{x^2}$  (in green) at each value $n$ of $e^x$ where $G(e^x)$ has a discontinuity,  for the first $80$ maximal prime gaps $g_{\pi(n)}$, excluding $n = 2,3$}
\label{SGx}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=70mm]{G_ex_M.png}
\caption{\centering Plot of $ \frac{G(e^x)}{x} = \widehat{g}_{\pi(n)}$ (in black) and $ \frac{G((e^x)^-)}{x}$ (in green) at each value $n$ of $e^x$ where $G(e^x)$ has a discontinuity,  for the first $80$ maximal prime gaps $g_{\pi(n)}$}
\label{MGx}
\end{figure}

Assuming that $\deg G = 0$,  the quantity $\dege_1 G$ is the lim sup of the points in black in Figure \ref{LGx}, while  $\underline{\dege}_1 G$ is the lim inf of the points in green in Figure \ref{LGx}, as $x \to \infty$.  Thus,  Maynard's conjecture holds if and only if all points, both black and green, tend to $0$ as $x \to \infty$.  In fact, by Proposition \ref{stepdeg}, one has $\dege G = \dege G|_X$ and $\underline{\dege} \, G = \underline{\dege} \, G(x^-)|_X$, where $X$ is the set of all points of discontinuity of $G$, i.e., where $X$ is the set of all primes $p_n$ for which $g_{n}$ is a maximal prime gap, so that $G(p_n) = g_{n}$ and $G(p_n^-) = \max_{k <n} g_k$ is the previous maximal prime gap, for each prime $p_n$ in $X$.
Thus,  one can restrict one's study of $G(x)$ to its points of discontinuity,  as we did in Figures \ref{Gx}--\ref{MGx}, at least for  the purpose of investigating  $\dege G$ and $\underline{\dege}\,  G$.   Recall that
$$\widetilde{G}(x) = \max_{p_{k+1} \leq x}  g_k = \max_{k < \pi(x)}  g_k, \quad \forall x \geq 0,$$
is a slight variant of $G(x)$ employed in place of $G(x)$ in some works, e.g.,  [96].
Note that
$$G(x^-) = \widetilde{G}(x), \quad \forall x \in X.$$   Thus, one has  $$\dege G = \dege \widetilde{G} \geq \dege \widetilde{G}|_X \geq \underline{\dege} \, \widetilde{G}|_X = \underline{\dege} \, G.$$
Since the function $G(x)$ is relatively ``stable,''  e.g., it is a nonincreasing step function,  in light of Maynard's conjecture, it is plausible that $G(x)$ has exact logexponential degree to all orders, which holds if and only if $\dege G = \underline{\dege}\, \widetilde{G}|_X$.   This holds, for example, if $G(x) \asymp r(x)\ (x \to \infty)$ for some Hardian function $r(x)$.

\section{Tables}

This section contains the tables discussed earlier in this chapter.

\begin{table}[h!]
\caption{\centering $f(x) = \li(x)-\pi(x)$ and $g(x) = \li(x)-\Ri(x)$ for $x = 10^n$ (https://oeis.org/A006880)}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline
$n$  & $\li(x)-\pi(x)$ & $\li(x)-\Ri(x)$ & $L_f(x)$ & $L_g(x)$ & $L_f[\frac{1}{2}](x)$ & $L_g[\frac{1}{2}](x)$ & $L_f[\frac{1}{2},\! \!-1](x)$ & $L_g[\frac{1}{2},\! \!-1](x)$ \\\hline \hline
1  &  2.165600 		&	1.601016    & 0.3356	&  0.2044	&  $-$0.4539	& $-$0.8161	&  $-$2.5095  & $-$0.8451  \\\hline
2  &   5.126142		&	4.464508  & 0.3549	&  0.3249	&  $-$0.4376 &  $-$0.5280	&  2.0286  & 1.7022 \\\hline
3  &  9.609658		&	9.250212   & 0.3276  &  0.3221	&  $-$0.6163	&  $-$0.6360 	&  1.1254   &1.0676  \\\hline
4  &  17.13722		&	19.20600  &	 0.3085 &  0.3209	&  $-$0.7944 	&  $-$0.7431	&  0.5722  & 0.7151 \\\hline
5  &  37.80900		&	42.37726  &	 0.3155 &  0.3254	&  $-$0.8692	&  $-$0.8225	&  0.3577  & 0.4853  \\\hline
6  &  129.5492		&	100.1497   &	 0.3521 &  0.3334	&  $-$0.7783	&  $-$0.8763	&  0.6030 & 0.3363  \\\hline
7  &  339.4050		&	250.9575  &	 0.3615 &  0.3428	&  $-$0.8028	&  $-$0.9114	& 0.5361   &  0.2408  \\\hline
8  &  754.3754		&	657.5081   &	 0.3597 &  0.3522	&  $-$0.8871	& $-$0.9342	& 0.3077   & 0.1792   \\\hline
9	 &	1700.957 &	1779.529 &	0.3590 &	0.3611 &	$-$0.9642 	& $-$0.9493 &	0.0051 &	0.0503 \\\hline
10	 &	3103.587 &	4931.280 &	0.3492 &	0.3693 &	$-$1.1071	& $-$0.9595 &	$-$0.2939 &	0.1111 \\\hline
11	&	11587.62 &	13905.99 &	0.3695 &	0.3767 &	$-$1.0231 &	$-$0.9667 &	$-$0.0636	& 0.0919 \\\hline
12	 &	38262.80 &	39738.58 &	0.3819 &	0.3833 &	$-$0.9832 &	$-$0.9718 &	0.0464 &	0.0779 \\\hline
13	 &	108971.05 &	114744.22 &	0.3875 &	0.3892 &	$-$0.9909 &	$-$0.9757 &	0.0254 &	0.0675 \\\hline
14	&	314889.95 &	334090.26 &	0.3927 &	0.3946 &	$-$0.9957 &	$-$0.9786 &	0.0120 &	0.0596 \\\hline
15	 &	1052618.58 &	979400.65 &	0.4015 &	0.3994 &	$-$0.9606 &	$-$0.9810 &	0.1103	& 0.0533 \\\hline
16	 &	3214631.79 &	2887579.61 &	0.4067 &	0.4038  &	$-$0.9531 &	$-$0.9828 &	0.1319 &	0.0482 \\\hline
17	 &	7956588.78 &	8554843.77 &	0.4059 &	0.4078	& $-$1.0041	& $-$0.9844 &	$-$0.0117 &	0.0441 \\\hline
18	 &	21949555.02 &	25450920.62 &	0.4079 &	0.4114	& $-$1.0254 &	$-$0.9857 &	$-$0.0719 &	0.0406  \\\hline
19		& 99877775.22 &	75993441.99 &	0.4210 &	0.4148 &	$-$0.9144 &	$-$0.9867 &	0.2433 &	0.0377 \\\hline
20	 &	222744643.55	 & 227636468.15 &	0.4174	& 0.4179 &	$-$0.9934 &	$-$0.9877 &	0.0190 &	0.0351  \\\hline
21	 &	597394254.33	& 683826458.60 &	0.4179 	& 0.4207 &	$-$1.0233 &	$-$0.9885 &	$-$0.0668 &	0.0329 \\\hline
22	 &	1932355208.15 &	2059487872.99 &	0.4221 &	0.4234 &	$-$1.0054 &	$-$0.9892 &	$-$0.0156 &	0.0310 \\\hline
23	 &	7250186215.78 &	6216886362.70 &	0.4287 &	0.4258  &	$-$0.9511  &	$-$0.9898 &	0.1408 &	0.0293  \\\hline
24		& 17146907278.15 &	18805896997.35	& 0.4264  &	0.4281 &	$-$1.0134  &	$-$0.9904  &	$-$0.0388 &	0.0277  \\\hline
25 &	55160980939.38	& 56995765653.55 &	0.4297 &	0.4302 	& $-$0.9990  &	$-$0.9909  &	0.0029  &	0.0263  \\\hline
26	 &  155891678120.79 &	173041013576.57 &	0.4305  &	0.4322 &	$-$1.0169 &	$-$0.9914 &	$-$0.0490 &	0.0251 \\\hline
27	&	508666658006.34 &	526202145940.76 &	0.4336 &	0.4341	& $-$1.0000 &	$-$0.9918 &	0.0000 &	0.0239 \\\hline
28  &  1427745660373.74  & 1602506180201.17   &  0.4341	& 	0.4359 & $-1.0199$	& $-0.9922$  &   $-0.0580$  & $0.0229$ \\\hline
29  &  4551193622464.34  & 4887005927157.18   &  0.4365	& 	0.4376 & $-1.0095$	& $-0.9925$  &   $-0.0277$  & $0.0219$ \\\hline
\end{tabular}\label{tab2}
\end{table}

\begin{table}[h!]
\caption{\centering Approximations of $f(x) = \li(x)-\pi(x)$ and $g(x) = \li(x)-\Ri(x)$ for $x = 10^n$ [235]}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline
$n$  & $\li(x)-\pi(x) \approx$   & $\li(x)-\Ri(x) \approx$ &  \ \ \ $L_f(x)$  & $L_g(x)$  & $L_{f}[\frac{1}{2}](x)$ &  $L_{g}[\frac{1}{2}](x)$  & $L_{f}[\frac{1}{2},\! \! -1]( x)$  & $L_{f}[\frac{1}{2},\! \! -1]( x)$ \\  \hline \hline

${26}$ & $1.5603 \cdot 10^{11}$  &  $1.7304\cdot 10^{11}$  & 0.4305 & 0.4322 & $-1.0167$ & $-$0.9914 &  $-0.0484$ &   0.0251 \\ \hline
${27}$ & $5.0811 \cdot 10^{11}$  &  $ 5.2620 \cdot 10^{11}$  & 0.4336 & 0.4341 & $-1.0003$ &$-$0.9918 & $-0.0008$ &  0.0239 \\ \hline
${28}$ &  $1.4271 \cdot 10^{12}$ &  $1.6025 \cdot 10^{12}$   & 0.4341  &  0.4359  &  $-$1.0200  &   $-$0.9922   &  $-$0.0584  & 0.0229  \\ \hline
${29}$ &  $4.5533  \cdot 10^{12}$ &  $4.8870 \cdot 10^{12}$  &  0.4365   &  0.4376  &  $-$1.0093  &   $-$0.9925   &  $-$0.0274   &   0.0219  \\\hline
${30}$ & $1.8779 \cdot 10^{13}$ &  $1.4922 \cdot 10^{13}$  &  0.4425  &  0.4391 &  $-$0.9386  &   $-$0.9928   &     0.1803   &   0.0210  \\\hline
${31}$ &  $3.9314 \cdot 10^{13}$ &  $4.5620 \cdot 10^{13}$  &    0.4385  & 0.4406   &  $-$1.0280  &   $-$0.9931   &   $-$0.0823  &  0.0202  \\ \hline
${32}$ & $9.0918 \cdot 10^{13}$  &   $1.3962 \cdot 10^{14}$  &  0.4362  & 0.4420   &  $-$1.0932   &   $-$0.9934   &  $-$0.2747  &   0.0194  \\\hline
${33}$ &  $3.9709 \cdot 10^{14}$ &   $4.2775 \cdot 10^{14}$  &  0.4424  &  0.4434   &  $-$1.0108   &   $-$0.9937   &  $-$0.0320   &   0.0187  \\\hline
${34}$ & $1.9605 \cdot 10^{15}$  &   $1.3118 \cdot 10^{15}$  &  0.4498   &  0.4446   &   $-$0.9017   &   $-$0.9939   &  0.2909   &   0.0181  \\\hline
${35}$ &  $4.8963 \cdot 10^{15}$ &    $4.0265 \cdot 10^{15}$ &  0.4483   &  0.4459   &  $-$0.9496   &   $-$0.9941   &     0.1497  &   0.0175  \\\hline
${36}$ & $7.4445 \cdot 10^{15}$  &   $ 1.2370 \cdot 10^{16}$ &  0.4409   &  0.4470  &   $-$1.1093  &   $-$0.9943   &   $-$0.3249    &   0.0169  \\\hline
${37}$ &  $2.2485 \cdot 10^{16}$ &   $3.8033 \cdot 10^{16}$  & 0.4419   &  0.4481   &  $-$1.1128   &   $-$0.9945   &   $-$0.3360  &   0.0163  \\\hline
${38}$ &  $1.1575 \cdot 10^{17}$ &    $1.1703 \cdot 10^{17}$  &  0.4490    &  0.4492   &  $-$0.9972   &   $-$0.9947  &   0.0085  &   0.0158  \\\hline
${39}$ &  $3.6121 \cdot 10^{17}$ &   $3.6036 \cdot 10^{17}$  &  0.4502   &  0.4502   &  $-$0.9943    &   $-$0.9949   &  0.0169  &   0.0153  \\\hline
${40}$ & $9.5267 \cdot 10^{17}$ &   $1.1104 \cdot 10^{18}$ &   0.4494   &  0.4511   &  $-$1.0291   &   $-$0.9950   &  $-$0.0866     &   0.0149  \\\hline
${41}$ & $ 2.8478 \cdot 10^{18}$ &   $3.4238 \cdot 10^{18}$   &  0.4501    &  0.4521   &  $-$1.0359   &   $-$0.9952   &  $-$0.1072  &   0.0145  \\\hline
${42}$ & $ 9.0106 \cdot 10^{18}$&    $ 1.0564 \cdot 10^{19}$  &  0.4513  &  0.4529   &   $-$1.0301   &   $-$0.9953   &   $-$0.0906   &   0.0141  \\\hline
${43}$ & $3.3434 \cdot 10^{19}$  &    $3.2612 \cdot 10^{19}$ &   0.4541  &  0.4538   &  $-$0.9900    &   $-$0.9955   &   0.0300  &   0.0137  \\\hline
${44}$ & $9.1921 \cdot 10^{19}$ &   $1.0073 \cdot 10^{20}$ &   0.4537   &  0.4546   &  $-$1.0154   &   $-$0.9956   &   $-$0.0465   &   0.0133  \\\hline
${45}$ & $2.2524 \cdot 10^{20}$ &    $3.1132 \cdot 10^{20}$  &  0.4523  &  0.4554   &  $-$1.0655   &   $-$0.9957   &  $-$0.1979  &   0.0130  \\\hline
${46}$ & $1.0325 \cdot 10^{21}$ &   $9.6266 \cdot 10^{20}$  &  0.4568   &  0.4562   &  $-$0.9808   &   $-$0.9958   &    0.0581    &   0.0126  \\\hline
${47}$ & $3.6107 \cdot 10^{21}$ &   $2.9782 \cdot 10^{21}$ &   0.4587  &  0.4569   &  $-$0.9548   &   $-$0.9959   &   0.1370  &  0.0123  \\\hline
${48}$ & $6.5485 \cdot 10^{21}$ &   $9.2178 \cdot 10^{21}$  &  $0.4545$   &  0.4576  &  $-$1.0687    &   $-$0.9960   &  $-$0.2088   &  0.0120  \\ \hline
${49}$ & $2.2397 \cdot 10^{22}$ &   $2.8543 \cdot 10^{22}$ &  $0.4561$    &  0.4583   &   $-$1.0475   &   $-$0.9961 &  $-$0.1444   &   0.0117  \\\hline
${50}$ & $8.2123 \cdot 10^{22}$ &    $8.8423 \cdot 10^{22}$  &  $0.4583$  &  0.4589   &  $-$1.0118   &   $-$0.9962 &   $-$0.0360   &  0.0115 \\ \hline
\end{tabular}\label{tab3}
\end{table}

\begin{table}[ht!]
\caption{\centering $f(x) = \li(x)-\pi(x)$ and $g(x) = \li(x)-\Ri(x)$ for $x = e^n$ (https://oeis.org/A040014)}
\tiny
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}\hline
$n$  & $\li(x)-\pi(x)$ & $\li(x)-\Ri(x)$ & $L_f(x)$ & $L_g(x)$ & $L_f[\frac{1}{2}](x)$ & $L_g[\frac{1}{2}](x)$ & $L_f[\frac{1}{2},\! \!-1](x)$ & $L_g[\frac{1}{2},\! \!-1](x)$ \\\hline \hline
1    & 0.895118      & 0.015939     & $-$0.1108 & $-$4.1390 & ---   & ---   & ---      & ---     \\ \hline
2   & 0.954234     & 1.263390      & -0.0234 & 0.1168  & $-$1.5103  & $-$1.1054 & 0.9650   & 0.1993 \\ \hline
3 & 1.933833     & 2.382997      & 0.2198 & 0.2895  & $-$0.7650 & $-$0.5749  & 2.7445   & 4.9653 \\ \hline
4   & 3.630874      & 3.608914      & 0.3224  & 0.3209  & $-$0.5125 & $-$0.5169 & 2.0689  & 2.0503 \\ \hline
5   & 6.185275       & 5.088573      &  0.3644  & 0.3254  & $-$0.4212 & $-$0.5424 & 1.9576  & 1.5475 \\ \hline
6  & 6.989762      & 6.984925      & 0.3241  & 0.3240  & $-$0.5891 & $-$0.5895 & 1.2624  & 1.2612 \\ \hline
7    & 8.504743     & 9.518112      & 0.3058  & 0.3219  & $-$0.6986 & $-$0.6407 & 0.8810 & 1.0501 \\ \hline
8   & 11.37990     & 13.00712     & 0.3040  & 0.3207   & $-$0.7541 & $-$0.6899 & 0.6984  & 0.8809 \\ \hline
9   & 18.87829    & 17.92737    & 0.3264  & 0.3207  & $-$0.7109 & $-$0.7344 & 0.8070  & 0.7413 \\ \hline
10 & 26.22898      & 24.99738     & 0.3267  & 0.3219  & $-$0.7527 & $-$0.7736  & 0.6828  & 0.6251 \\ \hline
11  & 23.40637     & 35.31241     & 0.2866  & 0.3240  & $-$0.9788  & $-$0.8073  & 0.0582  & 0.5284 \\ \hline
12  & 47.53267     & 50.55248    & 0.3218  & 0.3269 & $-$0.8606  & $-$0.8358 & 0.3805  & 0.4481 \\ \hline
13  & 69.68849     & 73.30739     & 0.3265  & 0.3304  & $-$0.8795 & $-$0.8598 & 0.3280  & 0.3818 \\ \hline
14  & 75.51363    & 107.58650   & 0.3089  & 0.3342  & $-$1.0139 & $-$0.8797 & $-$0.0377 & 0.3270 \\ \hline
15  & 100.8525   & 159.6199   & 0.3076  & 0.3382  & $-$1.0658  & $-$0.8963 & $-$0.1790 & 0.2819 \\ \hline
16  & 219.9987    & 239.1208    & 0.3371  & 0.3423 & $-$0.9401 & $-$0.9100 & 0.1630  & 0.2447 \\ \hline
17 & 404.8940     & 361.2780     & 0.3532   & 0.3464  & $-$0.8811  & $-$0.9213 & 0.3234  & 0.2140 \\ \hline
18  & 718.3306    & 549.9068    & 0.3654  & 0.3505  & $-$0.8383 & $-$0.9308 & 0.4403   & 0.1889 \\ \hline
19  & 561.2510   & 842.4401   & 0.3332  & 0.3545  & $-$1.0766  & $-$0.9386 & $-$0.2087  & 0.1674 \\ \hline
20  & 1090.664   & 1297.847   & 0.3497  & 0.3584  & $-$1.0032  & $-$0.9452 & $-$0.0089 & 0.1496 \\ \hline
21  & 2409.356  & 2009.211  & 0.3708  & 0.3622  & $-$0.8911 & $-$0.9507 & 0.2979   & 0.1347  \\ \hline
22 & 2774.300   & 3123.742   & 0.3604  & 0.3658  & $-$0.9938 & $-$0.9554 & 0.0170  & 0.1221 \\ \hline
23  & 2826.830   & 4874.654   & 0.3455  & 0.3692  & $-$1.1332 & $-$0.9594 & $-$0.3654  & 0.1114 \\ \hline
24 & 8547.849  & 7631.985  & 0.3772  & 0.3725  & $-$0.9272 & $-$0.9628 & 0.2002  & 0.1022 \\ \hline
25  & 14041.53       &  11983.74  & 0.3820  & 0.3757  & $-$0.9165 & $-$0.9658 & 0.2298  & 0.0942   \\ \hline
26  & 19730.90      &   18865.54 & 0.3804  & 0.3787  & $-$0.9546 & $-$0.9683   & 0.1253  & 0.0874 \\ \hline
27  & 26318.88       & 29767.99  & 0.3770  & 0.3815  & $-$1.0079 & $-$0.9706 & $-$0.0219 & 0.0814 \\ \hline
28  & 40913.31       & 47068.67 & 0.3793  & 0.3843  & $-$1.0146 & $-$0.9725 & $-$0.0404 & 0.0761 \\ \hline
29  & 81089.41       & 74564.04 & 0.3898  & 0.3869  & $-$0.9493 & $-$0.9743 & 0.1405 & 0.0714 \\ \hline
30   & 134842.27      & 118322.37 & 0.3937 & 0.3894  & $-$0.9374 & $-$0.9758 & 0.1740  & 0.0673 \\ \hline
31  & 184278.39      & 188052.43 & 0.3911 & 0.3918  & $-$0.9831 & $-$0.9772 & 0.0472  & 0.0636 \\ \hline
32  & 310474.79      & 299301.41 & 0.3952  & 0.3940  & $-$0.9678 & $-$0.9784 & 0.0898  & 0.0603  \\ \hline
33   & 462754.50      & 476986.81 & 0.3953 & 0.3962  & $-$0.9881 & $-$0.9795 & 0.0331  & 0.0573  \\ \hline
34   & 902506.27     & 761075.61 & 0.4033 & 0.3983   & $-$0.9321 & $-$0.9805 & 0.1899  & 0.0546 \\ \hline
35  & 1483541.60     & 1215721.63 & 0.4060  & 0.4003  & $-$0.9253 & $-$0.9814  & 0.2091 & 0.0522 \\ \hline
36  & 2551387.77     & 1943975.49 & 0.4098  & 0.4022  & $-$0.9063 & $-$0.9822 & 0.2630  & 0.0500 \\ \hline
37  & 4051058.40     & 3111478.72 & 0.4112  & 0.4041    & $-$0.9099 & $-$0.9830 & 0.2534  & 0.0479 \\ \hline
38   & 5477912.65     & 4984649.34 & 0.4083  & 0.4058  & $-$0.9577 & $-$0.9837 & 0.1191 & 0.0460\\ \hline
39  & 8227929.60     & 7992252.66    & 0.4083  & 0.4075  & $-$0.9764 & $-$0.9843 & 0.0667 & 0.0443 \\ \hline
40   & 12547993.58    & 12824726.72    & 0.4086 & 0.4092  & $-$0.9908 & $-$0.9849 & 0.0260  & 0.0427  \\ \hline
41  & 23321012.11   & 20594486.91    & 0.4138  & 0.4107  & $-$0.9519 & $-$0.9854 & 0.1360 & 0.0412 \\ \hline
42   & 38092380.59    & 33094760.86    & 0.4156  & 0.4123  & $-$0.9483 & $-$0.9859 & 0.1465  & 0.0399 \\ \hline
43  & 61485600.03    & 53217692.13    & 0.4171   & 0.4137  & $-$0.9480 & $-$0.9864  & 0.1476 & 0.0386 \\ \hline
44  & 108071721.51  & 85629989.66    & 0.4204  & 0.4151  & $-$0.9253 & $-$0.9869  & 0.2123 & 0.0374 \\ \hline
45   & 174585516.15   & 137865194.52   & 0.4217  & 0.4165  & $-$0.9252 & $-$0.9873 & 0.2129   & 0.0362 \\ \hline
46    & 204498265.00   & 222090120.72   & 0.4160  & 0.4178  & $-$1.0092  & $-$0.9877  & $-$0.0263 & 0.0352  \\ \hline
47   & 385623847.54  & 357962620.61 & 0.4206  & 0.4191   & $-$0.9687 & $-$0.9880 & 0.0894  & 0.0342 \\ \hline
48  & 651043126.55  & 577256516.55   & 0.4228  & 0.4203  & $-$0.9573 & $-$0.9884 & 0.1221 & 0.0332  \\ \hline
49   & 1164251352.07 & 931348981.39   & 0.4260   & 0.4215  & $-$0.9314 & $-$0.9887 & 0.1966 & 0.0324 \\ \hline
50    & 2005203489.62 & 1503345912.12  & 0.4284 & 0.4226    & $-$0.9154 & $-$0.9890 & 0.2427  & 0.0315 \\ \hline
51     & 2068331099.91  & 2427724401.02  & 0.4206  & 0.4237   & $-$1.0301  & $-$0.9893 & $-$0.0863  & 0.0307 \\ \hline
52   & 4005009209.52  & 3922162119.58  & 0.4252 & 0.4248  & $-$0.9843 & $-$0.9896 & 0.0452  & 0.0300 \\ \hline
53   & 6781992345.02 & 6339129875.41  & 0.4271  & 0.4258   & $-$0.9728 & $-$0.9898 & 0.0782  & 0.0292 \\ \hline
54    & 9918796068.45  & 10249542084.52 & 0.4263  & 0.4269  & $-$0.9983 & $-$0.9901 & 0.0048  & 0.0285 \\ \hline
55   & 20087997222.46 & 16578421917.45   & 0.4313  & 0.4278  & $-$0.9424 & $-$0.9903 & 0.1662  & 0.0279 \\ \hline
56 & 25498675636.03 & 26824982492.19 & 0.4279   & 0.4288  & $-$1.0032 & $-$0.9906 & $-$0.0092 & 0.0273 \\ \hline
57  & 47560462482.65 & 43419742391.88  & 0.4313  & 0.4297  & $-$0.9683 & $-$0.9908 & 0.0919  & 0.0267  \\ \hline
58   & 71057460397.51 & 70304144135.27 & 0.4308  & 0.4306  & $-$0.9884 & $-$0.9910 & 0.0337  & 0.0261 \\ \hline
59   & 94758718695.13 & 113871533611.46 & 0.4284  & 0.4315  & $-$1.0362 & $-$0.9912 & $-$0.1052 & 0.0255 \\ \hline
\end{tabular}\label{tab4}
\end{table}

\begin{table}[h!]
\caption{\centering Sup of $\Delta(x) = \left(\pi_0(x)- \Ri(x)  +\frac{1}{\log x} - \frac{1}{\pi} \arctan \frac{\pi}{\log x}\right) \frac{\log x}{\sqrt{x}}$ on $I_n = [10^n,10^{n+1}]$ [162]}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|r|} \hline
$n$  & $b_n$ & $\sup_{I_n} \Delta = \Delta(b_n^+)$ & $L_\Delta(b_n^+)$   & $L_\Delta[0](b_n^+)$ &  $L_\Delta[0, \! 0](b_n^+)$  \\\hline \hline
$0$ 		&	1			&1.0000000000		&	$-2.0000$	&	$0.0000$	& ---	\\ \hline
$1$ 		&	19			&0.5607597113		&	$-$0.1965	&	 $-$0.5357	&	$-$7.5237	\\ \hline
$2$ 		&	113			&0.7848341482		&	$-$0.05125	&	$-$0.1560	&	$-$0.5501	\\ \hline
$3$ 		&	1627			&0.6754517455		&	$-$0.05306	&	$-$0.1961	&	$-$0.5658	\\ \hline
$4$ 		&	24137		&0.7457431860		&	$-$0.02907	&	$-$0.1269	&	$-$0.3501	\\ \hline
$5$ 		&	355111		&0.7008073861		&	$-$0.02782	&	$-$0.1395	&	$-$0.3801	\\ \hline
$6$ 		&	3445943		&0.6809987397		&	$-$0.02552	&	$-$0.1417	&	$-$0.3851	\\ \hline
$7$ 		&	30909673	&0.7157292126		&	$-$0.01939	&	$-$0.1174	&	$-$0.3196	\\ \hline
$8$ 		&	110102617	&0.7878100197		&	$-$0.01288	&	$-$0.08171	&	$-$0.2227	\\ \hline
$9$ 		&	1110072773	&0.6833192028		&	$-$0.01828	&	$-$0.1254	&	$-$0.3429	\\ \hline
$10$ 	&	10016844407	&0.6386706267		&	$-$0.01947	&	$-$0.1429	&	$-$0.3922	\\ \hline
$11$ 	&	330957852107		&0.7533813432	&	$-$0.01068	&	$-$0.08639 	&	$-$0.2385		\\ \hline
$12$ 	&	2047388353069		&0.6808028098	&	$-$0.01356	&	$-$0.1150	&	$-$0.3185	\\ \hline
$13$ 	&	21105695997889		&0.6896466780	&	$-$0.01211	&	$-$0.1085	&	$-$0.3019	\\ \hline
$14$ 	&	117396942462053		&0.6789107425	&	$-$0.01195	&	$-$0.1113	&	$-$0.3107	\\ \hline
$15$ 	&	1047930291039067	&0.7042622330	&	$-$0.01014	&	$-$0.09894	&	$-$0.2771	\\ \hline
$16$ 	&	16452596773450399	&0.7144542025	&	$-$0.009005	&	$-$0.09288	&	$-$0.2614	\\ \hline
$17$ 	&	125546149553907317	&0.6572554320	&	$-$0.01066	&	$-$0.1143	&	$-0.3226$	\\ \hline
$18$ 	&	1325005986250807813		&0.7839983342	&	$-$0.005832	&	$-$0.06522	&	$-$0.1848	\\ \hline
$19$ 	&	11538454954199984761	&0.7574646817	&	$-$0.006329	&	$-$0.07346	&	$-$0.2088	\\ \hline
\end{tabular}\label{tab5}
\end{table}

\begin{table}[h!]
\caption{\centering Inf of $\Delta(x) = \left(\pi_0(x)- \Ri(x)  +\frac{1}{\log x} - \frac{1}{\pi} \arctan \frac{\pi}{\log x}\right) \frac{\log x}{\sqrt{x}}$ on $I_n = [10^n,10^{n+1}]$ [162]}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|r|} \hline
$n$  & $c_n$ & $\inf_{I_n} \Delta = \Delta(c_n^-)$ & $L_\Delta(c_n^-)$   & $L_\Delta[0](c_n^-)$ &  $L_\Delta[0, \! 0](c_n^-)$  \\\hline \hline
$0$ 		&	5			& $-$0.3952461978		&	$-$0.5768	&	$-$1.9506	&	1.2500	\\ \hline
$1$ 		&	11			& $-$0.5492343329		&	$-$0.2499	&	$-$0.6852	&	4.4719	\\ \hline
$2$ 		&	223			& $-$0.6051733874		&	$-$0.09289	&	$-$0.2976	&	$-$0.9596	\\ \hline
$3$ 		&	1423			& $-$0.7542604400		&	$-$0.03884	&	$-$0.1423	&	$-$0.4121	\\ \hline
$4$ 		&	19373		& $-$0.7278356754		&	$-$0.03218	&	$-$0.1387	&	$-$0.3835	\\ \hline
$5$ 		&	302831		& $-$0.6995719492		&	$-$0.02831	&	$-$0.1409	&	$-$0.3840	\\ \hline
$6$ 		&	1090697		& $-$0.6389660809		&	$-$0.03222	&	$-$0.1702	&	$-$0.4628	\\ \hline
$7$ 		&	36917099	& $-$0.7489165055		&	$-$0.01659	&	$-$0.1012	&	$-$0.2753	\\ \hline
$8$ 		&	516128797	& $-$0.6775687236		&	$-$0.01940	&	$-$0.1298	&	$-$0.3544	\\ \hline
$9$ 		&	7712599823	& $-$0.6889577485		&	$-$0.01637	&	$-$0.1192	&	$-$0.3270	\\ \hline
$10$ 	&	11467849447	& $-$0.7251609705		&	$-$0.01387	&	$-$0.1023	&	$-$0.2807	\\ \hline
$11$ 	&	110486344211	& $-$0.7355462679		&	$-$0.01208	&	$-$0.09492	&	$-$0.2616	\\ \hline
$12$ 	&	1635820377397		& $-$0.6892596608		&	$-$0.01323	&	$-$0.1115	&	$-$0.3088		\\ \hline
$13$ 	&	36219717668609		& $-$0.8360329846	&	$-$0.005736	&	$-$0.05204	&$-$0.1449		\\ \hline
$14$ 	&	348323506633621		& $-$0.6494959371	&	$-$0.01289	&	$-$0.1229	&$-$0.3436		\\ \hline
$15$ 	&	1212562524413153		& $-$0.7750460589	&	$-$0.007337	&	$-$0.07183	&	$-$0.2012	\\ \hline
$16$ 	&	18019655286689201	& $-$0.5710665212	&	$-$0.01497	&	$-$0.1547	&	$-$0.4353 \\ \hline
$17$ 	&	266175790131587543	& $-$0.7599282036	&	$-$0.006842	& $-$0.07436	&$-$0.2102		\\ \hline
$18$ 	&	5805523423155128399	& $-$0.6804259482	&	$-$0.008912	&	$-$0.1022 &	$-$0.2904	\\ \hline
$19$ 	&	55496658217283199013	& $-$0.8042730098		&	$-$0.004791	&	$-$0.05707	&	$-$0.1626	\\ \hline
\end{tabular}\label{tab6}
\end{table}

\begin{table}[h!]
\centering
\caption{\centering Sup of $|\Delta(x)|$  and upper bounds of $L_\Delta(x) $, $L_\Delta[0](x)$, and  $L_\Delta[0, \!0](x)$ on $I_n = [10^n,10^{n+1}]$ [162]}
\scriptsize
\bigskip
\begin{tabular}{|r|r|r|r|r|r|r|} \hline
$n$  & $a_n$ & $\sup_{I_n} |\Delta| = |\Delta(a_n)|$ & $L_\Delta(x) \leq$   & $L_\Delta[0](x) \leq$ &  $L_\Delta[0, \! 0](x) \leq$  \\\hline \hline
$0$ 		&	$1^+$			&1.0000000000		&	$-0.3906$	&	$\infty$	&	$\infty$	\\ \hline
$1$ 		&	$19^+$			&0.5607597113		&	$-$0.1618
&	 $-$0.4710
&	$\infty$\\ \hline
$2$ 		&	$113^+$			&0.7848341482		&	$-$0.05125
&	-$0.1559$
&	$-$0.5501
\\ \hline
$3$ 		&	$1423^-$		& 0.7542604400 	&	$-$0.03884
&	$-$0.1422
&	$-$0.4121
\\ \hline
$4$ 		&	$24137^+	$	&0.7457431860
&	$-$0.02907	&	$-$0.1269	&	$-$0.3500 \\ \hline
$5$ 		&	$355111^+$	&0.7008073861		&	$-$0.02267	&	$-$0.1294	&	$-$0.3504	\\ \hline
$6$ 		&	$3445943^+$		&0.6809987397		&	$-$0.02155	&	$-$0.1335		&	$-$0.3629	\\ \hline
$7$ 		&	$36917099^-$	& 0.7489165055		&	$-$0.01435	&	$-$0.09640	&	$-$0.2629	\\ \hline
$8$ 		&	$110102617^+$	&0.7878100197		&	$-$0.01144	&	$-$0.07853	&	$-$0.2146	\\ \hline
$9$ 		&	$7712599823^-$	& 0.6889577485		&	$-$0.01456	&	$-$0.1150	&	$-$0.3169	\\ \hline
$10$ 	&	$11467849447^-$	& 0.7251609705		&	$-$0.01260	&	$-$0.09924	&	$-$0.2734	\\ \hline
$11$ 	&	$330957852107^+$		&0.7533813432	&	$-$0.009744	&	$-$0.08408 	&	$-$0.2331		\\ \hline
$12$ 	&	$1635820377397^-$		& 0.6892596608		&	$-$0.01219	&	$-$0.1088	&	$-$0.3027		\\ \hline
$13$ 	&	$36219717668609^-$		& 0.8360329846	&	$-$0.005308	&	$-$0.05091	&$-$0.1423		\\ \hline
$14$ 	&	$117396942462053^+$		&0.6789107425	&	$-$0.01115	&	$-$0.1091	&	$-$0.3058	\\ \hline
$15$ 	&	$1212562524413153	^-$	& 0.7750460589	&	$-$0.006876	&	$-$0.07054	&	$-$0.1984	\\ \hline
$16$ 	&	$16452596773450399^+$	&0.7144542025	&	$-$0.008468	&	$-$0.09134	&	$-$0.2579	\\ \hline
$17$ 	&	$266175790131587543^-$	& 0.7599282036	&	$-$0.006452	& $-$0.07321	&$-$0.2076		\\ \hline
$18$ 	&	$1325005986250807813^+$		&0.7839983342	&	$-$0.005522	&	$-$0.06428	&	$-$0.1828	\\ \hline
$19$ 	&	$55496658217283199013^-$	&  0.8042730098		&	$-$0.004542	&	$-$0.05629	&	$-$0.1609	\\ \hline
\end{tabular}\label{tab7}
\end{table}

\begin{table}[h!]
\caption{\centering Prominent peak values of $\sum_{k = 1}^N \frac{2 \cos(\gamma_k \log x - \operatorname{Arg} \rho_k)}{|\rho_k|} \approx V(x) = \frac{ \Ri(x)-\pi(x) }{\li(x)-\Ri(x)}$  for $10^{20} <  x < 10^{10^{13}}$  \cite[Table 3]{stoll}}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|r|} \hline
$x$ & $V(x) \approx $ & $L_V(x)\approx$   & $L_V[0](x) \approx$ &  $L_V[0,\! 0](x)\approx$ &  $L_V[0,\! 0,\! 0](x)\approx $  \\\hline \hline
$2.08963594805312 \cdot 10^{31}$	&	0.85111265	&	$-$0.00224		&	 $-$0.0377	& 	$-$0.111	&  $-$0.431   \\ \hline
$4.489584432 \cdot 10^{41}$			&	$-$0.92040123	&	$-$0.000865	&	$-$0.0182	& 	$-$0.0546	&    $-$0.199  \\ \hline
$5.38952511422773 \cdot 10^{84}$		&	$-$0.93251498	&	$-$0.000358	&	$-$0.0132	& 	$-$0.0420	&   $-$0.137  \\ \hline
$1.25855876915892 \cdot 10^{179}$	&	$-$0.95469600	&	$-$0.000112		&	 $-$0.00770	& 	$-$0.0258	&  $-$0.0792   \\ \hline
$1.33793033022140 \cdot 10^{190}$	&	$-$0.96875484	&	$-$0.0000725	&	$-$0.00522	& 	$-$0.0176	&    $-$0.0537  \\ \hline
$1.9098756608800 \cdot 10^{215}$	&	0.99896402	&	$-2.091\cdot10^{-6}$	& $-$0.000167		& 	 $-$0.000568	&   $-$0.00172   \\ \hline
$1.397162914 \cdot 10^{316}$			&	$-$1.000	&	0	&	0	& 	0	&    0 	 \\ \hline
$1.397166708 \cdot 10^{316}$			&	$-$1.002	&	$2.74\cdot 10^{-6}$		&	 0.000303	& 	0.00106	&    0.00315 \\ \hline
$1.398215 \cdot 10^{316}$			&	$-$1.005	&	$6.85\cdot 10^{-6}$	&	 0.000757	& 		0.00265 &   0.00786  \\ \hline
$1.39822803460925 \cdot 10^{316}$	&	$-$1.01770266	&	0.0000241		&	0.0026627	& 	0.00931	&   0.0277  \\ \hline
$6.5769904020884 \cdot 10^{370}$	&	$-$1.04965574	&	0.0000568	&	0.00718	& 	0.0254	&    0.0746  \\ \hline
$2.56771067186843 \cdot 10^{807}$	&	1.04307053	&	0.0000227	&	0.00560	& 	0.0209	&    0.0600 \\ \hline
$1.592776 \cdot 10^{1165}$			&	$-$1.042	&	0.0000153	&	0.00521	& 	0.0199	&	0.0567     \\ \hline
$2.76934778827004 \cdot 10^{4943}$		&	$-$1.11199752	&	$9.33 \cdot 10^{-6}$	&	0.0114	& 	0.0475	&  0.132   \\ \hline
$1.303324322225356 \cdot 10^{651157}$	&	$-$1.26421901	&	$1.56 \cdot 10^{-7}$	&	0.0165	& 	0.0883		&  0.240   \\ \hline
$1.1952052716413331 \cdot 10^{1748085}$	&	$-$1.28300855	&	$6.19 \cdot 10^{-8}$	&	0.0164	& 		0.0916	&   0.249  \\ \hline
$1.382217859037819 \cdot 10^{6359808}$	&	$-$1.2851	&	$1.71 \cdot 10^{-8}$		&	0.0152	& 	0.0895	&   0.243  \\ \hline
$4.63188213923253 \cdot 10^{25462014}$	&	1.30691350	&	$4.57 \cdot 10^{-9}$	&	0.0150	& 	0.0928	&   0.253  \\ \hline
$6.5722 \cdot 10^{30802655}$			&	$-$1.332	&	$4.04 \cdot 10^{-9}$	&	0.0159	& 	0.0990	&  0.270   \\ \hline
$1.49741610590218 \cdot 10^{140451912}$	&	1.34051829	&	$9.06 \cdot 10^{-10}$	&	0.0150	& 		0.0985  &  0.269   \\ \hline
$1.14937434741219 \cdot 10^{263854435}$	&	1.36190288	&	$5.08 \cdot 10^{-10}$	&	0.0153	& 	0.103	&   0.281  \\ \hline
$3.3380935222734 \cdot 10^{1048348162}$	&	$-$1.36843786	&	$1.30 \cdot 10^{-10}$	&	0.0145	& 		0.102	&  0.279   \\ \hline
$5.60458099779389 \cdot 10^{1715984906}$		&	1.41803522	&	$8.84 \cdot 10^{-11}$	&	0.0158	& 	0.113	&    0.309 \\ \hline
$4.66560656850816 \cdot 10^{3926214944}$	&	1.45919474	&	$4.18 \cdot 10^{-11}$		&	0.0165	& 	0.121	&   0.331  \\ \hline
$5.51920707045967 \cdot 10^{16251858572}$	&	$-$1.50218678	&	$1.09 \cdot 10^{-11}$	&	0.0167	& 	0.127	&  0.351   \\ \hline
$7.48642968663951 \cdot 10^{1741296446971}$	&	1.51320794	&	$1.03 \cdot 10^{-13}$	&	0.0142	& 	0.123	&   0.341  \\ \hline
$1.26735548862278 \cdot 10^{2176190682786}$	&	$-$1.51154647	&	$8.24 \cdot 10^{-14}$	&	0.0141	& 	0.122	&    0.340 \\ \hline
$4.9055169754782 \cdot 10^{3648403832739}$	&	$-$1.55246992	&	$5.24 \cdot 10^{-14}$	&	0.0148	& 	 0.130	&   0.360  \\ \hline
$2.9778893050643 \cdot 10^{4026762218841}$	&	$-$1.51359970	&	$4.47 \cdot 10^{-14}$	&	0.0139 & 	0.122	&    0.339 \\ \hline
$2.30170547097451 \cdot 10^{7424180200481}$	&	$-$1.51177469	&	$2.42 \cdot 10^{-14}$	&	0.0136& 	0.121	&   0.336  \\ \hline
\end{tabular}\label{tab8}
\end{table}

\begin{table}[h!]
\caption{\centering Some prominent peak values of $M(n)$ [130] [164]}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|} \hline
$n$  & $M(n)$  & $\frac{M(n)}{\sqrt{n}}$  & \ \  $L_M(n)$  &  $L_{M}[\frac{1}{2}](n)$ & $L_{M}[\frac{1}{2},\!  0](n)$  \\  \hline \hline
179919749		&	$-$6226			&	$-$0.4642			&	0.4596		&	$-$0.2606		&  $-$0.7106	\\ \hline
6631245058		&	$-$31206	&	$-$0.3832	&	0.4576	&	$-$0.3076	     &  $-$0.8433   \\ \hline
7766842813		&	50286		&	{\bf 0.5706}	&	0.4754	&	{\bf $-$0.1795}	     &  {\bf $-$0.4923}   \\ \hline
15578669387		&	$-$51116		&	$-$0.4029	&	0.4620 	&	$-$0.2829	     &    $-$0.7768 \\ \hline
19890188718		&	60442		&	0.4286	&	0.4643	&	$-$0.2676	     &   $-$0.7352  \\ \hline
22867694771		&	$-$62880	&	$-$0.4158	&	0.4632	&	$-$0.2767 	     &    $-$0.7602 \\ \hline
38066335279		&	$-$81220	&	$-$0.4163	&	0.4640	&	$-$0.2745 	     &    $-$0.7549 \\ \hline
48638777062		&	76946		&	0.3489	&	0.4572	&	$-$0.3287	     &   $-$0.9045  \\ \hline
56808201767		&	$-$87995	&	$-$0.3692	&	0.4597	&	$-$0.3105	     &    $-$0.8545 \\ \hline
101246135617		&	$-$129332	&	$-$0.4065	&	0.4645	&	$-$0.2785	     &  $-$0.7673    \\ \hline
108924543546	&	170358		&	{\bf 0.5162}	&	0.4740	&	{\bf $-$0.2044}	     &  {\bf  $-$0.5632}  \\ \hline
148491117087  	&	$-$131461	&	$-$0.3412	&	0.4582 	&	$-$0.3312	     &   $-$0.9130  \\ \hline
217309283735	&	$-$190936	&	$-$0.4096 	&	0.4658	&	$-$0.2736	     &    $-$0.7549 \\ \hline
297193839495	&	207478		&	0.3806 	&	0.4634	&	$-$0.2951	     &     $-$0.8145	\\ \hline
330508686218	&	$-$294816	&	{\bf $-$0.5128}	&	 0.4748	&	{\bf $-$0.2037}	     &    {\bf $-$0.5625} \\ \hline
402027514338	&	271498		&	0.4282	&	0.4683	&	$-$0.2582	     &    $-$0.7131 \\ \hline
661066575037	&	331302		&	0.4075	&	0.4670	&	$-$0.2717	     &    $-$0.7512 \\ \hline
1440355022306	&	$-$368527	&	$-$0.3071	&	0.4578	&	$-$0.3543	     &   $-$0.9810  \\ \hline
1653435193541	&	546666		&	0.4251	&	0.4696	&	$-$0.2563	     &   $-$0.7098  \\ \hline
2087416003490	&	$-$625681	&	$-$0.4331	&	0.4705	&	$-$0.2502	     &    $-$0.6930 \\ \hline
2343412610499	&	594442		&	0.3883	&	0.4668	&	$-$0.2824	     &  $-$0.7826   \\ \hline
3270926424607	&	$-$635558	&	$-$0.3514	&	0.4637	&	$-$0.3112	     &    $-$0.8627 \\ \hline
4098484181477	&	780932		&	0.3857	&	0.4672	&	$-$0.2828	     &    $-$0.7843 \\ \hline
5191164528277	&	$-$668864	&	$-$0.2936	&	0.4581	&	$-$0.3630	     &  $-$1.00716   \\ \hline
5197159385733	&	$-$689688	&	$-$0.3025	&	0.4592	&	$-$0.3540	     &   $-$0.9824  \\ \hline
10236053505745	&	1451233		&	0.4536	&	0.4736	&	$-$0.2325	     &   $-$0.6460  \\ \hline
21035055623987	&	$-$1740201	&	$-$0.3794	&	0.4684	&	$-$0.2831	     &   $-$0.7875  \\ \hline
21036453134939	&	$-$1745524	&	0.3806	&	0.4685	&	$-$0.2822	     &     $-$0.7850  \\ \hline
23431878209318	&	1903157		&	0.3932	&	0.4697	&	$-$0.2724	     &    $-$0.7579  \\ \hline
30501639884098	&	$-$1930205	&	$-$0.3495	&	0.4661	&	$-$0.3060	     &    $-$0.8518  \\ \hline
36161703948239	&	2727852		&	0.4536	&	0.4747	&	$-$0.2297	     &    $-$0.6397  \\ \hline
36213976311781	&	2783777		&	0.4626 &		 0.4753	&	$-$0.2240	     &   $-$0.6238  \\ \hline
71578936427177	&	$-$4440015	&	{\bf $-$0.5248}	&	{\bf 0.4798}	& {\bf $-$0.1862}	     &   {\bf $-$0.5191}  \\ \hline
146734769129449	&	3733097		&	0.3082	&	0.4639	&	$-$0.3378	     &    $-$0.9428 \\ \hline
175688234263439	&	$-$5684793	&	$-$0.4289	&	0.4742	&	$-$0.2425	     &    $-$0.6772  \\ \hline
212132789199869	&	5491769		&	0.3771	&	0.4639	&	$-$0.3378	     &     $-$0.7792 \\ \hline
212137538048059	&	5505045		&	0.3780	&	0.4705	&	$-$0.2783	     &   $-$0.7773  \\ \hline
304648719069787	&	$-$5757490	&	$-$0.3299	&	0.4667	&    $-$0.3162	     &   $-$0.8839  \\ \hline
351246529829131	&	9699950		&	{\bf 0.5176}	&	{\bf 0.4803}	&	{\bf $-$0.1876}	     &   {\bf $-$0.5244}  \\ \hline
1050365365851491&	$-$13728339	&	$-$0.4236	&	 0.4752	&	$-$0.2424	     &   $-$0.6790 \\ \hline
1211876202620741	&	16390637	&	0.4708	&	{\bf 0.4783}	&	$-$0.2123	     &    $-$0.5949 \\ \hline
2458719908828794	&	$-$20362905	&	$-$0.4107	&	0.4749	&	$-$0.2494	     &    $-$0.6997 \\ \hline
3295555617962269	&	18781262		&	0.3272	&	0.4687	&	$-$0.3124	     &    $-$0.8768  \\ \hline
3664310064219561	&	$-$23089949	&	$-$0.3814	&	0.4731	&	$-$0.2693	     &    $-$0.7559 \\ \hline
4892214197703689	&	24133331		&	0.3450	&	0.4705	&	$-$0.2967	     &   $-$0.8331 \\ \hline
6287915599821430	&	$-$35629003	&	$-$0.4493	&	{\bf 0.4780}	&	$-$0.2226	     &  $-$0.6254   \\ \hline
7332940231978758	&	40371499	&	0.4715	&	{\bf 0.4794}	&	$-$0.2090	     &    $-$0.5873 \\ \hline
11609864264058592345 	&	$-$1995900927  & {\bf $-$0.5857}  &	{\bf 0.4878}	&	{\bf$- $0.1414}	& {\bf $-$0.4021}	 \\ \hline
\end{tabular}\label{tab9}
\end{table}

\begin{table}[h!]
\caption{\centering Supremum of $\frac{|M(x)|}{\sqrt{x}}$, $L_M(x)$,  $L_{M}[\frac{1}{2}](x)$, and $L_{M}[\frac{1}{2},\!  0](x)$ on intervals  $I$}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|} \hline
$I$    &   $\sup_{I} \!\! \frac{|M(x)|}{\sqrt{x} }= \frac{|M(a_I)|}{\sqrt{a_I}} $& \ \   $\sup_I \!L_M(x)$  &  $\sup_I \!L_{M}[\frac{1}{2}](x)$ & $\sup_I \!L_{M}[\frac{1}{2},\!  0](x)  $  \\  \hline \hline
$[10^0,10^1]$ 	&	$1.00000000\ldots$ \hfill at $1$	& $0.430676$ \hfill at $5$	& 	$
\infty$ at $e^-$ & ---	\\ \hline
$[10^1,10^2]$ 			&	$0.83205029\ldots$ \hfill at 13	&	$0.428317\ldots$ \hfill at 13	&	$-0.195195\ldots$	\hfill at 13 & 	$\infty$ \hfill at $(e^e)^-$ \\ \hline
$[10^2,10^3]$ 		&	$0.56710496\ldots$	\hfill at 199 &	$0.392843\ldots$ \hfill at 199 &	$-0.340372\ldots$	\hfill at 199 & 	$-1.110672\ldots$ \hfill at 199 \\ \hline
$[10^3,10^4]$ &	$0.47220269\ldots$	 \hfill at 2803 & $0.408988\ldots$ \hfill at 9861	&	$-0.362185\ldots$ \hfill at 2803 & $-1.030161\ldots$\hfill at 2803	\\ \hline
$[10^4,10^5]$ 		&	$0.46297703\ldots$ \hfill at 24185	&	 $0.429956\ldots$ \hfill at 59577	&		$-0.321227\ldots$ \hfill at 59577&  	$-0.880743\ldots$ \hfill at 59577\\ \hline
$[10^5,10^6]$ 		&	$0.43777620\ldots$ \hfill at 300551	&	 $0.434510\ldots$ \hfill at 300551	&		$-0.325888\ldots$ \hfill at 300551 &  	$-0.888129\ldots$ \hfill at 300551\\ \hline
$[10^6,10^7]$ 		&	$0.41825475\ldots$ \hfill at 1066854	&	 $0.444134\ldots$ \hfill at 6481601	&		$-0.318314\ldots$ \hfill at 6481601 &  	$-0.865335\ldots$ \hfill at 6481601\\ \hline
$[10^7,10^8]$ 		&	$0.46272869\ldots$ \hfill at 30919091	&	 $0.455318\ldots$ \hfill at 30919091	&		$-0.270616\ldots$ \hfill at 30919091 &  	$-0.736381\ldots$ \hfill at 30919091\\ \hline
$[33,10^{16}]$		&	$0.57059088\ldots$ \hfill at 7766842813	&	$\leq 0.490592$ \hfill \	&	$\leq-0.142501$ \hfill  \ & $\leq-0.405043$\hfill \	\\ \hline
\end{tabular}\label{tab10}
\end{table}

\begin{table}[h!]
\caption{\centering Increasingly large values of $q_{10^6}(x) \approx M(x)/\sqrt{x}$ (approximate IL$q$ values) \cite[Tables 2 and 3]{kuz}}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|} \hline
$\log x$  & $q_{10^6}(x)$  & $L_{q_{10^6}}(x)$  &  $L_{q_{10^6}}[0](x)$ & $L_{q_{10^6}}[0,  0](x)$  & $L_{q_{10^6}}[0, \! 0,\!0](x)$ \\  \hline \hline
43.898387   & $-0.58478$  & $-0.0122$  & $-0.142$   &   $-0.403$ &  $-$1.880  \\  \hline
97.526523 & 0.61863  &  $-0.00492$ & $-0.105$   & $-0.316$  &  $-$1.144  \\  \hline
140.373835  & $-0.58940$  &  $-0.00377$   &  $-0.107$  &  $-0.331$  &  $-$1.127  \\  \hline
853.851589  & $-0.67715$  & $-0.000455$   & $-0.0578$   &  $-0.204$  &    $-$0.603 \\  \hline
984.282019 & 0.62512 & $-0.000477$     &  $-0.0682$  &  $-0.243$ &   $-$0.714 \\  \hline
1625.698493 & 0.62687 &  $-0.000287$  &   $-0.0632$  & $-0.233$  &  $-$0.673  \\  \hline
1957.803133 & 0.62849 & $-0.000239$ &  $-0.0616$  & $-0.230$  &  $-$0.658  \\  \hline
2709.485814  & 0.65467 & $-0.000156$ &  $-0.0536$  & $-0.205$  &  $-$0.583   \\  \hline
2794.384965 & 0.65955 & $-0.000149$  &  $-0.0524$  &  $-0.201$ &  $-$0.572  \\  \hline
3005.762748  & $-0.68878$  & $-0.000124$   &   $-0.0466$ & $-0.179$   &  $-$0.509  \\  \hline
12277.362671 & 0.79344 &  $-0.0000188$ &  $-0.0246$  &  $-0.103$ &   $-$0.287  \\  \hline
102494.024866 &  $-0.69580$   &  $-3.54 \cdot 10^{-6}$  &  $-0.0314$  &  $-0.148$  &  $-$0.406  \\  \hline
150020.464414  &  $-0.70773$  &  $-2.30 \cdot 10^{-6}$  & $-0.0290$   &  $-0.139$  &  $-$0.381  \\  \hline
178259.151801   &  $-0.71541$  &  $-1.88 \cdot 10^{-6}$  &  $-0.0277$  &  $-0.134$  &   $-$0.367 \\  \hline
203205.659988  &  $-0.74083$  &  $-1.48 \cdot 10^{-6}$  &  $-0.0245$  &  $-0.120$  &   $-$0.327 \\  \hline
860440.495719 &  $-0.75254$  &    $-3.30 \cdot 10^{-7}$  &  $-0.0208$  & $-0.109$   &  $-$0.296  \\  \hline
1365643.292004  &  $-0.75927$  &  $-2.02 \cdot 10^{-7}$  & $-0.0195$   &   $-0.104$ &  $-$0.283  \\  \hline
2765781.628095  &  $-0.76041$  &  $-9.90 \cdot 10^{-8}$  &  $-0.0185$ &  $-0.102$  &   $-$0.276 \\  \hline
7078384.260482  &  $-0.76879$  &  $-3.71 \cdot 10^{-8}$  &  $-0.0167$  &   $-0.0953$ &  $-$0.259  \\  \hline
13670267.747472 &   $-0.78505$   & $-1.77 \cdot 10^{-8}$   &  $-0.0147$  & $-0.0865$   &  $-$0.235  \\  \hline
19371574.223934  &  $-0.78747$  & $-1.23 \cdot 10^{-8}$   &   $-0.0142$  & $-0.0847$   & $-$0.230   \\  \hline
57334128.09084  &   $-0.80765$  & $-3.73 \cdot 10^{-9}$   & $-0.0120$   &  $-0.0741$   &  $-$0.202  \\  \hline
86458087.131684 & 0.80443 & $-2.52 \cdot 10^{-9}$   &   $-0.0119$ &   $-0.0749$ & $-$0.204  \\  \hline
167211796.14902   &   $-0.82488$  &  $-1.15 \cdot 10^{-9}$  &  $-0.0102$  & $-0.0655$    &    $-$0.178	\\  \hline
249548703.533702 & 0.81472  & $-8.21 \cdot 10^{-10}$ & $-0.0106$  & $-0.0692$  &    $-$0.189 \\  \hline

405441986.398094  &   $-0.84497$  & $-4.15 \cdot 10^{-10}$    &  $-0.00850$  &   $-0.0564$ &  $-$0.155  \\  \hline
1467573228.501077  & 0.81702& $-1.38 \cdot 10^{-10}$ &  $-0.00957$  & $-0.0663$  &   $-$0.181 \\  \hline
2500922487.505913  & 0.82884   &   $-7.51 \cdot 10^{-11}$ &  $-0.00868$  &   $-0.0611$ &   $-$0.167 \\  \hline
3847517705.646364 & 0.83682 & $-4.63 \cdot 10^{-11}$ &  $-0.00807$  &  $-0.0576$ &   $-$0.158 \\  \hline
4016980126.87193  &  $-0.85146$  &  $-4.00 \cdot 10^{-11}$  &  $-0.00727$  &   $-0.0519$ &    $-$0.142 \\  \hline

10407545552.85608 & 0.842485 & $-1.65 \cdot 10^{-11}$ & $-0.00743$  &  $-0.0546$ &   $-$0.150 \\  \hline
21334043144.02927	& 0.86622	& $-6.73 \cdot 10^{-12}$	&$-$0.00604	& $-$0.0453	&   $-$0.125\\  \hline
86339883457.03526	&	$-$0.87296  &	$-1.57 \cdot 10^{-12}$	& $-$0.00540	&	$-$0.0421  &   $-$0.116 \\  \hline
187114096628.77484	& 0.88636	&$-6.45 \cdot 10^{-13}$	&$-$0.00465	&  $-$0.0370	& $-$0.102   \\  \hline
264421251554.46918	&	$-$0.89290  &	$-4.28 \cdot 10^{-13}$	&	$-$0.00431 &	$-$0.0346  &  $-$0.0956  \\  \hline
1278282683343.76520	&	$-$0.89907  &	$-8.32 \cdot 10^{-14}$	& $-$0.00382	& $-$0.0320	&  $-$0.0885  \\  \hline
1354137181464.62097 	& 0.90578	&	$-7.31 \cdot 10^{-14}$ & $-$0.00354 &	$-$0.0297 &  $-$0.0858  \\  \hline
3680547202477.03623	&	$-$0.91392  &	$-2.45 \cdot 10^{-14}$	&	$-$0.00311 & $-$0.0267	&  $-$0.0742  \\  \hline
6984497047106.74600	& 0.90744 &	 $-1.39 \cdot 10^{-14}$ &	$-$0.00328  &	$-$0.0287 &   $-$0.0796 \\  \hline
18747209824980.73961	&	$-$0.91405 &	$-4.79 \cdot 10^{-15}$	& $-$0.00294	& $-$0.0263	& $-$0.0731   \\  \hline
55714219637174.49540  &	$-$0.92025  &	$-1.49 \cdot 10^{-15}$	&	$-$0.00263 & $-$0.0241	&    $-$0.0670 \\  \hline

84594507546024.46719	& 0.92208	&$-9.59 \cdot 10^{-16}$ & $-$0.00253	& $-$0.0234 &  $-$0.0652  \\  \hline
117239588213313.90075	& 0.94102	& $-5.19 \cdot 10^{-16}$	&	$-$0.00188 &	$-$0.0175 &    $-$0.0488 \\  \hline
117892199597999.02070	&   $-$0.92078 	&	$-7.00 \cdot 10^{-16}$   &  $-$0.00255	&  $-$0.0237	&   $-$0.0662 \\  \hline
143697547951999.01914	&	$-$0.93038 &		$-5.02 \cdot 10^{-16}$	&	$-$0.00221  &	$-$0.0207  &   $-$0.0578 \\  \hline
258592103887306.71643	&	$-$0.94097 &		$-2.35 \cdot 10^{-16}$	& 	$-$0.00183  &	$-$0.0174  &   $-$0.0485 \\  \hline
505863698785929.24318 	&	$-$0.95652  &		$-8.79 \cdot 10^{-17}$	&	$-$0.00131  & $-$0.0126	 &   $-$0.0353  \\  \hline
\end{tabular}\label{tab11}
\end{table}

\begin{table}[h!]
\caption{\centering Increasingly large values of $|q_{10^6}(x)| \approx |M(x)|/\sqrt{x}$ (approximate IL$|q|$ values) \cite[Tables 2 and 3]{kuz}}
\scriptsize
\begin{tabular}{|r|r|r|r|r|r|r|} \hline
$\log x$  & $r_k = \lfloor\log_{10}x \rfloor$ (so $|q_{10^6}(x)| $  & $q_{10^6}(x)$  & $L_{q_{10^6}}(x)$  &  $L_{q_{10^6}}[0](x)$ & $L_{q_{10^6}}[0,  0](x)$  & $L_{q_{10^6}}[0, \! 0,\!0](x)$ \\
& $\approx \sup_{ [10^2,10^{r_{k+1}}]}|q_{10^6}(t)|)$  &   &  &   &   &  \\  \hline \hline
43.898387       &  19   &    $-0.58478$  & $-0.0122$  & $-0.142$   &   $-0.403$ &  $-$1.880  \\  \hline
97.526523     &   42  &    0.61863  &  $-0.00492$ & $-0.105$   & $-0.316$  &  $-$1.144  \\  \hline
853.851589      &  370   &    $-0.67715$  & $-0.000455$   & $-0.0578$   &  $-0.204$  &    $-$0.603 \\  \hline
3005.762748      &  1305   &    $-0.68878$  & $-0.000124$   &   $-0.0466$ & $-0.179$   &  $-$0.509  \\  \hline
12277.362671     &   5331  &    0.79344 &  $-0.0000188$ &  $-0.0246$  &  $-0.103$ &   $-$0.287  \\  \hline
57334128.09084      &  24899895   &      $-0.80765$  & $-3.73 \cdot 10^{-9}$   & $-0.0120$   &  $-0.0741$   &  $-$0.202  \\  \hline
167211796.14902       &   72619160  &      $-0.82488$  &  $-1.15 \cdot 10^{-9}$  &  $-0.0102$  & $-0.0655$    &    $-$0.178	\\  \hline
405441986.398094      &
176081217   &      $-0.84497$  & $-4.15 \cdot 10^{-10}$    &  $-0.00850$  &   $-0.0564$ &  $-$0.155  \\  \hline
4016980126.87193      &   1744552303  &     $-0.85146$  &  $-4.00 \cdot 10^{-11}$  &  $-0.00727$  &   $-0.0519$ &    $-$0.142 \\  \hline
21334043144.02927	     &  9265257214   &     0.86622	& $-6.73 \cdot 10^{-12}$	&$-$0.00604	& $-$0.0453	&   $-$0.125\\  \hline
86339883457.03526	       &  37496934953   &    	$-$0.87296  &	$-1.57 \cdot 10^{-12}$	& $-$0.00540	&	$-$0.0421  &   $-$0.116 \\  \hline
187114096628.77484	     & 81262619652    &     0.88636	&$-6.45 \cdot 10^{-13}$	&$-$0.00465	&  $-$0.0370	& $-$0.102   \\  \hline
264421251554.46918	     &  114836690448   &    	$-$0.89290  &	$-4.28 \cdot 10^{-13}$	&	$-$0.00431 &	$-$0.0346  &  $-$0.0956  \\  \hline
1278282683343.76520	     & 555151115688    &    	$-$0.89907  &	$-8.32 \cdot 10^{-14}$	& $-$0.00382	& $-$0.0320	&  $-$0.0885  \\  \hline
1354137181464.62097 	     &   588094305650  &     0.90578	&	$-7.31 \cdot 10^{-14}$ & $-$0.00354 &	$-$0.0297 &  $-$0.0858  \\  \hline
3680547202477.03623	     &   1598441340420  &    	$-$0.91392  &	$-2.45 \cdot 10^{-14}$	&	$-$0.00311 & $-$0.0267	&  $-$0.0742  \\  \hline
18747209824980.73961	     &   8141809778071  &    	$-$0.91405 &	$-4.79 \cdot 10^{-15}$	& $-$0.00294	& $-$0.0263	& $-$0.0731   \\  \hline
55714219637174.49540       & 24196378151970    &    	$-$0.92025  &	$-1.49 \cdot 10^{-15}$	&	$-$0.00263 & $-$0.0241	&    $-$0.0670 \\  \hline
84594507546024.46719	     &  36738927826561   &     0.92208	&$-9.59 \cdot 10^{-16}$ & $-$0.00253	& $-$0.0234 &  $-$0.0652  \\  \hline
117239588213313.90075	     &   50916506221651  &     0.94102	& $-5.19 \cdot 10^{-16}$	&	$-$0.00188 &	$-$0.0175 &    $-$0.0488 \\  \hline
505863698785929.24318 	     &  219693812977897   &    	$-$0.95652  &		$-8.79 \cdot 10^{-17}$	&	$-$0.00131  & $-$0.0126	 &   $-$0.0353  \\  \hline
\end{tabular}\label{tab12}
\end{table}

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