Rk.tex

\documentclass{article}
\title{Title of Document}
\author{Name of Author}

\begin{document}
\maketitle


\chapter[Building $\pi(X)$ knowing the spectrum]{How to build $\pi(X)$ knowing the spectrum (Riemann's way)}


  We have been dealing in Part~\ref{part3} of our book with $\Phi(t)$ a
  distribution\index{distribution} that---we said---contains all the essential information
  about the placement of primes among numbers. We have given a clean
  restatement of Riemann's hypothesis, the third restatement so far,
  in term of this $\Phi(t)$.  But $\Phi(t)$ was the effect of a series
  of recalibrations and reconfigurings of the original untampered-with
  staircase of primes.  A test of whether we have strayed from our
  original problem---to understand this staircase---would be whether
  we can return to the original staircase, and ``reconstruct it'' so to
  speak, solely from the information of $\Phi(t)$---or equivalently,
  assuming the \RH{} as formulated in Chapter~\ref{sec:tinkering}---can
  we construct the staircase of primes $\pi(X)$ solely
  from knowledge of the sequence of real numbers $\theta_1,
  \theta_2,\theta_3,\dots$?




  The answer to this is yes (given the \RH{}), and is discussed very
  beautifully by Bernhard Riemann\index{Riemann, Bernhard} himself in his famous 1859 article.

  Bernhard Riemann used the spectrum\index{spectrum} of the prime numbers\index{prime number} to provide
  an exact analytic formula that analyzes and/or synthesizes the
  staircase of primes.  This formula is motivated by Fourier's
  analysis of functions as constituted out of cosines.  Riemann\index{Riemann, Bernhard} started
  with a specific smooth function, which we will refer to as $R(X)$, a
  function that Riemann offered, just as Gauss\index{Gauss, Carl Friedrich} offered his $\Li(X)$,
  as a candidate smooth function approximating the staircase of
  primes.  Recall from Chapter~\ref{sec:rh1} that Gauss's guess is
  $\Li(X)= \int_2^{X}dt/{\rm log}(t).$ Riemann's guess for a better
  approximation to $\pi(X)$ is obtained from Gauss's, using the Moebius
  function $\mu(n)$, which is defined by
$$
 \mu(n) = \begin{cases}
    1 &
       \mbox{\begin{minipage}{0.6\textwidth}if $n$ is a square-free
       positive integer with an even number of distinct prime
       factors,\end{minipage}}\vspace{1em}\\
    -1& \mbox{\begin{minipage}{0.6\textwidth}if $n$ is a square-free
    positive integer with an odd number of distinct
    prime factors,\end{minipage}}\vspace{1em}\\
    0 & \mbox{if $n$ is not square-free.}
 \end{cases}
 $$
 See Figure~\ref{fig:moebius} for a plot of the Moebius function.\index{Moebius function}

\ill{moebius}{1}{The blue dots plot the values of the Moebius function\index{Moebius function} $\mu(n)$, which is only defined at integers.\label{fig:moebius}}


Riemann's\index{Riemann, Bernhard} guess is
$$
R(X) = \sum_{n=1}^{\infty}{\mu(n)\over n} \Li(X^{1\over n}),
$$
where $\mu(n)$ is the Moebius function\index{Moebius function} introduced above.

\ill{riemann_RX}{0.8}{Riemann\index{Riemann, Bernhard} defining $R(X)$ in his manuscript}

In Chapter~\ref{sec:pnt} we encountered the Prime Number Theorem,\index{prime number} which
asserts that $X/\log(X)$ and $\Li(X)$ are both approximations for
$\pi(X)$, in the sense that both go to infinity at the same rate.  That is, the ratio of any two of these three functions tends to $1$ as $X$ goes to $\infty$.
Our first formulation of the \RH{} (see page~\pageref{rh:first}) was
that $\Li(X)$ is an essentially square root accurate approximation of
$\pi(X)$.  Figures~\ref{fig:guess100}--\ref{fig:guess10000} illustrate
that Riemann's function $R(X)$ appears to be an even better
approximation to $\pi(X)$ than anything we have seen before.

\illtwo{pi_riemann_gauss_100}{pi_riemann_gauss_1000}{0.47}{Comparisons of $\Li(X)$ (top), $\pi(X)$ (middle), and $R(X)$ (bottom, computed using 100 terms)\label{fig:guess100}}


\ill{pi_riemann_gauss_10000-11000}{0.5}{Closeup comparison of  $\Li(X)$ (top), $\pi(X)$ (middle), and $R(X)$ (bottom, computed using 100 terms)\label{fig:guess10000}}


Think of Riemann's smooth curve $R(X)$ as the {\em fundamental}
approximation to $\pi(X)$.
Riemann\index{Riemann, Bernhard} offered much more than just a (conjecturally) better
approximation to $\pi(X)$ in his wonderful 1859 article.
He found a way to construct what looks like a Fourier series,
but with $\sin(X)$ replaced by $R(X)$ and spectrum\index{spectrum} the $\theta_i$, which
conjecturally exactly equals $\pi(X)$.
He gave an infinite sequence of improved guesses,
$$
R(X) = R_0(X),\quad R_1(X), \quad R_2(X), \quad R_3(X), \quad \ldots
$$
and he hypothesized that one and all of them were all
essentially square root approximations to $\pi(X)$,
and that the sequence of these better and better approximations converge to give an exact formula
for $\pi(X)$.

Thus not only did Riemann\index{Riemann, Bernhard} provide a ``fundamental'' (that is, a smooth curve
that is astoundingly close to $\pi(X)$) but he viewed this as just a
starting point, for he gave the recipe for providing an infinite
sequence of corrective terms---call them Riemann's {\em harmonics}\index{harmonics}; we
will denote the first of these ``harmonics'' $C_1(X)$, the second
$C_2(X)$, etc.  Riemann\index{Riemann, Bernhard} gets his first corrected curve, $R_1(X)$, from
$R(X)$ by adding this first harmonic to the fundamental, $$R_1(X) =
R(X) + C_1(X),$$ he gets the second by correcting $R_1(X)$ by adding
the second harmonic $$R_2(X) = R_1 (X) + C_2(X),$$ and so on $$R_3(X)
= R_2 (X) + C_3(X),$$ and in the limit provides us with an exact fit.

\ill{riemann_Rk}{0.8}{Riemann\index{Riemann, Bernhard} analytic formula for $\pi(X)$.}

The \RH{}, if true, would tell us that these correction
terms $C_1(X), C_2(X), C_3(X),\dots$ are all {\em square-root small},
and all the successively corrected smooth curves $$R(X), R_1(X),
R_2(X),R_3(X),\dots$$ are good approximations to $\pi(X)$.
Moreover,
$$
 \pi(X) = R(X) + \sum_{k=1}^{\infty} C_k(X).
$$

The elegance of Riemann's\index{Riemann, Bernhard} treatment of this problem is that the
corrective terms $C_k(X)$ are all {\em modeled on} the fundamental
$R(X)$ and are completely described if you know the sequence of real
numbers $\theta_1, \theta_2, \theta_3,\dots$ of the last section.


To continue this discussion, we do need some familiarity with complex numbers, for the definition of Riemann's\index{Riemann, Bernhard} $C_k(X)$
requires extending
the definition of the function $\Li(X)$ to make sense when given
complex numbers $X=a+bi$.  Assuming the \RH{}, the Riemann\index{Riemann, Bernhard} correction
terms $C_k(X)$ are then
$$
   C_k(X)= -R(X^{\frac{1}{2} + i\theta_k}),
$$
where $\theta_1 = 14.134725\dots, \theta_2 = 21.022039\dots$, etc.,
is the spectrum\index{spectrum} of the prime numbers\index{prime number} \bibnote{You may well ask how we propose to order these correction terms if RH
  is false. Order them in terms of
  (the absolute value of) their imaginary part, and in the unlikely
  situation that there is more than one zero with the same imaginary
  part, order zeroes of the same imaginary part by their real parts,
  going from right to left.}.

Riemann\index{Riemann, Bernhard} provided an extraordinary recipe that allows us to work
out the harmonics\index{harmonics}, $$C_1(X),\quad C_2(X),\quad C_3(X),\quad \dots$$ without our having
to consult, or compute with, the actual staircase of primes. As with
Fourier's modus operandi where both {\em fundamental} and all {\em
  harmonics}\index{harmonics} are modeled on the sine wave, but appropriately
calibrated, Riemann\index{Riemann, Bernhard} fashioned his higher harmonics\index{harmonics}, modeling them all
on a single function, namely his initial guess $R(X)$.

The convergence of $R_k(X)$ to $\pi(X)$ is strikingly illustrated
in the plots in Figures~\ref{fig:rkfirst}--\ref{fig:rklast} of $R_k$ for various values of $k$.


\ill{Rk_1_2_100}{.9}{The function $R_{1}$ approximating the staircase of primes up to $100$\label{fig:rkfirst}}

\ill{Rk_10_2_100}{.9}{The function $R_{10}$ approximating the staircase of primes up to $100$}

\ill{Rk_25_2_100}{.9}{The function $R_{25}$ approximating the staircase of primes up to $100$}

\ill{Rk_50_2_100}{.9}{The function $R_{50}$ approximating the staircase of primes up to $100$}


\ill{Rk_50_2_500}{.9}{The function $R_{50}$ approximating the staircase of primes up to $500$}

\ill{Rk_50_350_400}{.9}{The function $\Li(X)$ (top, green), the function $R_{50}(X)$ (in blue), and the staircase of primes on the interval from 350 to 400.\label{fig:rklast}}


\end{document}