are  {\it spike values}---as described in Chapter~\ref{ch:trigseries}---of the infinite trigonometric series\index{infinite trigonometric series}

    spectrum}\index{spectrum}. We consider the infinite trigonometric series\index{infinite trigonometric series}

\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}}

where $\mu(n)$ is the Moebius function\index{Moebius function} introduced above.

\bibnote{{\bf How not to factor the numerator of a Bernoulli\index{Bernoulli} number:}

is (up to sign) the $k$-th {\bf Bernoulli\index{Bernoulli} number}.  These numbers are

$-B_{200}/400$, where $B_{200}$ is the two-hundredth Bernoulli\index{Bernoulli} number.

the issue.  The factorization of the numerators of general Bernoulli\index{Bernoulli}

Wagstaff (\url{http://homes.cerias.purdue.edu/~ssw/bernoulli\index{Bernoulli}}).

(\url{http://homes.cerias.purdue.edu/~ssw/bernoulli/composite})\index{Bernoulli} which gives

attempts to date.  The two-hundredth Bernoulli\index{Bernoulli} number is 12th on the list.

Bernoulli\index{Bernoulli} number:

\item Sam Wagstaff maintains a table of factorizations of numerators of Bernoulli\index{Bernoulli}

  numbers at \url{http://homes.cerias.purdue.edu/~ssw/bernoulli/bnum}.\index{Bernoulli} According

  to this table, the 200th Bernoulli\index{Bernoulli} number is the 2nd smallest index with

  unfactored numerator (the first being the 188th Bernoulli\index{Bernoulli} number).

sage: 389 * 691 * 5370056528687 * c204 == -numerator(bernoulli\index{Bernoulli}(200))\\

the linear term of the polynomial $S_k(N)$ is the {\it Bernoulli\index{Bernoulli}

> 1$ this linear term vanishes.  For even integers $2k$ the  Bernoulli\index{Bernoulli} number

sums at even positive integers, where---surprise---the Bernoulli\index{Bernoulli}

Those ubiquitous Bernoulli\index{Bernoulli} numbers reappear yet again as

so since the Bernoulli\index{Bernoulli} numbers indexed by odd integers $>1$ all

    Part~\ref{part4} requires some familiarity with complex analytic functions, and returns to Riemann's original viewpoint.  In particular it  relates the ``Riemann spectrum'' that we discuss in Part~\ref{part3} to the {\it nontrivial zeroes of the Riemann zeta function}.\index{zeta function} We also provide a brief sketch of the more standard route taken by published expositions of the Riemann Hypothesis.

  ``nontrivial'' zeroes of the Riemann zeta function\index{zeta function}}.}.

nontrivial zeroes of his zeta function;''\index{zeta function} we will discuss this

 \item The Wikipedia entry for Riemann's Zeta Function\index{zeta function}

numbers come in again; and they and $\pi$ combine to yield the values of the zeta function\index{zeta function} at even positive integers:

values of this {\it extended} zeta function\index{zeta function} at negative integers:

vanish, the extended zeta function\index{zeta function} $\zeta(s)$ actually vanishes at

trivial zeroes} of the Riemann zeta function.\index{zeta function}  There are indeed

other zeroes of the zeta function,\index{zeta function} and those other zeroes could---in no way---be

  \begin{enumerate}\item {\bf Riemann's zeta function\index{zeta function} codes the

  Riemann zeta function\index{zeta function} doesn't entirely pin it down---you have to

      The zeta function,\index{zeta function} then, is the vise, that so elegantly clamps

the Riemann zeta function,\index{zeta function} plays a role with respect to $\pi(X)$ rather

\chapter{Companions to the zeta function\index{zeta function}}\label{ch:companions}

%product $\zeta(s)\cdot L(K,s)$ is called the {\it zeta function\index{zeta function} of

%the Riemann zeta function.\index{zeta function}  The ``nontrivial zeroes'' of $\zeta_K(s)$

The natural question to ask, then, is: how are the Gaussian prime numbers\index{prime number} distributed? Can one provide as close an estimate to their distribution and structure, as one has for ordinary primes?  The answer, here is yes: there is a companion theory,  with an analogue to the Riemann zeta function\index{zeta function} playing  a role similar to the prototype $\zeta(s)$.  And, it seems as if its ``nontrivial zeroes''\index{nontrivial zeroes} behave similarly: as far as things have been computed, they  all have the property that

%  band-pass, complex numbers, complex plane, Riemann Zeta function\index{zeta function},

 only ``units'' among the Gaussian integers\index{Gaussian integers} (i.e., numbers whose

 integers, in that the norm of a product of two Gaussian integers\index{Gaussian integers} is

 two Gaussian integers\index{Gaussian integers} of smaller size.  Given what we have just

    Part~\ref{part4} requires some familiarity with complex analytic functions, and returns to Riemann's original viewpoint.  In particular it  relates the ``Riemann spectrum'' that we discuss in Part~\ref{part3} to the {\it nontrivial zeroes\index{nontrivial zeroes} of the Riemann zeta function}.\index{zeta function} We also provide a brief sketch of the more standard route taken by published expositions of the Riemann Hypothesis.

nontrivial zeroes\index{nontrivial zeroes} of his zeta function;''\index{zeta function} we will discuss this

 $$I(\phi):= \int_1^{\infty}(1-{\frac{1}{x^3-x}})\phi(x)dx$$ (coming from the pole at $s=1$ and the ``trivial zeroes''). \item The more serious  summation on the left hand side of the equation is over the nontrivial zeroes\index{nontrivial zeroes} $\rho$, noting that if $\rho$ is a nontrivial zero so is ${\bar \rho}$. \end{itemize}

      nontrivial} zeroes. These nontrivial zeroes\index{nontrivial zeroes} are known to lie in

       All the nontrivial zeroes\index{nontrivial zeroes} of $\zeta(s)$ lie on the vertical

infinite collection of complex numbers, i.e., the nontrivial zeroes\index{nontrivial zeroes} of

%the Riemann zeta function.\index{zeta function}  The ``nontrivial zeroes''\index{nontrivial zeroes} of $\zeta_K(s)$

