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\input colordvi

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\def\cgy{\GreenYellow}     % GreenYellow  Approximate PANTONE 388
\def\cyy{\Yellow}	  % Yellow  Approximate PANTONE YELLOW
\def\cgo{\Goldenrod}	  % Goldenrod  Approximate PANTONE 109
\def\cda{\Dandelion}	  % Dandelion  Approximate PANTONE 123
\def\capr{\Apricot}	  % Apricot  Approximate PANTONE 1565
\def\cpe{\Peach}		  % Peach  Approximate PANTONE 164
\def\cme{\Melon}		  % Melon  Approximate PANTONE 177
\def\cyo{\YellowOrange}	  % YellowOrange  Approximate PANTONE 130
\def\coo{\Orange}	  % Orange  Approximate PANTONE ORANGE-021
\def\cbo{\BurntOrange}	  % BurntOrange  Approximate PANTONE 388
\def\cbs{\Bittersweet}	  % Bittersweet  Approximate PANTONE 167
%\def\creo{\RedOrange}	  % RedOrange  Approximate PANTONE 179
\def\cma{\Mahogany}	  % Mahogany  Approximate PANTONE 484
\def\cmr{\Maroon}	  % Maroon  Approximate PANTONE 201
\def\cbr{\BrickRed}	  % BrickRed  Approximate PANTONE 1805
\def\crr{\Red}		  % Red  VERY-Approx PANTONE RED
\def\cor{\OrangeRed}	  % OrangeRed  No PANTONE match
\def\paru{\RubineRed}	  % RubineRed  Approximate PANTONE RUBINE-RED
\def\cwi{\WildStrawberry}  % WildStrawberry  Approximate PANTONE 206
\def\csa{\Salmon}	  % Salmon  Approximate PANTONE 183
\def\ccp{\CarnationPink}	  % CarnationPink  Approximate PANTONE 218
\def\cmag{\Magenta}	  % Magenta  Approximate PANTONE PROCESS-MAGENTA
\def\cvr{\VioletRed}	  % VioletRed  Approximate PANTONE 219
\def\parh{\Rhodamine}	  % Rhodamine  Approximate PANTONE RHODAMINE-RED
\def\cmu{\Mulberry}	  % Mulberry  Approximate PANTONE 241
\def\parv{\RedViolet}	  % RedViolet  Approximate PANTONE 234
\def\cfu{\Fuchsia}	  % Fuchsia  Approximate PANTONE 248
\def\cla{\Lavender}	  % Lavender  Approximate PANTONE 223
\def\cth{\Thistle}	  % Thistle  Approximate PANTONE 245
\def\corc{\Orchid}	  % Orchid  Approximate PANTONE 252
\def\cdo{\DarkOrchid}	  % DarkOrchid  No PANTONE match
\def\cpu{\Purple}	  % Purple  Approximate PANTONE PURPLE
\def\cpl{\Plum}		  % Plum  VERY-Approx PANTONE 518
\def\cvi{\Violet}	  % Violet  Approximate PANTONE VIOLET
\def\parp{\RoyalPurple}	  % RoyalPurple  Approximate PANTONE 267
\def\cbv{\BlueViolet}	  % BlueViolet  Approximate PANTONE 2755
\def\cpe{\Periwinkle}	  % Periwinkle  Approximate PANTONE 2715
\def\ccb{\CadetBlue}	  % CadetBlue  Approximate PANTONE (534+535)/2
\def\cco{\CornflowerBlue}  % CornflowerBlue  Approximate PANTONE 292
\def\cmb{\MidnightBlue}	  % MidnightBlue  Approximate PANTONE 302
\def\cnb{\NavyBlue}	  % NavyBlue  Approximate PANTONE 293
\def\crb{\RoyalBlue}	  % RoyalBlue  No PANTONE match
%\def\cbb{\Blue}		  % Blue  Approximate PANTONE BLUE-072
\def\cce{\Cerulean}	  % Cerulean  Approximate PANTONE 3005
\def\ccy{\Cyan}		  % Cyan  Approximate PANTONE PROCESS-CYAN
\def\cpb{\ProcessBlue}	  % ProcessBlue  Approximate PANTONE PROCESS-BLUE
\def\csb{\SkyBlue}	  % SkyBlue  Approximate PANTONE 2985
\def\ctu{\Turquoise}	  % Turquoise  Approximate PANTONE (312+313)/2
\def\ctb{\TealBlue}	  % TealBlue  Approximate PANTONE 3145
\def\caq{\Aquamarine}	  % Aquamarine  Approximate PANTONE 3135
\def\cbg{\BlueGreen}	  % BlueGreen  Approximate PANTONE 320
\def\cem{\Emerald}	  % Emerald  No PANTONE match
\def\cjg{\JungleGreen}	  % JungleGreen  Approximate PANTONE 328
\def\csg{\SeaGreen}	  % SeaGreen  Approximate PANTONE 3268
\def\cgg{\Green}	  % Green  VERY-Approx PANTONE GREEN
\def\cfg{\ForestGreen}	  % ForestGreen  Approximate PANTONE 349
\def\cpg{\PineGreen}	  % PineGreen  Approximate PANTONE 323
\def\clg{\LimeGreen}	  % LimeGreen  No PANTONE match
\def\cyg{\YellowGreen}	  % YellowGreen  Approximate PANTONE 375
\def\cspg{\SpringGreen}	  % SpringGreen  Approximate PANTONE 381
\def\cog{\OliveGreen}	  % OliveGreen  Approximate PANTONE 582
\def\pars{\RawSienna}	  % RawSienna  Approximate PANTONE 154
\def\cse{\Sepia}		  % Sepia  Approximate PANTONE 161
\def\cbr{\Brown}		  % Brown  Approximate PANTONE 1615
\def\cta{\Tan}		  % Tan  No PANTONE match
\def\cgr{\Gray}		  % Gray  Approximate PANTONE COOL-GRAY-8
\def\cbl{\Black}		  % Black  Approximate PANTONE PROCESS-BLACK
\def\cwh{\White}		  % White  No PANTONE match


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\def\lra{$\Leftrightarrow$ }


\ctln{\bf Math 445 Number Theory}

\smallskip

\ctln{October 22, 2004}

\medskip

Since the Jacobi symbol has essentially the same properties as
the Legendre symbol, we can compute them in essentially the same
way; extract factors of 2 from the top (and $-1$), and use reciprocity
to compute the rest. The advantage: we don't need to factor the top
any further, any odd number will work fine.

\msk

Example: $\Big({{2225}\over{3333}}\Big)$ = 
$ \Big({{3333}\over{2225}}\Big)(-1)^{1666\cdot 1112} =
\Big({{2225+1108}\over{2225}}\Big) = \Big({{2^2\cdot 277}\over{2225}}\Big) =
(\Big({{2}\over{2225}}\Big) )^2\Big({{277}\over{2225}}\Big) =
\Big({{2225}\over{277}}\Big)(-1)^{1112\cdot 138} = 
\Big({{277\cdot 9+182}\over{277}}\Big) = \Big({{182}\over{277}}\Big) =
\Big({{2}\over{277}}\Big)\Big({{91}\over{277}}\Big) =
(-1)^{{{277^2-1}\over{8}}}\Big({{277}\over{91}}\Big)(-1)^{138\cdot 45} = 
(-1)^{9591}\Big({{91\cdot 3+4}\over{91}}\Big) =
(-1)\Big({{4}\over{91}}\Big) = (-1)(\Big({{2}\over{91}}\Big) )^2 = -1$ 

\msk

One basic result coming from reciprocity: for a fixed (odd) $a$, we can determine for which primes $p$
the equation $x^2\equiv a\pmod{p}$ will have solutions.

\ssk

$1= \Big({{a}\over{p}}\Big) = \Big({{p}\over{a}}\Big)(-1)^{{{p-1}\over{2}}{{a-1}\over{2}}}$
is determined by $\Big({{p}\over{a}}\Big)$ (which only depends on $p$ mod $a$) and (if
$a\equiv 3\pmod{4}$) on $p$ mod 4 (to determine the parity of ${{{p-1}\over{2}}{{a-1}\over{2}}}$ - if
$a\equiv 1\pmod 4$ it is always even). So $\Big({{a}\over{p}}\Big)$ depends on $p$ mod $a$
and on $p$ mod 4 (when $a\equiv 3\pmod{4}$), so it depends at most on $p$ mod $4a$ 
(by the Chinese Remainder Theorem). So the primes
for which $x^2\equiv a\pmod{p}$ have solutions fall {\it precisely} into certain equivalence classes mod
$a$ or $4a$, depending upon $a$.  If we include even values for $a$, then we need to extract 2's, and the
result will depend upon $p$ mod 8 (for the $\Big({{2}\over{p}}\Big)$'s) and, at worst, on $p$ mod $a/2$, and 
so it still depends at most on $p$ mod $4a$ .

\bsk

A brief interlude: we know that there are infinitely many primes. But how are they distributed? For example,
$\sum_{n=1}^{\infty} {{1}\over{n}}$ diverges, but 
$\sum_{n=1}^{\infty} {{1}\over{n^2}} = {{\pi^2}\over{6}}<\infty$. So how about
$\sum_{p \text{ prime}}{{1}\over{p}} $ ? We will show that this sum diverges, so that we know that,
in some sense, primes are more common than perfect squares....

\msk

To show this, pick a positive number $N$, and let $p_1,\ldots p_k$ be the primes $\leq N$. Then let 

$A =\sum_{i_1,\ldots i_k=0}^{\infty} {{1}\over{p_1^{i_1}\cdots p_k^{i_k}}} = 
(\sum_{i_1=0}^{\infty} ({{1}\over{p_1}})^{i_1})\cdots(\sum_{i_k=0}^{\infty} ({{1}\over{p_k}})^{i_k})=
{{1}\over{1-{{1}\over{p_1}}}}\cdots {{1}\over{1-{{1}\over{p_k}}}} = 
{{p_1}\over{p_1-1}}\cdots{{p_k}\over{p_k-1}}$ . 

But the initial sum includes all denomenators $\leq N$, since
every $k\leq N$ is a product of primes $\leq N$ , i.e, is a product of the primes $p_1,\ldots ,p_k$.
So $A\geq \sum_{n=1}^{N}{{1}\over{n}} \geq \int_{1}^{N} {{1}\over{x}}\ dx = \ln(N) $
by the integral test. So ${{p_1}\over{p_1-1}}\cdots{{p_k}\over{p_k-1}}\geq \ln(N)$ .
Taking logs of both sides, we have

$\sum_{i=1}^{k}\ln({{p_i}\over{p_i-1}}) = \sum_{i=1}^{k}\ln(1+{{1}\over{p_i-1}})\geq \ln(\ln(N))$ .
But from power series we know that for $|x|<1$, $\ln(1+x) = x-{{x^2}\over{2}}+{{x^3}\over{3}}-\ldots \leq x$
(since it is an alternating series with terms decreasing to 0 (or, if you prefer, by using ${{1}\over{1+x}}\leq 1$
and integrating from 1 to $x$)), so 
$\sum_{i=1}^{k}{{1}\over{p_i-1}})\geq \sum_{i=1}^{k}\ln(1+{{1}\over{p_i-1}}\geq \ln(\ln(N))$ .
But ${{1}\over{p_i-1}}\leq {{p_i+2}\over{p_i^2}}={{1}\over{p_i}}+{{2}\over{p_i^2}}$
(since $(p_i-1)(p_i+2) = p_i^2+p_1-2\geq p_i^2$), so 
$\sum_{i=1}^{k}{{1}\over{p_i}}+{{2}\over{p_i^2}}\geq \sum{i=1}^{k}{{1}\over{p_i-1}})
\geq \ln(\ln(N))$ . So 

$\sum_{i=1}^{k}{{1}\over{p_i}}\geq \ln(\ln(N))-\sum_{i=1}^{k}{{2}\over{p_i^2}}
\geq \ln(\ln(N))-\sum_{i=1}^{\infty}{{2}\over{n^2}} = \ln(\ln(N))-{{\pi^2}\over{3}}\geq \ln(\ln(N))-4$ . 
So the sum of the reciprocals of the primes $\leq N$ is $\geq \ln(\ln(N))-4$ .
Since $\ln(\ln(N))$ tends to $\infty$ as $N\rightarrow\infty$ (albeit very slowly), the sum of the 
reciprocals of the primes diverges.

\msk

It is in fact true that as $n\rightarrow \infty$, $(\sum_{p\text{ prime},p\leq n}{{1}\over{p}})-\ln(\ln(n))$
converges to a finite constant $M$, known as the {\it Meissel-Mertens constant}. It's value is, 
approximately, 0.26149721284764278... .

\vfill\end



$\displaystyle \sum_{n=1}^{\infty} $

\Big({{}\over{}}\Big)



