,calculus ,a1 ,lab #8

    ,9troduc;n4
  ,? lab is 9t5d$ 6get u />t$ on ! o!r
pro#m ( calculus1 ! pro#m ( -put+ ! >ea
"u a curve4 ,! basic task (! lab is
6-pute ! >ea "u a curve />t+ at a fix$
po9t ;a & 5d+ a v>i\s po9ts ;x_4 ,?
gives u a new func;n1 ! >ea func;n
,f;a"(x)_4 ,we'll !n -pute ! derivative
( ? >ea func;n1 & see if we c "u/& :at
we obs]ve4
  ,i want$ u 6d ? lab us+ n j simple
func;ns ": we mi<t 2 a# 6d "ey?+
algebraic,y1 b al r&om func;ns ": we _h
6"w m geometric,y4 ,if \r obs]v,ns &
3jectures hold = f(x) .k x^2, x mi<t 2 a
co9cid;e1 b if !y "w =a re,y -plicat$
r&om func;n1 !n we ga9 m* m 3fid;e 9 !m4
  ,! -plicat$ func;n ,i pick$ is !
,riemann ,zeta func;n1 .z(s)_4 ,? turns
\ 6be "o (! mo/ important func;ns 9 pure
ma!matics1 b x's "o y've probably n"e
met4 ,z a t1s]1 ,i've li/$ a few (! ?+s
                                      #j
e ma!matician "\ 6"k ab ! ,zeta func;n 9
a app5dix 6? lab4 ,= u "r n[1 ?\<1 !
only important ?+ is t we're "w+ )a
func;n t ,maple calls ,zeta(;t)_4 ,if y
l1 y mi<t />t \ j 0plott+ y .k .z(t) on
a few 9t]vals1 j 6get a s5se ( :at x
looks l4

    ,>ea ,func;ns4
  ,let y .k f(t) 2 any func;n4 ,! >ea
func;n y .k ,f;a"(x) is def9$ 6be ! >ea
abv ! ;t axis1 2l ! graph ( y .k f(t),
6! "r (! l9e t .k a, & 6! left (! l9e
t .k x_4 ,if "o ?9ks ( ;a z a 3/ant1 !n
? >ea is a func;n ( ;x, : m>ks ! "r h&
$ge (! region ^: >ea ,f;a"(x) m1sures4
,figure #1 %[s a sket*4
  #1_4 ,let f(t) .k #2t+3, & let
a .k #1_4 ,! >ea func;n y .k ,f;a"(x) is
"!=e ! >ea abv ! ;t axis1 2l ! graph (
y .k #2t+3, 6! "r (! l9e t .k #1, & 6!
left (! l9e t .k x_4
  (a) ,make a sket* %[+ ! graph (
y .k #2t+3 &! region ^: >ea is
                                      #a
,f;a"(x)_4
  (;b) ,-pute ! values ( ,f;a"(2),
,f;a"(3), ,f;a"(.p), ,f;a"(1), &
,f;a"(x)_4
  (;c) -pute ! derivative ,f';a"(x)_4 ,d
y notice any?+8

  #2_4 ,let f(t) .k .z(t) 2 ! ,riemann
,zeta func;n1 & let a .k -#20_4 ,figure
#2 is a plot (! ,zeta func;n ne> ;a_4
,"h is a ta# ( "s (! values (tak5 f
,maple)_4












                                      #b
----------------------------
.z(-20.0) .k #0
----------------------------
.z(-19.9) .k #11. 707 237 1
----------------------------
.z(-19.8) .k #20. 561 821
----------------------------
.z(-19.7) .k #26. 871 821 69
----------------------------
.z(-19.6) .k #30. 963 861 72
----------------------------
.z(-19.5) .k #33. 168 325 79
----------------------------
.z(-19.4) .k #33. 807 683 32
----------------------------
.z(-19.3) .k #33. 187 616 64
----------------------------
.z(-19.2) .k #31. 590 624 88
----------------------------
.z(-19.1) .k #29. 271 766 88
----------------------------
.z(-19.0) .k #26. 456 212 12
----------------------------
.z(-18.9) .k #23. 338 283 64
----------------------------
.z(-18.8) .k #20. 081 698 92
----------------------------
.z(-18.7) .k #16. 820 741 07
----------------------------
.z(-18.6) .k #13. 662 121 68
----------------------------
.z(-18.5) .k #10. 687 327 07
----------------------------
                                      #d
  (a) ,e/imate ! value ( ,f;a"(x) at ea*
(! po9ts li/$ abv4 ,try 6keep yr total
]ror less ?an ab #1 if y c4 ,i'd e/imate
! >ea 0a4+ up a bun* ( trapezoids4
  (;b) ,e/imate ! value ( ,f';a"(x) at
ea* (! po9ts li/$ abv4 ,try 6keep yr
]ror less ?an ab #1 if y c4 ,i'd -pute !
derivatives 0rememb]+ t if ;h is small1
!n
  ,f';a"(x)
    ".k<*] ?,f;a"(x+h)-,f;a"(x)_/h#_4
,use ! values = ,f;a"(x) f "p (a) 6d yr
calcul,ns4 ,d y notice any?+8
  (;c) ,draw a sket* %[+ t z l;g z ;h is
small1 ! num]ator (! di6];e quoti5t
  ?,f;a"(x+h)-,f;a"(x)_/h#
is v ne>ly ! >ea (a trapezoid4
  (;d) ,n[ try 6get a v a3urate e/imate
( ,f';a"(-19) 0us+ ! =mula =! >ea (a
trapezoid 6-pute z a3urately z y c !
di6];e quoti5ts
  ?,f;a"(-19+h)-,f;a"(-19)_/h#
= h .k #0.1, h .k #0.01, h .k #0.001, &
h .k #0.0001_4 (,rememb] t ,maple c
                                      #e
-pute ! ,zeta func;n_4) ,does yr value =
,f';a"(-19) agree ) :at e>li] pro#ms h
l$ y 63jecture8

  #3_4 ,! previ\s pro#ms %d h l$ y 6make
a 3jecture ab ! rel,n 2t ! func;n f(t)
&! derivative ,f';a"(x) (! >ea func;n4
,/ate ? 3jecture z precisely z y c1 &
write a few s5t;es expla9+ :y y ?9k yr
3jecture is correct4 ,y ne$ n give a
=mal algebraic >gu;t---?\< ,i wd 2
deli<t$ if y did---b y %d try 6expla9
geometric,y :y yr 3jecture w "w reg>d.s
( :at yr />t+ func;n ;f mi<t 24

  #4_4 ,suppose we _h d"o ,pro#m #1 />t+
) f(t) .k #2t+3 &) a .k -#4_4 ,2t
t .k -#4 & t .k -#2, ! graph ( f(t) lies
2n ! ;t axis3 f(t) "k #0_4
  (a) ,h[ wd y def9e ! >ea func;n
,f;a"(x) 9 ! region -#4 "k: x "k: -#2_8
  (;b) ,us+ yr def9i;n ( ,f;a"(x), -pute
,f;a"(-2) & ,f';a"(-2)_4

                                      #f
  (;c) ,is yr 3jecture "w+8
  (;d) ,if yr 3jecture fails 9 ? case1
!n try 6?9k (a new way 6def9e ,f';a"(x)
: wd pres]ve yr 3jecture4 ,if yr
3jecture su3e$s1 !n try 6imag9e o!r
s5si# def9i;ns ( ,f;a"(x), & see if yr
3jecture wd "w = !m4 ,9 ei case1 :at wd
y say 6advocates (! -pet+ def9i;n8


    ,app5dix3 ,fun ,facts ,ab ! ,riemann
    ,zeta ,func;n4
  #1_4 ,= r1l numb]s s .1 #1, ! ,zeta
func;n is giv5 by
  .z(s)
    .k ?1_/1^s"#+?1_/2^s"#+?1_/3^s"#+?1
      _/4^s"#+ ''' +?1_/n^s"#+ '''
    .k (1+?1_/2^s"#+?1_/2^2s"#+?1_/
      2^3s"#+ ''')*(1+?1_/3^s"#+?1_/
      3^2s"#+?1_/3^3s"#+ ''')*
    (1+?1_/5^s"#+?1_/5^2s"#+?1_/5^3s"#
      + ''')* '''
    .k (?2^s"_/2^s"-1#)(?3^s"_/3^s"-1#)(
      ?5^s"_/5^s"-1#) ''' (?p^s"_/p^s"-
                                      #g
      1#) '''
": ! products run ov] all primes ;p_4 ,!
9f9ite product repres5t,n m1ns t ! ,zeta
func;n 3ta9s a />tl+ am.t ( 9=m,n ab !
4tribu;n (! prime numb]s1 : is :y !
,zeta func;n is s important4
  ,= numb]s : >e n r1l numb]s s .1 #1,
.z(s) does n h s 3v5i5t a repres5t,n4

  #2_4 ,= ev5 positive 9teg] values (
;s, ! ,zeta func;n has elegant expres.ns
9volv+ .p_4 ,= 9/.e1
  .z(2) .k ?.p^2"_/6#,
  .z(4) .k ?.p^4"_/90#_4
,no s* =mulas >e "kn = odd positive
9teg] values ( ;s_2 ev5 ! "q ( :e!r
.z(3) is a r,nal multiple ( .p^3 is op54

  #3_4 ,= negative ev5 values ( ;s,
  .z(-2) .k .z(-4) .k .z(-6) .k '''
    .k #0_4
,e o!r root (! ,zeta func;n is a -plex
numb] (! =m x+it, ": i .k >-1], & ":
#0 "k: x "k: #1_4 ,= 9f9itely _m ( ^!
                                      #h
roots1 x .k ?1_/2#_4
  ,x is >guably true t ! mo/ important
unsolv$ pro#m 9 ma!matics is ! ,riemann
,hypo!sis1 ! 3jecture t e non-trivial
root (! ,zeta func;n has ! =m
?1_/2#+it_4 ,"! is a huge numb] ( !orems
9 analytic numb] !ory (! =m1 8,if !
,riemann ,hypo!sis is true1 !n ''' 0
,prov+ ? 3jecture wd "!=e settle 9 "o
/roke ! tru? ( all ^! o!r results1
resolv+ a v l>ge family ( "qs 9 numb]
!ory4
  ,david ,hilb]t1 ! grte/ ma!matician (!
2g9n+ (! la/ c5tury1 sd t if he 7 6awake
9 #500 ye> l ,b>b>ossa1 ! f/ "q he wd
ask wd 21 8,has ! ,riemann ,hypo!sis be5
prov$80 ,i'm less poetic1 b if a g5ie 7
6grant me #3 wi%es1 x wd 2 h>d n 6make
"o ( !m a reque/ =a pro( or 4pro( (!
,riemann ,hypo!sis4




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