Thomas J. Osler Mathematics Department Rowan University Glassboro NJ Introduction

Transcription

1 Ot Euler s little summtio formul d speil vlues of te zet futio Toms J Osler temtis Deprtmet Row Uiversity Glssboro J 0608 Osler@rowedu Itrodutio I tis ote we preset elemetry metod of determiig vlues of te zet futio ζ ( z) for z = 0 We use little kow summtio formul due to Euler Te zet futio defied by te series ζ ( z) = Re( z ) > z is oe of te most importt speil futios i mtemtis Te futiol equtio for te zet futio is z π z ζ( z) = ( π) os Γ( z) ζ( z) Euler foud losed form epressios for ζ ( z) we z is eve turl umber He 4 6 π π π sowed tt ζ () = ζ (4) = ζ (6) = d i geerl ( ) ζ( p) = = p+ p p p π p ( p)! (See Kopp [3 pge 37]) Here te umbers B B re lled Beroulli s umbers d tey re ll rtiol Te first few re B0 = B = B = B4 = B6 = d B3 = B5 = B7 = = d teir geertig futio is B = e! 0

2 Tese ll be lulted reursively by strtig wit B 0 = () B0 + B+ B + + B = 0 0 for = 3 4 d usig Usig te futiol equtio d te speil vlues of te zet futio t positive eve itegers sow bove we lulte te vlues + ζ (0) = B d ζ ( ) = for = 3 + B A seod metod [ p ] of fidig tese vlues of te zet futio is to use te teory of residues pplied to otour itegrl represettio of ζ ( z) I tis pper we sow tt tese vlues of te zet futio be foud witout usig eiter te futiol equtio or te speil vlues of te zet futio t positive eve itegers or do we eed otour itegrls d te teory of residues We use oly rrely see elemetry summtio formul due to Euler tt we ll Euler s little summtio formul Tis little summtio formul [] ws foud by Euler s itermedite item i te derivtio of is big result [3 pp ] tt we ll tody te Euler-luri summtio formul I te et setio we will sow simple derivtio of te little summtio formul d i setio 3 we will use it to fid te vlues of ζ ( z) for z = 0 Euler s little summtio formul for ifiite series Defiitio: Let b d be ostts d let Sb f( ) = f( ) + f( + ) + f( + ) + + f( b)

3 3 were b= + wit = 0 Here we red S f( ) s summtio of f( ) from to b wit iremet Usully we sum wit iremet = d i tis se we suppress te d write S f( ) = S f( ) b b Teorem (Euler s little summtio formul): Let f( ) be futio wit + otiuous derivtives over te itervl < d let us use te ottio f( ) = lim f( ) Te b () ( ) S f ( ) S f '( ) = f( ) f( ) R! were d ( ( + ) ( + ) ( + R ( ) ( ) = f + f + ) + f ( + ) + ) ( + )! is i te itervl < < + Proof: From te ypotesis f( ) is futio of wit + otiuous derivtives Tus (for < ) we epd f( ) i Tylor s series wit + terms d remider: ( ) f ( ) f( + ) = + r! 0 were r = ( + ) f ( ) ( + )! + d is i te itervl < < + Summig from to b= + we ve (3) S f ( ) ( ) ( ) ( ) b Sb f + = Sb f + + R! were R = Sb r is desribed by

4 4 d d so + ( + ) ( + ) ( + ) R = ( f ( ) + f ( + ) + + f ( b) ) ( + )! is i te itervl < < + otie tt Sb f( + ) = f( + ) + f( + ) + + f( b) + f( b+ ) Sb f( ) = f( ) + f( + ) + f( + ) + + f( b) S f( + ) S f( ) = f( b+ ) f( ) b b From tis lst reltio d (3) we get ( ) ( ) ( ) ( ) S b f f b+ f = + R! d isoltig te first term i te sum d dividig by we get (4) S f ( ) ( ) '( ) ( ) ( ) b Sb f = f b+ f R! ow let = d let b te tis lst reltio beomes () d te teorem is proved 3 Euler s little summtio formul for ifiite series d te zet futio We use () to determie reursively te vlues of ζ ( z) for z = 0 3 Let z+ f( ) = z + Te t derivtive is f ( ) = ( ) ( ) + zz ( + )( z+ ) ( z+ ) z+

5 5 If Re{} z > te lim f( ) = f( ) = 0 d we ve from () ( ) zz ( + )( z+ ) ( z+ ) (5) S = + S R z z + z+ ( z )! were R zz ( + )( z+ ) ( z+ ) = ( ) z+ z+ z+ ( )! ( ) ( + ) ( + ) d < < + otie tt < ζ ( z+ ) ( ) z + ( ) z + ( ) z + we ve + + d tus (6) R zz ( + )( z+ ) ( z+ ) ζ ( z+ ) ( + )! Let = i (5) to get (7) ( ) zz ( + )( z+ ) ( z+ ) ζ( z) = + ζ( z+ ) + R z! Wile (7) ws proved wit te restritio Re{} z > by te teory of lyti otiutio (6) is vlid for ll z eept z = For z = wit oegtive iteger te remider i (7) is zero if = + otie lso tt ( z ) ( z ) ( z ζ ) ( ) ( ) z A B z C z + + = d tus lim ( z+ ) ζ ( z+ ) = z ( ) Returig to (7) we see tt for = 0 we ve

6 6 + ( ) ( )( + )( + ) ( + ) lim ζ( z) = + ζ( + )! z + ( ) ( )( + )( + ) ( ) lim( z ( ) ) ζ ( z ( ) ) z ( + )! Tus we ve + ( )( )( ) ( + ) ζ( ) = ζ( ) +! + ( + )( + ) wi simplifies to (8) + ( + )( + ) ζ( ) = ( + ) ζ( ) Wit = 0 we use (8) to fid ζ (0) = et usig tis vlue of ζ (0) wit = i (8) we lulte ζ( ) = Cotiuig i tis wy we fid suessively ζ( ) = 0 ζ( 3) = ζ( 4) = 0 ζ( 5) = ζ( 6) = We lso use (8) to prove te geerl formul + (9) ζ (0) = B d ζ ( ) = for = 3 + B We ve sow tt ζ (0) = B By idutio ssume (9) is true for - Settig ζ( ) = B i (8) for 3 we get + + ( + )( + ) ζ( ) = ( + ) ζ(0) + B Addig d subtrtig o te rigt we get

7 7 + + ( + )( + ) ζ( ) = { + ζ(0) } + B wi we rewrite s + + ( + ) ζ ( ) = B + 0 It follows t oe from () tt ( + ) ( ) = B d (9) is proved ζ + Referees [] Edwrds H Riem s Zet Futio Dover Publitios ew York 974 [] Euler L Eerpts o te Euler-luri summtio formul from Istitutioes Cluli Differetilis (trsl Dvid Pegelley) t mtmsuedu/~dvidp ew eio Stte Uiversity 000 [3] Kopp Kord Teory d Applitio of Ifiite Series Dover Publitios ew York 990 (A trsltio by R C H Youg of te 4 t Germ dditio of 947)