On Elementary Methods to Evaluate Values of the Riemann Zeta Function and another Closely Related Infinite Series at Natural Numbers

Transcription

1 Global oural of Mathematical Sciece: Theory a Practical. SSN 97- Volume 5, Number, pp teratioal Reearch Publicatio Houe O Elemetary Metho to Evaluate Value of the Riema Zeta Fuctio a aother Cloely Relate fiite Serie at Natural Number Dhruhil Baai baaihruhil@gmail.com Abtract thi paper, a elemetary metho to fi the value of the Riema Zeta fuctio at eve atural umber, a to fi value of a cloely relate erie at o atural umber i preete. Aother metho, pecifically for the evaluatio of ζ, i alo preete. Keywor: Riema Zeta Fuctio, ζ, fiite erie, Elemetary metho trouctio The zeta fuctio i probably the mot challegig a myteriou object of moer mathematic, i pite of it utter implicity. ~ M.C Gutzwiller. The Riema Zeta fuctio i oe of the mot importat fuctio i Mathematic - it i eeply relate to the prime umber theorem a ha wie-ragig applicatio i phyic, probability theory, aalytic umber theory a other fiel of mathematic. The Riema Hypothei, oe of the Milleium Prize problem, i cloely relate to the Riema Zeta fuctio. thi paper,the oly coiere cae are where the argumet i purely real. Firtly, we ugget a alterate metho to evaluate ζ i.e. by ettig up a efiite itegral a evaluatig it uig metho of calculu. Secoly, we ugget a elemetary metho by mea of which we geeralize the proceure to fi the um of the followig ifiite erie: Whe i a eve atural umber, ζ

2 5 Dhruhil Baai a, Whe i a o atural umber, let Φ A Alterate Metho to Calculate ζ thi ectio, we ue the tool of itegral calculu to et up a efiite itegral for ζ, a evaluate it. ζ trouce a parameter a coier the followig fuctio f : f Differetiate both ie of thi equatio with repect to : f f Alo, coier the Taylor Serie epaio of l : l f we replace with i thi epaio, we have, l Thu, l f f l f l tegrate both ie of thi equatio with repect to from to : f f l Now itrouce a parameter a et l Thu, l

3 O Elemetary Metho to Evaluate Value of the Riema Zeta Fuctio 5 l l A, l l l Now, et l Thu, Let. Thu, l l [Chagig the variable of itegratio oe ot alter the value of a efiite itegral] l l l l From, 5 Now, l Sice, a are cotiou i the iterval, a are boue, we ca ifferetiate uer the itegral ig. Differetiatig uer the itegral ig with repect to, we have:.

4 5 Dhruhil Baai O evaluatig the itegral, we have: arcta arcta arcta arcta tegrate both ie of the equatio with repect to, from to. Sice arcta i a o fuctio, it itegral over thi iterval will be. arcta From a 5, Thu, arcta Let arcta u Thu, u Whe, u a whe, u. u u But,. f Thu,

5 O Elemetary Metho to Evaluate Value of the Riema Zeta Fuctio 55 f Sice, f it follow that f ζ Thu, ζ A Elemetary Metho To Calculate Value Of ζ A Other Relate Fuctio thi ectio, a elemetary metho i ue to calculate the aforemetioe value. A appropriate fuctio i choe a the followig value are obtaie for N : Whe i a eve atural umber, ζ a, Whe i a o atural umber, let Φ Now, coier ζ for N,, ζ Sice, ca be either o or eve, ζ ζ ζ ζ Thu, ζ The Fourier Serie of a perioic fuctio f, itegrable o [, ] i give: a f a co b i

6 5 Dhruhil Baai where, for a f co a, for b f i Coier the followig efiitio of f which i perioic i : f f k >, < where k N. Thu, a a a A, a co a co co a [ i co ] Sice N, i co. Thu, a [ ] Alo, b i Sice i i a o fuctio, it itegral over [, ]. Thu, b Therefore, the Fourier Serie epaio for f efie above i: a a co b i [ ] co

7 O Elemetary Metho to Evaluate Value of the Riema Zeta Fuctio 57 Whe i eve, [ ] Whe i o, [ ] Thu, [, ], co[ ] what follow, we hall coier oly.,. Thu, [, ], co[ ] Rearragig the term of the equatio, we have, co[ ] 7 8 tegrate both ie of thi equatio, with repect to, from to a. a co[ ] a 8 a co[ ] a a 8 8 i[ a] a a tegrate both ie of thi equatio, with repect to a, from a to a. i[ a] a a a a 8 8 i[ a]a 9 co[ ] 9 tegrate both ie of thi equatio, with repect to, from to a. a co[ ] a 9 a co[ ] a a a

8 58 Dhruhil Baai i[ a] a a a 9 tegrate both ie of thi equatio, with repect to a, from i[ a] a 5 a a 9 8 a 9 a a to a. i[ a]a co[ ] tegrate both ie of thi equatio, with repect to, from to a. 5 a co[ ] a a co[ ] 5 a a a a i[ a] a a a a the calculatio performe above, the itegral a ummatio ig have bee iterchage which ca be jutfie by uig a corollary of the mootoe a omiate covergece theorem. The omiate covergece theorem implie that we may itegrate the equece of partial um term-by-term, which i tatamout to ayig that we may witch itegratio a ummatio. equatio 7, put. 8 From, ζ. 8 equatio 9, put. 9 From, ζ equatio, put. 9

9 O Elemetary Metho to Evaluate Value of the Riema Zeta Fuctio 59 From, ζ equatio 8, a, put. Sice, i, We have, Φ A, 5 5 Φ5 5 5 A, 7 Φ7 7 8 Thu, the value of ζ, ζ a ζ alog with Φ for,5,7 have bee erive. Cocluio Thu, elemetary metho have bee ue to arrive at value of the Riema Zeta fuctio at eve atural umber a a cloely relate ifiite um for o atural umber. The firt metho ivolve the techique of Differetiatio Uer The tegral Sig while the eco ue a Fourier Serie epaio to arrive at the eee value. The eco metho ca be ue to attai the eee value for other iteger a well by cotiuig the ame proce, i the ame patter i.e. by itegratig over a iterval a ubtitutig the require value. The eco metho i efficiet i a way that you get the um of two erie uig oe proceure, a ca be cotiue elely to attai value that oe ee. t may eve be programme to obtai reult for large iteger. Referece [] Lag, S., 98, Uergrauate Aalyi, Seco Eitio. [] Toltov, G., 97, Fourier Serie Dover Book. [] Appel, W., 7, Mathematic For Phyic a Phyicit. [] Ewar, H.M.,, Riema Zeta Fuctio. [5] PlaetMath.org,

10 Dhruhil Baai