Euler-type formulas. Badih Ghusayni. Department of Mathematics Faculty of Science-1 Lebanese University Hadath, Lebanon

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1 Iteratioal Joural of Mathematics ad Computer Sciece, 7(), o., 85 9 M CS Euler-type formulas Badih Ghusayi Departmet of Mathematics Faculty of Sciece- Lebaese Uiversity Hadath, Lebao badih@future-i-tech.et (Received May,, Accepted July, ) Abstract Fidig the eact value that coverges to is oe of the most otorious problems that did ot eve yield to Euler. A less difficult problem to cosider is to fid a represetatio of ζ() i terms of ( ) oly which would be a beautiful result similar to Euler s ζ() = ( ) ot to metio the fact that it is faster covergig. I this paper we fid a ew represetatio of ζ() i terms of ad si d. More precisely, we prove the followig ew formula: si y ζ() = π dy ). y 4 ( Key words ad phrases: Euler, Euler-type, Zeta fuctio. AMS (MOS) Subject Classificatios: 4A5, M.

2 8 B. Ghusayi Itroductio The formula ζ() = 7 π log + 7 log(si )d () was discovered by Euler [5] who by the time he proved this i 77 had bee blid for years accordig to [], p. 84. From ([7], formula ): ( ) = log( si( ))d which is equivalet to ( ) = 8 u log( si u)du = 9 π log 8 u log(si u)du suggests a possible coectio betwee ζ() ad A variat of formula () is ([4], formula 89) ζ() = 9 π log + log(si )d π Euler [] showed that oly. Usig itegratio by parts we have log(si )d = π log. cot d= log(si ) Sice log(si ) = log si result, t cot d= t log(si t) log(si ) d + log we have lim log si =. As a log(si ) d

3 Euler-type fomulas 87 which for t = π yields ad cosequetly we get cot d= log(si ) d ζ() = 7 π log 8 cot d, 7 a formula where the itegrad is free of log. Agai, a variat of this formula ([4], formula 4) is ζ() = 9 π log 9π cot d. It is worth metioig here the well-kow Euler itegral ([4], p. 54) ad ([9], p. 8) log(cos )d = cot d = π log () O the other had ([4], formula,) ( ) t d = t log ta si Agai, it is worth metioig here that log(si )d = π log () + = si d = π log. si( +)t ( +) (4) However, d = G, Catala s costat which is still ukow. si Together with [], formula (5) yield the curiously lookig formula si d =πg 7 ζ() ζ() = 7 (π ) si d.

4 88 B. Ghusayi The Mai problem I [7] I have obtaied the followig represetatio ζ() = si π 4 ( ) which i [] I have improved to ζ() = 8 π + 4 π ( ) 4 ( ) (5) The simply-lookig series, however, is hard to evaluate as I came ( ) to realize. Ideed, it appeared ofte eve i Ramauja s writigs (see, for eample, []) ad specifically as ta t dt = π 5π log t The alterative veue i lieu of the value that epress i terms of either ζ() or ( ) k= ( ) (k +) () coverges to is to thus attaiig the objective of epressig ζ() i terms of oly. However, I was ot able to do that mathematically or eve by usig the Computer Algebra System Maple. This gridlock motivated this paper to obtai a formula similar to formula (5) with the term replaced with a easier oe. ( ) Lookig at formula () the idea is to use the LLL algorithm [] icorporated i Maple to try to fid a iteger relatio amog say ζ(), ta d ad which ca the possibly prove mathematically. The previous discussio suggested the followig Maple worksheet: > a := evalf((pi*sqrt())*(sum(/(*+), =.. ifiity))) a :=

5 Euler-type fomulas 89 > b := evalf(zeta()) b :=.59 > c := evalf(sum(/( *biomial(*, )), =.. ifiity)) c :=.5949 f := evalf(pi*(it(arcta(t)/t, t =.. /sqrt()))) f := > A := truc( *a) A := > B := truc( *b) B := 59 > C := truc( *c); F := truc( *f) C := 5949 F := > v:=[a,,,,] v := [795887,,,, ] > v:=[b,,,,] v := [59,,,, ] > v:=[c,,,,];v4:=[f,,,,] v := [5949,,,, ] v4 := [75585,,,, ] > with(itegerrelatios) [LLL, LiearDepedecy, P SLQ] > LLL([v, v, v, v4]); [[-,,, -, ], [5, 84, -45, -8, 4], [4, -,, 4, 98], [-7, -89, -5, 7, 44]] Ufortuately, the presece of the relatively large umbers, say,,, did ot look promisig. Luckily, whe I chaged ta to si with i the right iterval (, ) (clearly, si =ta for < ), the gloomy picture tured aroud as the followig worksheet shows (icidetally, I added more digits for a chage-but that is ot required-ad I checked the result with Maple after small umbers came out): > Digits := Digits:= >a := evalf(pi*sqrt()*sum(/(*+), =.. ifiity) a := >b:= evalf(zeta()) b :=

6 9 B. Ghusayi >c:= evalf(sum(/( *biomial(*, )), =..ifiity)) c := > f := evalf(pi*(it(arcsi(t)/t, t =../))); f := >A:= truc( *a) A := > B := truc( *b) B := 59 > C := truc( *c); F := truc( *f) C := 5949 F := > v:=[a,,,,] v := [795885,,,, ] > v:=[b,,,,] v := [59,,,, ] > v := [C,,,, ]; v4 := [F,,,, ] v := [5949,,,, ] v4 := [594547,,,, ] > with(itegerrelatios) [LLL, LiearDepedecy, PSLQ] > LLL([v, v, v, v4]) [[,, 4,, 4], [78, 48, 45, 449, 998], [75, 49, 7, 889, 8], [98, 48, 55, 8, 95]] Thus the formula we obtaied with Maple as a tool is: 4ζ() ( ) +4π si t dt =. t To test the result obtaied usig Maple we have: > evalf(-4*zeta()-*(sum(/( biomial(*,)),..ifiity))+ 4*Pi*(It(arcsi(t)/t, t=../))). 59 Note that the last step is reassurig.

7 Euler-type fomulas 9 We ow supply a mathematical proof of the discovered idetity: si y ζ() = π dy ( ) y 4 By the mai result i [7], it is eough to show that si y dy = si π. y Clearly, u cot udu w=si u {}}{ = si w w dw. I particular, si t si y t dy = cot d y (Usig formula () we get the curious special case Therefore, usig a formula i ([4], p. 5) si y dy = y = cot d = si()d + = si y dy = π log.) y cos si( +)d cos si( +)d where the first term evaluates to π + ad the secod term yields, upo 4 8 itegratio by parts, ( si π π cos π (+)π si + 4 4( +) π cos (+)π + ) Now by ([7], p. 74) for (, π), cos = log( si ) ad so cos π =ad si (+)π 4( +) = = cos (+)π = cos π = + si π 4 = si π 4. I additio, 8.

8 9 B. Ghusayi Cosequetly, si y dy = y si π. Refereces [] R. Ayoub, Euler ad the zeta fuctio, Amer. Math. Mothly, 8, (974), 7-8. [] D. M. Bradley, Represetatios of Catala s costat, (a upublished catalogue of formulae for the alteratig sum of the reciprocals of the odd positive squares), [] B. C. Bredt, P. T. Joshi, Chapter 9 of Ramauja s Secod Notebook, AMS, Providece, RI, USA, 98. [4] D. F. Coo, Some ifiite series ivolvig the Riema zeta fuctio, It. J. Math. Comput. Sci., 7, o., (), -8. [5] L. Euler, Eercitatioes Aalyticae, 77. [] L. Euler, De summis serierum umeros Beroulliaos ivolvetium, Novi commetarii acadamiae scietiarum Petropolitaae, 4,(79), 9-7. [7] B. Ghusayi, Some represetatios of ζ(), Missouri Joural of Mathematical Scieces,, o., Fall 998, Math Review: b:. mjms/998-p.html [8] B. Ghusayi, Eplorig ew idetities with Maple as a tool, WSEAS Trasactios o Iformatio Sciece ad Applicatios,, o. 5, (4), [9] B. Ghusayi, Number Theory from a aalytic poit of view,. [] A. K. Lestra, H. W. Lestra, L. Lovasz, Factorig Polyomials with Ratioal Coefficiets, Math. A.,, (98),