A Ramanujan enigma involving the first Stieltjes constant. Donal F. Connon. 5 January 2019

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1 A Ramauja eigma ivolvig the firt Stieltje cotat Doal F. Coo 5 Jauary 9 Atract We provide a rigorou formulatio of Etry 7(v) i Ramauja Noteook ad how how thi relate to the firt Stieltje cotat.. Itroductio A curiou item appear i Ramauja Noteook ([] ad [3]) which were writte y him i Idia more tha year ago. Etry 7(v) tate: If, the (.) ( ) ( ) = [ + log( )]cot( ) + i( )log = where ( ) i defied a (.) log log( + ) ( ) = = + Ramauja alo ote that (.3) = i( ) = cot( ) I hi commetary o Ramauja Noteook, Berdt [, p.] ha writte: Of coure, Etry 7(v) i meaigle ecaue the erie o the right ide diverge for. I the midt of hi formula, after cot( ), Ramauja iert a parethetical remark for the ame limit, the meaig of which we are uale to dicer. Uurpriigly, Berdt goe o to ay that (.3) i alo devoid of meaig. I elieve that I may have dicovered Ramauja modu operadi i Etry 7(v) ad ca cofirm that hi two related formulae may e rigorouly epreed for a i( )log (.4) ( ) ( ) = [ + log( )]cot( ) + lim =

2 ad i( ) (.5) lim = cot( ) = I view of thi, we ow kow the meaig of Ramauja parethetical remark for the ame limit. We hall ee later i thi paper that ( ) ( ) = ( ) ( ) where ( ) are the geeralied Stieltje cotat defied y (3.) elow. Thi reult i i( )log (.6) ( ) ( ) = [ + log( )]cot( ) + lim Firt of all, we eed to et out our uildig lock.. The Hurwitz zeta fuctio (, ) = The Hurwitz zeta fuctio (, ) i iitially defied for Re ad y (.) (, ) = = ( + ) Note that (, ) may e aalytically cotiued to the whole plae ecept for a imple pole at =. For eample, Hae ( ) howed that [3] k ( ) (.) (, ) = = + k= k ( k + ) i a gloally coverget erie for (, ) ad, ecept for =, provide a aalytic cotiuatio of (, ) to the etire comple plae. It may e oted from (.) that (,) = ( ). We eaily ee from (.) that lim[( ) (, ) = ( ) = + k= k k

3 k ad, ice ( ) = ( ) =,, we have k= k (.3) lim[( ) (, )] = which how that (, ) ha a imple pole at =. Thi eale u to write the Lauret epaio for the Hurwitz zeta fuctio how i (3.) elow. 3. The geeralied Stieltje cotat ( ) The geeralied Stieltje cotat ( ) are the coefficiet i the Lauret epaio of the Hurwitz zeta fuctio (, ) aout = (3.) ( ) (, ) = = + ( )( ) ( + )! = = We have (3.) ( ) = ( ) where ( ) i the digamma fuctio which i the logarithmic derivative of the gamma d fuctio ( ) = log ( ). It i eaily ee from the defiitio of the Hurwitz zeta fuctio du that (,) = ( ) ad accordigly that () =. The Stieltje cotat (or the Euler-Macheroi cotat) are the coefficiet of the Lauret epaio of the Riema zeta fuctio ( ) aout =. (3.3) ( ) ( ) = + ( )! = Sice lim ( ) it i clear that. A elemetary proof of ( ) = ( ) = = wa recetly give y the author i [] (thi formula wa firt otaied y Berdt [] i 97). We ote from (3.) that (3.4) [( ) (, )] = ( ) = ( ) ad it i eay to ee that = 3

4 ( ) ( m) ( m) m (, ) (, y) = [ ( ) ( y)] ( )...( m + )( ) =! I the limit thi ecome (3.5) ( m) ( m) m lim[ (, ) (, y)] = ( ) [ m( ) m( y)] ad hece we have for m (3.6) m m log ( + ) log ( + y) m( ) m( y) = = + + For eample, we have (3.7) m m log ( + ) log ( + ) m( ) m = = + + ad m = give u (3.8) ( ) = = + + It i eaily ee from (.) that (3.9) (, + ) (, ) = ad (3.5) give u (3.) m log m( + ) m( ) = I the particular cae m =, we have the familiar formula for the digamma fuctio (3.) ( + ) = ( ) + We have for eample uig (.) for ad (i.e. ) (3.) ( +, ) ( +, ) = ( ) + + = + ( + ) I the limit a we otai 4

5 ( ) ( ) = = + + ad uig (3.) we have (3.3) ( ) ( +, ) ( +, ) = [ ( ) ( )]! = We ee from (.) that for Re( ) (3.4) (, ) (, ) = + ad, y aalytic cotiuatio, thi hold for all. Uig (3.) we fid that (3.5) k ( ) ( +, ) = + k ( ) k! k = k + Differetiatio reult i k ( ) [ ( +, )] = k ( )( k + ) k( k )...( k ) k! + + k = k or equivaletly + + k = k k ( ) (, ) = k ( )( k + ) k( k )...( k )! k We ote that the partial derivative commute i the regio where (, ) i aalytic ad hece we have (, ) = (, ) Therefore we otai (3.6) (, ) = ( + )( ) ( ) ( + ) + Itegratio reult i (3.7) u d ( ) u + + ( + ) ( + ) ( ) = (, ) () It i immediately ee that Lerch formula 5

6 (3.8) (, ) = log ( ) () arie i the cae = ecaue ( ) = ( ). We have for u = (3.9) + ( ) ( + ) ( + ) ( ) = (,) () d + ad, ecaue (,) = (), we deduce that ( ) ( ) (3.) ad (3.) ( ) d = ( + ) d = We ee from (3.) that (3.) [( ) (, )] = ( ) ( ) = 4. The Ramauja eigma Propoitio 4. We have for i (4.) lim = cot( ) = Proof We have the well-kow Hurwitz formula for the Fourier epaio of the Riema zeta fuctio (, ) a reported i Titchmarh treatie [7, p.37] co i (4.) (, ) = ( ) i + co = ( ) = ( ) where Re ( ) < ad. I, Boudjelkha [4] howed that thi formula alo applie i the regio Re ( ) <. It may e oted that whe = thi reduce to Riema fuctioal equatio for (). Lettig i (4.) we get for 6

7 co i (4.3) (, ) = ( ) i co = ( ) = ( ) ad we therefore ee that for co (4.4) (, ) + (, ) = 4 ( )i ( ) = i (4.5) (, ) (, ) = 4 ( )co ( ) = With regard to (4.5) we ote that lim i (, ) (, ) = lim ( ) 4 ( )co ( ) = ad we ee that ( ) = ( + [ ]) We have = ( ) ( ) ( ) co ( )co ( ) = ( ) ad we firt of all coider the limit ( ) co ( ) lim ( )co = lim = where we have employed L Hôpital rule. Hece, we have lim i (, ) (, ) = lim ( ) 4 ( )co ( ) = ( ) ( ) = ( ) ( ) = where we have ued (3.3) i the umerator. It i well kow [6, p.4] that for 7

8 ( ) ( ) = cot( ) ad we fid that Sice i lim cot( ) = = ( ) lim i i = lim ( ) = = we otai a valid iterpretatio of Ramauja formula (.3) i (4.6) lim = cot( ) = It hould e pecifically oted that i i (4.7) lim lim = = ecaue the latter erie oviouly diverge (ice limi ). I am aware of at leat two iteretig mathematical paper where the author have otaied correct reult y oldly tartig off with the erroeou aumptio that i = cot( ). We refer to (4.6) i more detail i the arrative followig Propoitio 4.3 elow. Propoitio 4. We have for i( )log (4.8) ( ) ( ) = lim [ log( )]cot( ) + + = where ( ) are the geeralied Stieltje cotat defied y (3.). = Proof We recall (4.4) i (, ) (, ) = 4 ( )co ( ) = ad utitutig the idetity 8

9 ( ) = ( ) we have the firt derivative (4.9) ( ) co i log( ) (, ) (, ) = 4 ( ) ( ) = ( ) d ( ) co co i 4 ( ) + ( ) d ( ) = For computatioal coveiece we write co( ) i ( [ ] ) ( ) d ( ) d co i [ ] = d d [ ] = o that Employig the Maclauri erie for the ie fuctio we eaily deduce that d i ( [ ] ) co ( ) = ad, a oted aove, we have lim =. d [ ] = Therefore, takig the limit of (4.9) a reult i i log( ) i ( ) ( ) = lim () lim ( ) = = ( ) where we have ued (3.5). Sutitutig () = ad uig (4.) we otai i( )log (4.) ( ) ( ) = lim [ log( )]cot( ) + + = Thi formula doe ot provide ay ew iformatio uder the chage. Referrig to (3.6) we ee that Ramauja defied ( ) a log( + ) log( + y) ( ) ( y) = = + + log log( + ) log( + ) log( + + ) ( ) = = = + =

10 ad thu we have log( + ) log( + ) ( ) ( ) = = + + = ( ) ( ) Hece, we otai a rigorou formulatio of Ramauja formula (.) i( )log ( ) ( ) = lim [ log( )]cot( ) + + = We how elow how (4.) may e employed to formally derive a kow reult due to Deiger []. Formal itegratio of (4.) over [, u ], where u, reult i (4.) u u u u i( )log ( ) d ( ) d = lim d + [ + log( )] cot( ) d = ( ) Uig the elemetary itegral u u ( ) d = ( ) d ( ) d ad u u ( ) d = ( t) dt u = ( t) dt + ( t) dt we may write the left-had ide of (4.) a u u ( ) d ( ) d + ( ) d ad uig (3.7) u ( ) d = [ (, u) ()]

11 thi ecome [ (, u ) + (, u )] + (, ) Uig the well-kow formula (, ) = ( ) ( ) ad Lerch formula (3.8) it i readily foud that (, ) = log ( ) log( ) = (, ) log log( ) log o that the left-had ide of (4.) may therefore e writte a [ (, u) + (, u)] log log( )log We ow aume that formal itegratio reult i u i( ) log i( ) log lim d = lim d ( ) ( ) = = u [( ) co( u)]log = lim ( ) = + ( ) log log co( u ) = = = It i well kow that [5] + ( ) log a () = = log log = ad the right-had ide of (4.) ecome log log log co( u) + [ + log( )]log i( u) = With a little algera we ed up with a kow reult due to Deiger [] (4.) = log co( u) = [ (, u) + (, u)] + [ + log( )]log(i( u))

12 A elemetary itegratio of (4.) how that (4.3) (, u) du = which wa oted y Deiger []. Propoitio 4.3 We have for co (4.4) lim = = Proof We write (4.4) a (, ) + (, ) co = 4i ( ) ( ) = ad we have the limit (, ) + (, ) co lim = 4lim ( ) ( ) = Notig that (, t) ( ) (, t) = ( ) ( ) we eaily determie the limit ( t, ) lim = ( ) a oted i [8, p.66]. We therefore otai co lim = = ( ) or equivaletly co lim = =

13 Comiig thi with (4.) we otai co i lim i [ i cot( )] = + = + or equivaletly i e lim [ icot( )] = + = A good eplaatio for thi iteretig (ad o-ituitive) reult i cotaied i [4, p.] ad [6]. We have hitherto tacitly aumed that i a real umer. Now let u aume that = u + iv i a comple umer with v. Thi give u e = e i v = = e iu We ee that thi erie coverge for all ad therefore i e lim = e = = e v iu e = e i i By aalytic cotiuatio, thi hold true for every RZ. Havig regard to (4.6) i lim cot( ) = = we ote that Niele [5, p.8] report a Fourier erie for the cotaget fuctio (4.5) = i( )i( ) = cot( ) where i( ) i the ie itegral fuctio. Thi could e epreed a i( ) lim i( ) = cot( ) = or a i( ) lim i( ) = cot( ) = ( ) 3

14 where we have ow alo icluded i the factor i( ). It might e worthwhile eplorig thi apect further. We multiply (4.6) y a Riema itegrale fuctio f( ) ad formal itegratio o [ a, ] with a reult i a f ( )i f ( )cot( ) d = lim d a = Thi may e compared with [8] ad [9] where we howed that (4.6) a f ( )cot( ) d = f ( )i( ) d = for uitaly ehaved fuctio. Similarly, we alo howed there that (4.7) a f ( ) d = f ( )co( ) d = ad thi correpod with the itegratio of (4.4) multiplied y a Riema itegrale fuctio f( ) f ( )co f ( ) d lim d = a a = Oe would eed to coider the circumtace whe lim f (, y) d = lim f (, y) d y c y c a a i valid. I thoe two earlier paper [8] ad [9] we foud the followig idetity to e etremely ueful (which i eaily verified y multiplyig the umerator ad the deomiator y the comple cojugate ( e i ) ) i i i = = + cot( / ) e i co 4

15 We have (, ) (, ) lim = ( ) Therefore we have i lim co = ( ) = ( ) co i lim ( ) = = ( ) i lim( ) = ( ) = which cocur with the oviou limit lim( )cot( ) = The erie (.3) feature promietly i the firt chapter of Hardy ook, Diverget Serie, [] where he demotrate how it may e ued to geerate variou mathematical formulae. We ow differetiate (4.4) with repect to ad otai i [ (, ) + (, )] = 4 ( )i ( ) = Referrig to (.) we ee that (, ) (, ) = + (, ) (, ) = + ad we otai i [ ( +, ) ( +, )] = 4 ( )i ( ) = Dividig y give u 5

16 ( ) i i (, ) (, ) 4 ( ) ( ) + + = = Takig the limit i lim[ ( +, ) ( +, )] = lim ( ) = we ee that thi correpod with (4.). Propoitio 4.4 We have for (4.8) log( )co lim = ( ) + cot( ) + [ + log( )] = ( ) Proof We differetiate (4.) with repect to to otai log( ) co log( )i (, ) = ( ) i + co = ( ) = ( ) co i + ( ) co i = ( ) = ( ) co i ( ) i + co = ( ) = ( ) ad the differetiate thi with repect to log( )i log( ) co (, ) = ( ) i co + = ( ) = ( ) Lettig give u i co + ( ) co i = ( ) = ( ) i co ( ) i + co = ( ) = ( ) 6

17 log( )co i co (, ) lim lim () lim = ( ) ( ) ( ) ad hece = = = log( )co (, ) = lim cot( ) ( ) = where we have employed (4.) ad (4.4) co i lim i [ i cot( )] ( ) + = ( ) = + Uig Lerch formula (3.8) (, ) = log ( ) () we ee that (, ) ( ) = We the otai (4.9) log( )co ( ) = lim cot( ) ( ) = which may e writte a log( )co ( ) = lim cot( ) [ log( )] ( ) + = Lettig we otai log( )co ( ) = lim cot( ) ( ) + = ad utractio reult i the familiar idetity ( ) ( ) = cot( ) We alo ee that (4.) log( )co ( ) + ( ) = + 4lim ( ) = 7

18 Formal itegratio of (4.) over [, ] reult i or = ( ) + log ( ) log ( ) 4 lim = ( ) + log ( ) log ( ) log( )i ( ) = log( ) i = ad thi correpod with the formula reported y Deiger [8] i 984 (which i effectively Kummer Fourier erie for the log gamma fuctio). Chakraorty et al, [7, p.46] metio i paig that = [ + log( )]co + i = ut thi doe ot agree with (4.9) which may e writte a [ + log( )]co + i ( ) = lim ( ) = 5. Ope acce to our ow work Thi paper cotai referece to variou other paper ad, rather urpriigly, mot of them are curretly freely availale o the iteret. Surely ow i the time that all of our work hould e freely acceile y all. The mathematic commuity hould lead the way o thi y pulihig everythig o arxiv, or i a equivalet ope acce repoitory. We thik it, we write it, o why hide it? You kow it make ee. REFERENCES [] B.C. Berdt, O the Hurwitz Zeta-fuctio. Rocky Moutai Joural of Mathematic, vol., o., pp. 5-57, 97. [] B.C. Berdt, Ramauja Noteook. Part I, Spriger-Verlag, 989. [3] B.C. Berdt, Chapter eight of Ramauja Secod Noteook. J. Reie Agew. Math, Vol. 338, -55, [4] M.T. Boudjelkha, A proof that eted Hurwitz formula ito the critical trip. Applied Mathematic Letter, 4 () [5] W.E. Brigg ad S. Chowla, The power erie coefficiet of (). Amer. Math. Mothly, 6, 33-35,

19 [6] K. Chakraorty, S. Kaemitu ad H. -X. Wag, The modular relatio ad the digamma fuctio. Kyuhu J. Math. 65 (), [7] K. Chakraorty, S. Kaemitu ad H. Tukada, Arithmetical Fourier erie ad the modular relatio. Kyuhu J. Math. 66 (), 4 47 [8] D.F. Coo, Some erie ad itegral ivolvig the Riema zeta fuctio, iomial coefficiet ad the harmoic umer. Volume V, 7. arxiv:7.447 [pdf] [9] D.F. Coo, Some applicatio of the Dirichlet itegral to the ummatio of erie ad the evaluatio of itegral ivolvig the Riema zeta fuctio.. arxiv:.44 [pdf] [] D.F. Coo, Some itegral ad erie ivolvig the Stieltje cotat, 8. [] C. Deiger, O the aalogue of the formula of Chowla ad Selerg for real quadratic field. J. Reie Agew. Math., 35 (984), [] G.H. Hardy, Diverget Serie. Chelea Pulihig Compay, New York, 99. [3] H. Hae, Ei Summierugverfahre für Die Riemache - Reithe. Math.Z.3, , [4] S. Kaemitu ad H. Tukada, Vita of pecial fuctio. World Scietific Pulihig Co. Pte. Ltd., 7. [5] N. Niele,Theorie de Itegrallogarithmu ud verwater trazedete, [6] H.M. Srivatava ad J. Choi, Serie Aociated with the Zeta ad Related Fuctio. Kluwer Academic Puliher, Dordrecht, the Netherlad,. [7] E.C. Titchmarh, The Zeta-Fuctio of Riema. Oford Uiverity (Claredo) Pre, Oford, Lodo ad New York, 95; Secod Ed. (Revied y D.R. Heath- Brow), 986. [8] E.T. Whittaker ad G.N. Wato, A Coure of Moder Aalyi: A Itroductio to the Geeral Theory of Ifiite Procee ad of Aalytic Fuctio; With a Accout of the Pricipal Tracedetal Fuctio. Fourth Ed., Camridge Uiverity Pre, Camridge, Lodo ad New York, 963. Wee Houe, Devize Road, Upavo, Wilthire SN9 6DL 9