Some integrals involving the Stieltjes constants: Part II. Donal F. Connon. 11 April 2011

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1 Soe itegral iolig the Stielte cotat: Part II Doal F. Coo April Abtract Soe ew itegral iolig the Stielte cotat are deeloped i thi paper. CONTENTS Page. Itroductio. A ueful forula for the Stielte cotat 3. A applicatio of the Abel-Plaa uatio forula 6 4. A faily of itegral repreetatio of the Stielte cotat 6 5. A applicatio of the alteratig Hurwitz zeta fuctio 35. Itroductio The Stielte cotat γ p ( x ) are the coefficiet of the Lauret expaio of the Hurwitz zeta fuctio ς (, x) about = (.) p ( ) ς(, x) = = + γ p ()( x ) ( + x) p! = p= p where γ ( x ) are ow a the geeralied Stielte cotat ad we hae [4] p (.) γ ( x) = ψ ( x) where ψ ( x) i the digaa fuctio. With x = equatio (.) reduce to the Riea zeta fuctio p ( ) ς() = + γ p ( ) p! p= p

2 A preiouly oted i [], uig (.) it i eaily ee that the differece of two Stielte cotat ay be repreeted by p p (.3) γ p( x) γ p( y) = ( ) li [ ς(, x) ς(, y )] p. A ueful forula for the Stielte cotat We recall Hae forula [3] for the Hurwitz zeta fuctio which i alid for all C except = (i thi for, it i alid i the liit a ) ( ) (.) ( ) ς (, x) = ( x ) = + = + ad differetiatio with repect to x gie u ( ) ς (, x) = x ( x+ = + = ) ad we the hae (.) ( ) log + ( x + ) ς (, x) = () x = + = ( x+ ) We ote that the partial deriatie coute i the regio where ς ( x, ) i aalytic ad hece we hae ς(, x) = ς(, x) x x Ealuatio of (.) at = reult i (.3) (, x) = ( ) ( ) log ( x+ ) ( ) + ς x = + = We ay write (.) a (.4) ( ) ς( +, x) = + γ( x)! = ad we hae the Maclauri expaio

3 (.5) where = + ( ) R ( x) ς (, x) = +! R x x + ( ) = ( ) (, ) ς = R( x) = R ( x) i referred to a the Deiger R fuctio, after Deiger [8] who itroduced it i 984. Differetiatig (.5) with repect to x gie u (.6) We ote that ς (, x) = x = + ( ) R ( x)! (, x) (, x) x ς = ς + ad, uig (.4), thi i equal to (.7) + ( ) = + γ ( x)! = + ad coparig the coefficiet of (.6) ad (.7) Charaborty, Kaeitu ad Kuzuai [] deduced the iportat idetity (.8) R x x + ( ) = ( ) (, ) ( ) x ς = γ x = Thi ay be copared with the ore failiar forula for the Stielte cotat where the liit i ealuated at = (.9) + ς x = ( ) (, ) = γ ( x) We ee fro (.3) that (.8) i equal to = ( ) log ( x + ) = + = 3

4 ad hece we eaily deduce that (.) γ ( x) = ( ) log ( x+ ) = = which wa preiouly obtaied i 7 i []. I [, Eq. (4.3.3)] we alo oted that R ( x) = γ ( x) i the equialet for for x > x t dt ( ) γ = ς x ς + + ( + ) ( + ) ( ) (, ) () but the uefule of thi iple idetity wa ot the fully appreciated by the author. The forula (.8) feature throughout the ret of thi paper where it i ued to iplify the deriatio of oe ow idetitie ad alo to produce oe ew oe. Rear (i) The equialet forula to (.5) a reported by Charaborty et al. [] did ot iclude the ter ; it ee to e that it hould be o icluded if oly to cocur with the aalyi preiouly carried out by Sitaraachadrarao [37] i 986 where he coidered the Maclauri erie for the Riea zeta fuctio (.) ( ) δ ς () + =! = where (.) δ = li log log xdx log = = + ( ) ( ) ς ()! Thi additioal ter of coure aihe whe equatio (.5) i differetiated with repect to x ad thu (.8) cotiue to reai alid. 4

5 Rear (ii) We ote fro (.8) that with = R ( x) = (, x) ( x) ( ) x ς = γ = ψ x = or equialetly (, x) ( x) x ς = ψ ad itegratio the reult i (.3) ς (, t) ς () = log Γ ( t) We hae Legedre duplicatio forula for the gaa fuctio [38, p.7] log Γ ( t) = log Γ ( t) + log Γ t+ + (t) log logπ ad ubtitutig (.3) gie u ς (, t) ς (, t) ς = +, t+ ς () + (t) log logπ ad with t = / we obtai ς, = ς () + logπ We recall the idetity [3] ς, = [ ] ς ( ) ad differetiatio reult i o that ς, = [ ] ς ( ) + ς( ) log ς, = ς()log = log 5

6 Thi the gie u the well-ow reult (.4) ς () = log( π ) ad hece we hae obtaied Lerch idetity [7] i a ery direct aer without the eed to reort to the fuctioal equatio for the Riea zeta fuctio (.5) ς (, t) = log Γ( t) log( π ) Rear (iii) It hould be oted that the Stielte cotat γ referred to i Eq. (.5) of Deiger paper [8] hould be icreaed by a factor of (thi differece arie becaue Deiger [8, p.74] eployed a differet defiitio i the Lauret expaio of the Hurwitz zeta fuctio; thi alue wa alo icoitetly eployed i Eq. (.6) of the paper by Charaborty et al. []). 3. A applicatio of the Abel-Plaa uatio forula Adachi [] ha recetly reported that the Herite itegral for the Hurwitz zeta fuctio ay be deried fro the Abel-Plaa uatio forula [38, p.9] f( ix) f( ix) (3.) f ( ) = f() + f( x) dx+ i x e π = dx which applie to fuctio which are aalytic i the right-had plae ad atify the π coergece coditio li e y f( x+ iy) = uiforly o ay fiite iteral of x. y Deriatio of the Abel-Plaa uatio forula are gie i [39, p.45] ad [4, p.8]. The Herite itegral for the Hurwitz zeta fuctio ay be deried a follow. Lettig f ( ) = ( + u) we obtai (3.) u u ( u+ ix) ( uix) x ( u) e π = + ς (, u) = = + + i dx The, otig that iθ iθ ( u + ix) ( u ix) = ( re ) ( re ) 6

7 iθ iθ = r [ e e ] (3..) u+ ix u ix = x u)) iu ( + x) ( ) ( ) i( ta ( / / we ay write (3.) a Herite itegral for the Hurwitz zeta fuctio ς ( u, ) (3.3) u u i( ta ( x/ u)) / x ( u + x ) ( e π ) ς (, u) = + + dx Differetiatig (3.) with repect to u (3.4) ( ) ( ) x e π u u + ix u ix ς (, u) = u i dx u We ote that if f ( ) = g( ) the the Leibiz differetiatio forula reult i o that (3.5) ( + ) ( + ) ( ) f = g + + g () () ( ) () ( + ) ( ) f () = ( + ) g () ad hece we obtai ς (, u) = ()log u() log u + u u = i( ) ( + ) ( u ix)log ( u+ ix) ( u+ ix)log ( uix) dx x ( u + x )( e π ) ad coparig thi with (.8) we readily obtai the itegral forula origially obtaied by Coffey [7] i 7 for the Stielte cotat (3.6) γ ( u) = log u log u+ i x u + ( u + x )( e π ) + ( u ix)log ( u+ ix) ( u+ ix)log ( uix) dx Thi deriatio i lightly ore direct tha the oe origially proided by Coffey [7] (ad i alo ipler tha y preiou proof i []). Che [3] ha recetly how that for u > ad > 7

8 (3.7) uy e y i( xy ) dy i[ = Γ () ( u + x ) ta ( x/ u)] / ad hece we hae uig Herite itegral (3.3) u u 4 uy (, ) = + + i( ) x Γ( ) e π ς u dx e y xy dy uy x Γ u u 4 i( xy ) = + + e y dy dx ( ) e π Uig Legedre relatio [39, p.] i( xt) t (3.7.) dx = + = coth π x t e e t t thi reult i u u uy ς (, u) = + + e y + dy y Γ( ) e y With the ubtitutio = y thi becoe (3.8) u u u u = + + e + d Γ( ) e ς (, ) which i reported i [38, p.9] a beig alid for Re () >. With (3.9) u = we hae ς () = + + e + Γ e d ( ) which i alo reported i [38, p.] a beig alid for Re () >. We ay write (3.7) a Γ() Γ ( + ) uy uy e y i( xy ) dy = e y i( xy ) dy 8

9 ad we hae the liit li i( ) li i( ) Γ() Γ ( + ) uy uy e y xy dy = e y xy dy uy e i( xy) = li dy Γ ( + ) y We hae the well-ow itegral (rigorou deriatio of which are cotaied i [3, p.85] ad [6, p.7]) (3.) ta ( x / u) ad hece we ee that uy e i( xy) = dy y uy li e y i( xy ) dy = Γ() howig that Che reult i alid for. For copletee we preet aother proof of (3.7). Uig the defiitio of the gaa fuctio we hae zy Γ e y dy = z () ad with z = u± ix we obtai uy e y xy i xy dy = u+ ix Γ [co( ) i( )] ( ) ( ) uy e y xy + i xy dy = uix Γ [co( ) i( )] ( ) ( ) Thi gie u uy i e y xy dy = u+ ix uix Γ i( ) ( ) ( ) ( ) 9

10 uy e y xy dy = u+ ix + uix Γ co( ) ( ) ( ) ( ) The, otig (3..) we obtai ad uy e y i( xy) dy i[ t = Γ () ( u + x ) uy e y co( xy) dy a ( x/ u)] / co[ t = Γ () ( u + x ) a ( x/ u)] / We hae fro (3.8) (3.) Γ() (, ) = + u u u ς u e d e ad we coider the liit a. With iple algebra we ay write u u u u ς( u, ) = ς( u, ) ς(, u) + u = ς(, u) ς(, u) + u ( u ) u + ad we hae ( ) (, ) (, ) u u u ς u ς u u Γ() ς (, u) =Γ ( + ) + u Taig the liit a we ee that u u li Γ( ) ς(, u) = ς (, u) + u logu+ u ad hece we obtai (3.) u e ς (, u) + u log u+ u = + e d

11 Uig Lerch idetity (.5), it ay be ee that thi i equialet to Biet firt forula for l og Γ( u) [39, p.49] (3.3) uy e log Γ ( u) = u log u u+ log( π ) + dy y + e y y We ote Alexeiewy theore [38, p.3] (3.4) t log Γ ( udu ) = t( t) + tlog( π ) log G( + t) + tlog Γ( t) where Gx ( ) i the Bare double gaa fuctio Γ ( x) / Gx ( ) = defied, iter alia, by the Weiertra caoical product [38, p.5] (3.5) G x x x x x x x x ( + ) = ( π) exp ( γ + + ) exp + = ad we ote that G() = G() =. Hece itegratig (3.3) reult i t e log G( + t) = tlog Γ ( t) + t t t log t + d (3.6) ( ) 4 e ad with t = we iediately obtai e d + = e 4 Subtractig thi fro (3.6) reult i t e e log G( + t) = tlog Γ ( t) + ( t ) t t log t+ + d (3.7) ( ) 4 e It ay be poible eploy Prighei artifice with thi itegral (a wa eployed i [39, p.49]). We obtai by differetiatig (3.)

12 ta ( x e π x/ u) ς (, u) = u log u u+ d x ad differetiatig thi with repect to u gie u x ς (, u) = log u dx u u u + x e π x ( )( ) Sice fro (.8) (, u) ( ) u ς = γ u we obtai x γ ( u) = logu dx u u x e π x ( + )( ) ad ice γ ( u) ( = ψ u ) thi i equialet to x ψ ( u) = logu dx u u x e π a reported i [39, p.5]. Siilarly, we hae o that x ( + )( ) + (, u) u log u ulog u u x e π ( u x ) ( x u) log ta / ς = + dx ς (, ) = log log u u u u u x xlog ( u + x ) ta ( u/ x) πx x ( )( ) π u x e ( u x )( e ) πx + + ( u + x )( e ) π dx + dx + 4u dx The uig (, u) = γ () u we obtai u ς =

13 (3.8) xlog ( u + x ) ta ( x/ u) πx πx ) dx γ ( u) = logu log u+ dxu ( + )( ) ( + )( u u x e u x e It ay be oted that Shail [36, p.799] ade referece to thee eeigly itractable itegral i. We hae the well-ow Hurwitz forula for the Fourier expaio of the Hurwitz zeta fuctio ς ( t, ) a reported i Titcharh treatie π co πt π i πt ς (,) t = Γ( )i + co = ( π) = ( π) where Re ( ) < ad < t. I, Boudelha howed that thi forula alo applie i the regio Re ( ) <. It ay be oted that whe t = thi reduce to Riea fuctioal equatio for ς ( ). Ufortuately, it appear that (.8) caot be applied to Hurwitz forula for the Fourier expaio of the Hurwitz zeta fuctio becaue thi would reult i dierget erie. Thi howeer ay gie a idicatio of Raaua erroeou thiig i thi area a exeplified i Berdt boo, Raaua Noteboo, Part I [8, p.]. It i itructie to coider the origial deriatio of the forula for the Stielte cotat which wa deried by Stielte hielf. Thi i recorded i a letter dated Jue 885 fro Stielte to Herite [3] ad the proof proceed a follow: Fro (3.9) we hae x ς ( + ) = + e + x Γ ( + ) e x x dx x x e = + e + x x Γ ( + ) dx e x o that (3.9) x = + e x x Γ ( + ) e x dx x ς ( + ) = e x Γ ( + ) e x x dx 3

14 Uig the Maclauri expaio x log x =! = we ee that a ς ( + ) = Γ ( + )! = where x log a = e xd x e x x. Let () tx ( ) x F t = e log xdx. We hae Γ () = e log xdx ad with x tx thi becoe We write F ( ) t ( ) tx Γ = + a () t e ( log x log t) dx ( ) tx F t = e (log x+ log tlog t) dx ad we the hae = Γ t = ( ) ( ) ()log t (3.) ( ) log t tx ( ) Γ () = e log x = t dx We hae the uatio r r ( ) log t tx ( ) Γ () = e log x = t= t dx t= x ( r+ ) x e e = log x e x dx ad the right-had ide ay be writte a a + f ( r) g ( r) where 4

15 (3.) (3.) ( r+ ) x () = log x e x g r e xdx x ( r+ ) x e e f() r = log xdx x x where we ote that li g ( r) =. Now treatig r a a cotiuou ariable we obtai r log ( r + ) r + ( ) () = ( + ) = () Γ () = f r F r ad ice f () = we hae by itegratio (3.3) o f r + + ( ) () r = () Γ () = + log ( ) r + ( r+ ) x ( ) log log ( r+ ) a e log xdx ( ) () x e x = Γ = = + r log log ( ) log ( ) log r+ r+ r = ( ) () Γ = = + Therefore lettig r we obtai a r + ( ) log log r = ( ) Γ ()li r = = + ice (3.4) where + + li log ( r+ ) log r =. Thi ay be writte a r a C = C Γ = ( ) ( ) () r + log log r = li r = + Eq. (3.4) reid u of the Cauchy product of two erie 5

16 ( ) a C Γ () = ( )!!! = = = where ( ) Γ () Γ ( + ) =.! = We hae fro (3.9) (3.5) x ς ( + ) Γ ( + ) = e x e x dx x ad we ote fro (.4) that ( ) ς( + ) =! = γ ad we coclude that (3.6) γ r + log log r = C = li r = + A ore direct deriatio of (3.6) i gie, iter alia, by Boha ad Fröberg []. 4. A faily of itegral repreetatio of the Stielte cotat With a iew to utiliig (.8), we firt of all differetiate (3.8) with repect to u u u ς (, u) = u e + d u Γ( ) e u to gie ad differetiatig thi tie with repect to (with the aitace of (3.5)) gie u (4.) ( ) u ς ( u, ) = ( ) log u( ) log u F (, e ) + d u u e = where, for coeiece, we hae deigated F (, ) a (4.) F (,) = = Γ() Γ ( + ) 6

17 ad we ote that F (, ) = F(, ) =. We eploy the otatio ( = F ) (, ). F F ( ) (, ) = (, ) ad The uig (.8) (, u) ( ) () u u ς = γ = we hae thu deteried that for (4.3) γ ( ) u ( u) log u log u F (, ) e ( ) = u + e d ad, i particular, for we hae a faily of itegral repreetatio of the Stielte cotat + ( ) ( ) u (4.4) γ = γ () = F (, ) e d + e ad for = we obtai () F (, ) e d e γ = + + We hall iitially coider the firt two deriatie of repect to reult i F (, ) = log ψ ( ) ( ) + + Γ + F (, ). Differetiatig (4.) with ad o we hae = ψ Γ ( + ) [ log ( ) ] (4.5) F (, ) = F(, )log [ ψ () ] Sice ψ ( ) = ψ ( + ) we ee that li ψ ( ) = ad hece we deduce that (4.6) F (, ) = 7

18 With = i (4.3) we hae u γ ( u) = + logu e u + e d or equialetly, ice ψ ( u) = γ ( u), we obtai the well-ow itegral (4.7) ψ u ( u) = + logu e u + e d Thi itegral, which appear i [38, p.6], ay be eaily erified by differetiatig Biet firt forula for lo g Γ( u) [39, p.49], which we aw aboe i (3.3). We ow coider the ecod deriatie [ ψ ] F (, ) = F (, ) ψ () + F (, )log () = F (, ) ψ ( ) + F (, ) log ψ( ) log + ψ ( ) ad we hae the liit F F F (,) = li (,) ψ () ψ () log li (,) ψ() ψ ( + ) li F (, ) ψ ( ) ψ ( ) log = ψ ( + ) = li F (, ) ψ ( ) ψ ( ) log Sice F(, ) = thi gie u ψ ( + ) = li F (, ) log + ad we therefore obtai (4.8) F (, ) = ( γ + log ) Referrig bac to (4.3) we hae with = ( ) log u () log (, ) u γ u = u u F e d + e 8

19 ad ubtitutig (4.8) thi becoe ( ) log u log ( γ log ) u γ u = u u + e d + e Uig (4.7) we obtai (4.9) u ( u) = logu log u+ ( u) logu e log u + u + e γ ψ γ d ad for u = we hae (4.) γ = γ + γ e log + e d ( ) u ( ) Sice ( u) e log d we hae () e log d ad, i particular, we Γ = ee that Γ () = γ = e log d. Hece we ay write (4.) a (4.) γ γ γ Γ = = + e log e d We ote that e = = + e e e with the reult that e log d = e log d + e log d e e = e log dγ e We the ee that (4..) γ =γ e e log d 9

20 a preiouly deteried by Coppo [7] i 999. Haig obtaied itegral for γ ad γ, we ow coider the geeral cae for γ. We will approach thi by referece to the (expoetial) coplete Bell polyoial, the aliet feature of which are uaried i the attached Appedix. It i well ow that [34] d dx f ( x) f ( x) () () ( ) (4.) e = e Y ( f ( x), f ( x),..., f ( x) ) where the (expoetial) coplete Bell polyoial Y ( x,..., x ) are defied by Y = ad for! x x x (4.3) Y( x,..., x) =... π!!...!!!! ( ) where the u i tae oer all partitio π ( ) of, i.e. oer all et of iteger that = 3 uch For exaple, with which reult i = we ee that the oly poibility i = ad = Y( x ) = x With =, we ee that the poible outcoe are ( = ad = ) ad ( = ad =) which reult i Y ( x, x ) = x + x Suppoe that h ( x) = h( x) g( x) ad let f ( x) = log h( x). We ee that h ( x) f ( x) = = g( x ) hx ( ) ad the uig (4.) aboe we hae d dx log h( x) () ( ) (4.4) hx ( ) = e = hxy ( ) ( gx ( ), g ( x),..., g ( x) ) d dx

21 A a ariatio of (4.4) aboe, uppoe that ( x) = ( x)[ g( x) + α] where α i idepedet of x ad let f ( x) = log ( x ). We ee that ( x) f ( x) = = g( x) + α x ( ) ad ( + ) ( + ) f x g x ( ) = ( ) for ad therefore we obtai d dx log ( x) () ( ) (4.5) ( x) = e = ( x) Y ( g( x) + α, g ( x),..., g ( x )) d dx = ad therefore we obtai We ote fro (4.8) aboe that F (, ) F(, )log [ ψ () ] ( ) () ( ) (4.6) F (, ) = F(, ) Y ( log ψ( ), ψ ( ),..., ψ ( ) ) We aw i (4.3) aboe that (4.7) γ ( ) u ( u) log u log u F (, ) e ( ) = u + e d ad hece we deduce that (4.8) + () ( ) γ F Y( ψ ψ ψ ) e d e = ( ) = (, ) log ( ), ( ),..., ( ) + Pria facie, it i quite rearable that thi liit actually exit coiderig that the digaa fuctio ad all of it deriatie dierge at =. It i how i Appedix A that Y( x+ α, x,..., x) = α Y( x,..., x) = ad we the deterie that d dx (4.9) ( ) ( ) ( ( ), () ( ),..., ( x = x α Y ) g x g x g ( x) ) = Now, referrig bac to (4.5), we ee that with g ( ) = log ψ ( )

22 (4.9.) ( ) (, ) (, ) log ( ( ), () ( ),..., ( F = F Y ) ψ ψ ψ ( ) ) ad thu we hae (4.) γ = ( ) u ( u) = log u log u Y( ) F(, )log e d u = + e = where, for coeiece, we deote Y ( ) a () ( ) ( ψ ψ ψ ) Y ( ) = Y ( ), ( ),..., ( ) ad we ote that Y ( ) i idepedet of the itegratio ariable. We ow adopt a lightly differet approach o a to eliiate all of the apparetly ( troubleoe factor ψ ) () i the liit a. To thi ed we write (4.) i the equialet for F (,) = = Γ() Γ ( + ) ad firt of all we eploy the Leibiz differetiatio forula to obtai d d F (, ) = [ ] = Γ ( + ) We ee that d ( ) d Γ ( + ) = Γ ( + ) ψ + ad applyig (4.4) we deterie that d d Γ ( + ) Γ ( + ) (4.) ( () ( = Y ( ), ( ),..., ) ψ + ψ + ψ ( + ) ) We alo hae [ ] = log + log

23 o that F (, ) = ( log + ( )log ) Y ( + ) ( ) Γ + = where, a before, we deote () ( ) ( ψ ψ ψ ) Y ( + ) = Y ( + ), ( + ),..., ( + ) Whe = thi becoe () ( F(, ) = ( )log Y (), (),..., () = ) ( ψ ψ ψ ) = ( )log Y (), (),..., () = Uig the eleetary bioial idetity [9, p.57] ( ) = () ( ) ( ψ ψ ψ ) thi becoe = log (), (),..., () = () ( ) Y ( ψ ψ ψ ) We therefore coclude that (4.) γ u u u ( ) = log log u () ( ) u + ( ) Y ( ψ(), ψ (),..., ψ () ) log e d = + e ad (4.3) () ( ) γ = ( ) Y ( ψ(), ψ (),..., ψ () ) log e d = + e We ote that 3

24 ( log e d ) =Γ () ad fro (A.6) we hae ( ) Γ () =Y ψ(), ψ (),..., ψ () ( ) () ( ) We will ee i (A.9) that for ( = iplie a alue of ) (4.4) Y ( x,..., x ) Y ( x,..., x ) = δ =, ad hece we ay eliiate the factor of ½ i the itegrad of (4.) to obtai (4.5) γ u u u ( ) = log log u () ( ) u + ( ) Y ( ψ(), ψ (),..., ψ () ) log = e e d (4.6) () ( ) γ = ( ) Y ( ψ(), ψ (),..., ψ () ) log = e e d or equialetly for = Y e d ( ) (), (),..., () log e = () ( ) (4.7) γ ( ψ ψ ψ ) We could alo repreet thi i ter of the partial expoetial Bell polyoial but thi oly ee to add a extra layer of coplexity. With the ubtitutio t = e i (4.7) we obtai (4.8) ( ) () ( ) t γ = ( ) Y ψ(), ψ (),..., ψ () log [log(/ t)] = + dt t log t ( ) () ( ) = ( ) Y ψ(), ψ (),..., ψ () log [log(/ t )] = + + dt t log t 4

25 Sice ( ) log ( log(/ t)) dt =Γ () the ter iolig we ee, i the ae aer a before, that for i the itegrad cacel out ad we coclude that ( ) Y (), (),..., () log [log(/ )] = log t t (4.9) ( ) () ( ) γ = ψ ψ ψ t + dt I hi diertatio, Brede [] howed that there exit a polyoial p( z ) of degree uch that (4.3) γ = p[ log log(/ t)] + dt log t t Brede [] tated, for exaple, that p ( z ) = p ( z) = z γ p z z z ( ) = γ + γ ς() but he did ot pecify the precie for of the geeral polyoial p ( z ). Uig thee polyoial for =,,3 we hae γ = + dt log t t o that γ = log[log(/ t)] + dt γ + log t t log t t dt log[log(/ )] t + dt =γ log γ t t γ = log [log(/ t)] + dt γ log[log(/ t)] log t t + + log t t dt 5

26 o that + [ γ ς()] + dt log t t log [log(/ t)] + dt = γ γ + γ [ γ ς()] γ γ log t t Reidexig to = we ay write (4.9) a γ = ( ) ψ(), ψ (),..., ψ () log [log(/ )] + log ( ) () ( ) Y t dt = t t ad ice = thi becoe after reerig the order of uatio (4.3) () ( ) γ = ( ) Y ( ψ(), ψ (),..., ψ () ) ( log[log(/ t)] ) = + log t t dt Coparig thi with Brede repreetatio (4.3) we are therefore able to pecify the precie for of Brede polyoial (4.3) ( ) ( ) ( (), () (),..., ( p ) z = Y ψ ψ ψ () ) or equialetly = (4.33) ( ) ( ) ( (), () (),..., ( p ) z = Y () ) = With regard to the aboe, we ay ote that ψ ψ ψ z z d d = Y Γ() ( () ( ψ( ), ψ ( ),..., ψ ) ( )) Differetiatig (4.33) gie u p ( z) = ( ) ( ) Y ( ψ(), ψ (),..., ψ () ) z = () ( ) 6

27 = ( ) ( ) Y ( ψ(), ψ (),..., ψ () ) z = () ( ) ( ) = Y (), (),..., () = ( ψ ψ ψ ) () ( ) ad we therefore ee that (4.34) p ( z) = p ( z) Sice p ( z) = p ( z) we ee that p( z ) i a Appell polyoial ad therefore we hae the relatio p( z) = p() z = p( x+ y) = p( x) y = Thi cocur with Brede reult [] z z x = p ( xlog z) e dz We alo ote that p ( z) = ( ) Y (), (),..., () = ( ψ ψ ψ ) () ( ) z Haig expeded oe eergy gettig to (4.), it wa oewhat diappoitig to ubequetly dicoer that thi reult could hae bee deried i a ore uccict aer uig (3.8). Subtractig a factor of fro both ide of (3.8), we ay write that equatio i the followig for (4.5) u u u Γ ς (, u) = + + e + d ( ) e 7

28 which we the differetiate ad ealuate at = thi tie to obtai ( ) ( ) u (, ) log () ς u u f e d = + + ( ) + = = u Γ e where, a a ueful artifice, we hae deoted f () a u f() = We ca repreet f () by the followig itegral o that u f () = = u x dx u ( ) f () =() x log xdx ad thu we ee that f u ( ) log x () =( ) x dx =( ) Hece ubtitutig (.9) + log u + ς(, u) = () γ() u = we obtai + ( ) log u u γ u u e u + Γ( ) e d = ( ) ( ) = log ( ) + + Referrig to the defiitio (4.) of F (, ) we ee that u u e e d F(, ) d ( ) + = + = = Γ e e 8

29 ad uig (4.9.) thi becoe () ( ) Y u ( ( ), ( ),..., ( )) e = ψ ψ ψ log d = + e Hece we obtai (4.5) γ ( u ) log log u u + = u + () ( ) ( ) Y u ( ( ), ( ),..., ( )) e + ψ ψ ψ log d = + e which correpod with (4.). Writig (4.5) a (4.5.) u u u ς (, u) Γ () = e + d e ad, uig the Leibiz forula to differetiate thi, we obtai a ierio forula (4.5) d + log u ( ) u ( ) γ ( u) log u+ Γ () = e log + = u + e ad with u = we hae (4.53) Γ Γ = + e d ( ) ( ) ( ) γ () () e log = or equialetly (4.54) Γ = d e ( ) ( ) γ () e log = Differetiatig (4.5.) with repect to u gie u u u ς ( +, u) + + u Γ ( ) = e d + e which ay be writte a 9

30 ς ( + ) Γ ( + ) = e + e d ad differetiatig thi will alo reult i (4.5) (4.55) + log u ( ) u ( ) γ ( u) log u+ Γ () = e log + = u + e d Referrig bac to the ethod origially eployed by Stielte, i particular (3.), we hae ( u+ ) e e f( u) = log d log ( u + ) u + ( ) ( ) = ( + ) = ( ) Γ () = f u F u ad ice f () = we hae by itegratio o that f f u + + ( ) ( u) = ( ) Γ () = ( ) ( u ) = ( ) Γ () = + log ( ) + log u + u + e e ( ) log u log d= ( ) Γ () = + Referrig bac to (4.5) d + log u ( ) u ( ) γ ( u) log u+ Γ () = e log + = u + e we ee that (4.56) = γ u u u ( ) ( ) ( ) log Γ () 3

31 u u e e = e log + d log e d ad with u = u e = e + e we coe bac to (4.53) aboe. log d Γ Γ = + e d ( ) ( ) ( ) γ () () e log = With =,, we obtai γ = e d + e γ γ + γ = e log d + e γ [ γ + ς()] + γγ + γ = e log + e d The approxiate alue of the firt three Stielte cotat are [5] γ =.577 γ =.78 γ =.96 ad iertig thee alue ito the aboe three equatio uerically deotrate that the correpodig three itegral hae poitie alue. We ow wih to how that the itegral I, which i defied below, i trictly poitie for all poitie iteger ad for = We ee that I = log e + d e log log I = e + d+ e + d e e 3

32 J K = + With the ubtitutio t e we obtai for the firt copoet = J / e log e + d= log (log(/ t) dt e + + / t logt ad with the ubtitutio t = / u thi becoe (4.56) / e du log (log(/ t) + + dt log (log u) / t logt = + u logu e u It i clear that log (log u) for all u e ad we ow coider the other part of the itegrad i (4.56). Let φ ( u) = u logu + ( u+ )loguu+ = ( u ) logu ad we ote that φ () e = >. The deoiator of φ( u) i poitie for all u e e ad we ow coider the uerator hu ( ) = ( u+ )logu u+ We ote that he () = 3e > ad we hae the deriatie h ( u) = logu+ u ad h () e = e >. We ee that u h ( u) = u ad therefore h ( u) > for u >. Thi eable u to coclude that h ( u) i ootoic icreaig for u > e. Sice h ( e) i poitie we the deduce that hu ( ) i alo ootoic icreaig for u > e. Accordigly, hu ( ) for all u e. Fially, we hae φ( u) for all u e. Hece, ice the itegrad i poitie, we coclude that J i poitie. 3

33 We ow coider K ad we wih to deterie the ig of f () where f() = e + + ( )( e ) = ( e ) The deoiator of f ( ) i poitie for all ad we ow coider the uerator We ote that g ( ) = + ( )( e ) g () = 3e g ( ) = ( ) e + > ad we hae the deriatie o that g ( ) > for. Accordigly, g ( ) for all. Fially, we hae f( ) for all. Hece, ice the itegrad i poitie, we coclude that i poitie. K It ha therefore bee deotrated that all poitie iteger ad for =. I = log e + d e i poitie for Haig regard to (3.7.) we eaily ee that li + = e ; alteratiely, thi ay be eaily deteried uig L Hôpital rule. It i poible that uch repreetatio for the Stielte cotat ay be ueful i coectio with the proof of the Riea Hypothei ia the Li/Keiper cotat (ee for exaple [6]). Thi i becaue we ote fro (4.53) that (4.57) Γ Γ = + e d ( ) ( ) ( ) γ () () e log = > ad referece to the Appedix the how that if i ee the thu ( ) (4.58) ( ) γ Γ () > = ( Γ ) () i poitie ad 33

34 Whilt ot pecifically eployed i thi paper, I accidetally cae acro the followig lea whilt I wa tryig to proe that I wa poitie Suppoe that f( t) ad f ( t) for all t [, ). We ee that I ()log = f t tdt dt = f ()log t tdt+ f()log t t ad with the ubtitutio t =/ y we hae f ( t) log t dt = ( ) f (/ y) y log y dy We therefore obtai ( ) ( ) I = f t + t f(/ t) log tdt ad it i clear that the itegrad i egatie i the iteral [, ) i the cae where i a ee iteger. We ow coider the cae where i a odd iteger ad we wat to proe that the followig itegral i alo egatie () (/ ) log + I + = f t t f t tdt A a coequece of the Mea Value Theore of calculu we hae where α i uch that t >α > ( ) f() t f(/ t) = t t f ( α) t. We ee that f t t f t f t f t f t t f t ( ) (/ ) = ( ) (/ ) + (/ ) (/ ) ( ) = f ( t) f(/ t) + f(/ t) t = t f ( α) + f(/ t) t t ( ) 34

35 [ tf ( α) f(/ t) ]( t ) = + ad thi i clearly egatie i the iteral (, ). We therefore coclude that egatie. I + i alo We could alo expre I a ( ) ( ) I = f t + t f(/ t) log tdt ad hece I will alo be egatie if f( t) ad f ( t) for all t (,]. 5. A applicatio of the alteratig Hurwitz zeta fuctio The alteratig Hurwitz zeta fuctio ς ( x, ) i defied by a ς (, x) = a = ( ) ( + x) Upo a eparatio of ter accordig to the parity of we ee that for Re( ) > ( ) ς (, x) = = + a = ( + x) = ( + x) = (+ x) = = ( + x/) = ( + ( x+ )/) ad we therefore ee that ς ( t a, ) i related to the Hurwitz zeta fuctio by the wellow forula [3] (5.) x + x ςa (, x) = ς, ς, Differetiatio gie u x x a (, x),, x ς ς ς + = ς (, ) ( ) [ (, ) ] a x = + f x + x = = 35

36 ad uig the Leibiz forula we obtai = + = ( ) ( ) log ( ) f (, x ) = where x + x f(, x) = ς +, ς +, We hae ( ) ( ) x ( ) + x f (, x) = ς, ς, ad therefore x + x = ( ) γ γ (5.) x + x (, ) ( ) ( ) log a x x ς + γ + = + γ + = = We ow refer to the Hae idetity for the alteratig Hurwitz zeta fuctio (ee equatio (4.4.79) i []) i i ( ) (5.3) ς (, x) = + i= = ( x + ) Differetiatio gie u a i i i ( ) ς (, x) = x x+ ) a i+ + i= = ( i ( ) log ( x + ) x x+ + i + ς (, ) () ( ) a x = + + i+ i = = = ad equatig thi with (5.) gie u (5.4) i x + x i ( ) log ( x+ ) log γ γ i+ = = i= = x+ We hae 36

37 ς ( ) log ( x + i) i ( ) log ( x+ ) ( x i) ( x ) i ( ) i a (, x) = () = () i+ i= + i= = + ( ad hece we ay obtai expreio for ς ) (, x) i ter of the Stielte cotat a (5.5) ς ( ) a x + x (, x) = log γ γ = With x = ad = i (5.4) we obtai i i ( ) () i γ γ + = i= = + which gie u i i ( ) log = i+ i= = + Thi i equialet to With x = ad = li ς ( ) = log. a i (5.4) we get or i i ( ) log( + ) γ γ() log γ γ i+ + = i= = + i ( ) log(+ ) + i log + γ γ = i+ i= = We hae the relatiohip [3] (5.6) p ( ) ( ) log q p+ p γ r p log q p p q q p r q γ = + + p γ = + = q ad with q = thi becoe (5.7) p+ p γ p log p p p ( ) ( ) p log γ = + + p γ + = ad i particular we hae 37

38 (5.8) γ = log log γ + γ γ We the obtai i i ( ) log( + ) (5.9) γ = log i+ log i= = + Thi expreio for Euler cotat wa origially deried by Coffey [7] i 6 (a differet deriatio i cotaied i equatio (4.4.6g) i [])). Siilarly we ay alo obtai with = γ i i i i log( + ) log ( + ) = log + ( ) ( ) i+ i+ i= = + log i= = + which wa alo preiouly deteried by Coffey [7] i a differet aer. Lettig x = ad x = i (5.4) reult i i i ( ) log ( + ) log γ γ i+ = = i= = + i 3 i ( ) log ( + ) log γ γ i+ = = i= = + We ote fro [38, p.89] that for N ς(, + x) = ς(, x) = ( + x) ad thu + log ( + x) [ ς(, + x) ς( x, )] = ( ) ( + x) = Therefore, referrig to (.3) we obtai γ ( + x) γ ( x) = = log ( + x) + x 38

39 ad i particular we hae log γ( + x) γ( x) = x x Therefore we hae 3 γ γ = + ( ) log APPENDIX Soe apect of the (expoetial) coplete Bell polyoial It i well ow that [33] d dx f ( x) f ( x) () () ( ) (A.) e = e Y ( f ( x), f ( x),..., f ( x) ) where the (expoetial) coplete Bell polyoial ad for Y ( x,..., x ) are defied by Y = (A.)! x x x Y( x,..., x) =...!!...!!!! π ( ) where the u i tae oer all partitio π ( ) of, i.e. oer all et of iteger that = uch For exaple, with which reult i = we ee that the oly poibility i = ad = Y( x ) = x With =, we ee that the poible outcoe are ( = ad = ) ad ( = ad =) which reult i Y ( x, x ) = x + x Suppoe that h ( x) = h( x) g( x) ad let f ( x) = log h( x). We ee that 39

40 h ( x) f ( x) = = g( x ) hx ( ) ad the uig (A.) aboe we hae d dx log h( x) () ( ) (A.3) hx ( ) = e = hxy ( ) ( gx ( ), g ( x),..., g ( x) ) d dx A a exaple, lettig hx ( ) =Γ( x) i (A.3) we obtai d dx log Γ( x) ( ) () ( ) (A.4) e =Γ ( x) =Γ( x) Y ( ψ( x), ψ ( x),..., ψ ( x )) = x t t e log tdt ad ice [6, p.] (A.5) ψ ( x) = ( ) p! ς ( p+, x) ( p) p+ ( we ay expre Γ ) ( x) i ter of ψ ( x) ad the Hurwitz zeta fuctio. I particular, Kölbig [33] oted that (A.6) () Y (, x,..., x ) ( ) Γ = γ p+ ( where xp = ( ) p! ς ( p+ ). Value of Γ ) () for are reported i [38, p.65] ad the firt three are Γ () () = γ () Γ () = ς () + γ (3) 3 Γ () = [ ς (3) + 3 γς () + γ ] ( A how i [], we ote that Γ ) ( x) ha the ae ig a ( ) for all x (, α] where α i the uique poitie root of ψ ( x) = (Gau deteried that α ). Thi wa alo reported a a exercie i Apotol boo [3, p.33] for the particular cae of ( Γ ) (). We hae fro (A.5) ad (4.) 4

41 d dx Γ ( + x) Γ ( + x) (A.6.) = Y ( ψ( + x),! ς(, + x),..., ( ) ( )! ς(, + x) ) o that with x = d dx Γ ( + x) (A.6.) = Y ( γ,! ς(),..., ( ) ( )! ς( ) ) x= The coplete Bell polyoial hae iteger coefficiet ad the firt ix are et out below [9, p.37] (A.7) Y( x) = x Y ( x, x ) = x + x Y ( x, x, x ) = x + 3x x + x Y ( x, x, x, x ) = x + 6x x + 4x x + 3x + x Y ( x, x, x, x, x ) = x + x x + x x + 5x x + 5x x + x x + x Y ( x, x, x, x, x, x ) = x + 6x x + 5x x + x + 5x x + 5x + 6x x x x x + 45x x + 5x x + x The coplete Bell polyoial are alo gie by the expoetial geeratig fuctio (Cotet [9, p.34]) t t (A.8) exp x = Y( x,..., x ) =! =! We ee that d dt t exp (,..., ) x = Y x x =! t= We ote that t t t t Y ( ax,..., ax ) = exp ax = exp a x = exp x =! =! =! =! a 4

42 ad thu we hae a t t Y( x,..., x) = Y( ax,..., ax) =! =! We ee with a = that t t = Y ( x,..., x ) Y ( x,..., x )!! = = ad uig the Cauchy product forula thi becoe t = Y( x,..., x) Y( x,..., x ) = =! Hece we deduce that for (A.9) Y ( x,..., x ) Y ( x,..., x ) = = Fro the defiitio of the (expoetial) coplete Bell polyoial we hae Y ( ax, a x,..., a x ) a Y ( x,..., x ) = r ad thu with a = we hae Y ( x, x,...,( ) x ) = ( ) Y ( x,..., x ) We alo ote that Y ( x, x,...,( ) x ) = ( ) Y ( x,..., x ) + but o dicerible ig patter eerge here. We hae the recurrece relatio [35] Y ( x,..., x ) = Y ( x,..., x ) x = Y ( x,..., x ) x = = = x + Y x x + (,..., ) x + = 4

43 We ote that t t t Y( x+ y,..., x + y) = exp ( x + y) = exp x exp y =! =! =! t t = Y ( x,..., x ) Y ( y,..., y )!! = = ad, a before, we apply the Cauchy erie product forula to obtai (A.) Y( x+ y,..., x + y) = Y( x,..., x) Y( y,..., y) = ad we ote that Y ( α,,...,) = α With y = x we obtai fro (A.) (A.) Y(,...,) = = Y( x,..., x) Y( x,..., x) = a i (A.9) aboe. REFERENCES [] V.S.Adachi, A Cla of Logarithic Itegral. Proceedig of the 997 Iteratioal Sypoiu o Sybolic ad Algebraic Coputatio. ACM, Acadeic Pre, -8,. [] V.S. Adachi, O the Hurwitz fuctio for ratioal arguet. Applied Matheatic ad Coputatio, 87 (7) 3-. [3] T.M. Apotol, Matheatical Aalyi. Secod Ed., Addio-Weley Publihig Copay, Melo Par (Califoria), Lodo ad Do Mill (Otario), 974. [4] T.M. Apotol, Itroductio to Aalytic Nuber Theory. Spriger-Verlag, New Yor, Heidelberg ad Berli, 976. [5] T.M. Apotol, Forula for Higher Deriatie of the Riea Zeta Fuctio. Math. of Cop., 69, 3-3,

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