A Class of Logarithmic Integrals. Victor Adamchik. Wolfram Research Inc. 100 Trade Center Dr. April 10, 1997

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1 A Class of Logarithmic Itegrals Victor Adamchik Wolfram Research Ic. Trade Ceter Dr. Champaig IL 68 USA April 997 Abstract. A class of deite itegrals ivolvig cyclotomic polyomials ad ested logarithms is cosidered. The results are give i terms of derivatives of the Hurwitz Zeta fuctio. Some special cases for which such derivatives ca be epressed i closed form are also cosidered. The itegratio procedure is implemeted i Mathematica V.. Itroductio The aim of the paper is to develop a approach for evaluatig a class of itegrals ivolved cyclotomic polyomials ad the ested logarithms log log. This class of itegrals arose from the research regardig the Potts model o the triagular lattice (see [] []). The Potts model ecompasses a umber of problems i statistical physics ad lattice theory. It geeralizes the Isig model so that each spi ca have more tha two values. It icludes the ice-verte ad bod percolatio models as special cases. It is also related to graph-colorig problems. Bater Temperley ad Ashley (see []) derived the followig geeratig fuctio for the Potts model o the triagular lattice: sih(( y)) sih ( y P (y) ) () sih() cosh(y) Performig a logarithmic substitutio the itegral ca be rewritte i the "algebraic" form: z y ( z y )(z z y ) P (y) ( z )( + z y ) log(z) dz Although it is't kow whether the fuctio P (y) has a closed-form epressio for all values of y it ca be evaluated eplicitly for ay y that is a ratioal multiple of. Let y p where p ad are positive itegers the z p ( z p )(z z p ) P (y) ( + z p )( z ) log(z) dz

2 This itegral belogs to the more commo class of itegrals R(z) log(z) dz where R(z) is a ratioal fuctio. We assume that the itegral is coverget. The above itegral ca be evisaged i a alterative form. Performig a itegratio by parts we obtai Z Q(z) log log dz () where Q(z) is a ratioal fuctio. It is ot kow whether the above itegral is doable for ayq(z). However if the deomiator of Q(z) is a cyclotomic polyomial the the itegral ca be always epressed i terms of derivatives of the Hurwitz Zeta fuctio. Usig the Graee procedure for determiig if a give polyomial is cyclotomic (see []) ad the covertig a cyclotomic polyomial to the form allows us to reduce the problem of itegratio of () to the followig two classes of itegrals: p log log ( + ) ad z p log log assumig that p ad provide the covergece of the itegrals. A few such itegrals (with p ad ; ) ca be foud i Gradzhtey ad Ryzhyk's hadbook (see [] pp. 7-7) ad i [6]. Derivatives of the Hurwitz Zeta Fuctio It is well-kow (see [7]) that (s; z) s (z) log p However if the rst argumet of (s; z) is ot zero o eact formulas were developed. I this sectio we cosider the dierece of derivatives of the Zeta fuctios s; p s; p where (s; z) for ease of otatio deotes (s; z) ad p ad are positive itegers ad show that () ca be represeted i ite terms of other fuctios. Throughout the paper we will freely use the otatio for the limit of () whe s!. ; p ; p () ()

3 Propositio Let p ad be positive itegers ad p< the ; p ; p p X cot (log()+) j j jp log si () Proof. The idetity () follows straightforwardly from Rademacher's formula (see [8]): X z; p ( z)() z j z; (6) j z si + jp by dieretiatig it with respect to z ad the settig z to. We have ; p (log()+) X j ; p Takig ito accout () alog with X j jp si ; j (;z) z X j j si jp jp si cot p X j jp si ; j ; p< we arrive at the idetity (). QED. The propositio was rst proved by G. Almkvist ad A. Meurma [9]. Let us cosider several particular cases: ; 6 ; ; ; + log() + log() log ; ( log() + 8 log() log (( ))) p ; 6 (6 + log(98) log (( )) + log (( ))) p (7) (8) (9) ; ; log()( + log() + log())+ 6 () p + log + 8 log() log

4 Propositio Let be apositive iteger ad << the (; )+() (; ) i B +() + i + e! () Li +(e i ) () Proof. From Lerch's trasformatio formula for the fuctio (z; s; v) (see [7]): e is e iv ; s; log(z) (z; s; v) iz v () s ( s) i with v s s ad z e i we obtai (s; )+e is (s; ) e is e i( s +v) e iv ; s; log(z) i () s (s) Li s(e i ) () where we assume that < < ad s is real. Dieretiatig the fuctioal euatio () with respect to s settig s to where is a positive iteger ad makig use of () lim! () ()+! (; ) B +() + where B + () deotes the Beroulli polyomials we complete the proof. QED. The idetity () ca be rewritte i the alterative form by meas of the Clause fuctio that is deed by Cl (z) Hece we have the followig <(Li (e iz )); is odd (Li (e iz )); is eve Corollary Let be apositive iteger ad << the (; )+() (; ) ()b c! () Cl + () () where bc is the oor fuctio. The followig idetities pop up immediately from (): ; ; ; G + ; () 6

5 where G is Catala's costat (see [7] ad []). Moreover usig the multiplicatio property of the Zeta fuctio (s; kz) k s k X i s; z + i k ad the Propositio (or Rademacher's formula (6)) oe ca easily deduce that ; 6 log() ) p + ( 8 p + 6 () () ; ; ; G 8 () () log() 7 8 p + ( ) p () (6) log() ; () (7) log() p ( ) p () (8) G 8 () (9) ; ; 6 log() + ) p ( 8 p + 6 () () Remark. The special case of () whe was also obtaied by W. Gosper (see []). Remark. Notice that () is related to Glaisher's costat A (see []) as Itegrals () log(a) Propositio Let <(p) > ad <() > the p log log + + log() ( ( p ) + ( p )) + p ( (; ) (; + p )) () where (s; z) deotes (s; z).

6 Proof. We shall proceed with the well-kow idetity X p + k () k k (k + p) Dieretiatig both sides of () times with respect to p we obtai p + log () X k () k () k (k + p) () Dieretiatig both sides of the ew idetity () with respect to ad the settig we d that p + log log p + X k () k log(k + p) k (k + p) Now we cosider the limitig case!. It is clear that the iite sum i the righthad side of () is diverget whe. However the sum ca be aalytically cotiued to the whole comple plae of the parameter by meas of the Lerch fuctio. We have X k () k log(k + p) k (k + p) lim s! Hece whe teds to we obtai sice Thus lim X! k p log log + () k log(k + p) k (k + p) X k lim () k k (k + p) s s! ( ;s; p )s (s; p p lim s! ( ;s; p ) s (s; p ) (s; + p )! () s ) (s; + p ) + lim + s! (s; p ) (s; + p )! () s Performig further evaluatios ad takig ito accout the asymptotic epasio we ally arrive at (). QED. (s; )! ()+(s);s! () Note if p i terms of elemetary fuctios. Let p r r N the is a positive iteger the the right-had side of () ca be epressed r log log + r (log()+) ( ) r + + (; r ) ; r + () 6

7 Now makig use of the followig reductio formulas (s; v) log(v ) (s; v ) + (v ) s ad the idetity (s) (s ) + s ; (; ) (log() + ) log() log () it is easy to see that the right-had side of () ca be trasformed to the combiatio of logarithmic fuctios. Thus Corollary If p ad <() > the log log log() log( ) + (6) Corollary If p ad <() > the log log log () + (log() ) log() + (7). Here we cosider several particular itegrals. log log + From () with p ad we obtai log log + + log() By meas of (7) we have + ; ; p log log ( + log )! ( ) (8). log log + 7

8 Takig p ad i () we have log log + 8 (log(8) + ) + ; ; 8 Usig the idetity (7) we obtai p log log ( + log )! ( ). log log + From () with p ad we obtai log log + 6 (log(6) + ) + 6 ; ; 6 6 Performig further simplicatios ad usig the formula () we d that log log +. log() 6 log + log log ( log() 8 log() + log (( ))) 6 p We observe that a give itegral ca be rewritte as + log log log log + + Applyig Propositio twice we obtai + log log ; ; 6 + ; 6 log log + (log(6) + ) ; p 6 The takig ito accout formulas (8) ad (9) we ally d that + log log ( log() 6 log (( ))) 6 p (9) () () 8

9 Propositio Let <(p) > ad <() the p log log p p + (log()+)( ( ) ( ))+ (; p ) (; p + () ) where (s; z) deotes (s; z): If the the formula () simplies to Proof. We observe that sice p p log log log(p)+ X k p X k (k + p) X p k k k (k + p) k (k + p +) Dieretiatig both sides of () times with respect to p we obtai p log () X k k (k + p) () X k k (k + p +) Dieretiatig () with respect to ad settig we d that p log log p X k log(k + p) k (k + p) + X k log(k + p +) k (k + p +) () () () (6) The iite sums i the right side of (6) ca be represeted i terms of the Lerch fuctio. We have X k log(k + p) k (k + p) lim s! Therefore settig we obtai X k log(k + p +) k + p + log() ( p X k k (k + p) s log(k + p) lim k + p ) p + s! + (; p lim s! (s; p+ ) s ) ; p + ( ;s; p ) s (s; p )! s 9

10 sice (;s; p )(s; p ) ad Thus (s; )! ()+(s);s! p log log p log() ( ) p + Evaluatig the elemetary itegral we complete the proof. QED.. Sice p ( ) Here are some particular cases. + + log log + + log log the from () with p ad wehave p ( )+ + (; p ) ; p + ( p + ) ( p ) log log + + log log (log() + ) p + (; ) (; ) Hece by virtue of (8).6 I view of + + log log ( log() + 8 log() log (( ))) 6 p log log ( ) log log log log from () with p ad it follows that log log (log() + ) + ; ;

11 Applyig Propositio we obtai log log cot ( )! log() log ( ) si ( )+log ( )! ( )! (7) si Propositio Let <(p) > ad <() > the p ( + ) log log + log() log ( p where (s; z) deotes (s; z) If p the ( p)(log()+)( ( p ) ( +p )!! + ( + ) log log ) ( +p )) ( p) ( p (; ) (; + p )) (8) + log Proof. Dieretiatig both sides of () with respect to we obtai p ( + ) log log p ( + ) X k () k (k + ) log(k + p) k (k + p) We epress the iite sum i the left-had side of () by meas of the Lerch fuctio X k lim s! () k (k + ) log(k + p) k (k + p) s ( p ) ;s; p lim s! X k () k (k +) k (k + p) s ;s ; p Therefore the limitig value of the sum whe approaches is lim X! k () k (k + ) log(k + p) k (k + p) log() ( p) log() + ( ( + p ) ( p ))+ ( (; + p ) (; p )) + p ( (; + p ) (; p )) (9) ()

12 The takig ito accout the idetity () ad we arrive at (8). QED. p ( + ) + ( p) ( ( + p ) ( p )) Here are a few ice-lookig itegrals that follow immediately from the above propositio p ( + ) log log log() + log ( )! ( ) + log p ( )! () ( ) p ( + ) log log 6 p 6( log )! ( ) ( + ) log log log 6p! + 6 log ( ) ( )! () p + log() 6 log 7 6 () Propositio 6 Let <(p) > ad <() > the p ( + ) log log If p the ( p)(log()+) p + log 8 ( p)( p)(log()+) ( ( ; p ) ( ; + p ))+ ( p )! ( +p ) + ( ( p ) ( + p ))+ ( p)( p) ( (; p ) (; + p )) () ( + ) log log log() 9 log() + 6 log() 9 6 () ()

13 Proof. The proof is similar to that for Propositio. QED. The followig itegrals follow immediately from (): p ( + ) log log G + 8 log p ( )! ( ) (6) ( + ) log log 6 log() + log()+ 8G 8 + ( 8) log ( ) ( )!! (7) Z ( 6 + ) log log ( + ) sech(z) tah(z) dz G (8) Let us cosider the particular case of the geeratig fuctio for the Potts model o the triagular lattice whe y. Other special cases of the itegral () are described i []. From () we have Z P ( ) tah() (9) ( cosh( )) Origially the itegral was calculated by R.J. Bater (see []) by usig the Fourier aalysis ad the residue theorem. Performig a epoetial substitutio ad the itegratig a correspodet itegral oe time by parts we trasform (9) to P ( )6 Fially applyig Propositio 6 we d that ( + ) ( + ) ( + ) log log( ) z P ( ) p log() + p 6 ( 6 ) Ackowledgemet. I'd like to thak G. Almkvist R. Bater ad A. Meurma for helpful discussios ad commets ad also the referee who poited out a dieret approach to evaluatig itegrals (). Refereces [] R. M. Zi S. R. Fich ad V. Adamchik Number of clusters i D percolatio:values ite-size correctios uctuatios ad eplicit evaluatio of eact results Phys. Rev. Letters (submitted for publicatio).

14 [] V. Adamchik S. R. Fich ad R. M. Zi The Potts model o the triagular lattice i Web site [] R. J. Bater H. N. V. Temperley ad S. E. Ashley Triagular Potts model at its trasitio temperature ad related models Proc. Royal Soc. Lodo A 8 (978) -9. [] R. J. Bradford ad J. H. Daveport Eective Tests for Cyclotomic Polyomials Proceedigs of ISSAC '88 p.-. [] I. S. Gradzhtey ad I. M. Ryzhyk Table of Itegrals Series ad Products Academic Press New York 98. [6] I. Vardi Itegrals a Itroductio to Aalytic Number Theory Amer. Math. Mothly 9(988). [7] H. Batema ad A. Erdelyi Higher Trascedetal Fuctios Vol. McGraw- Hill 9. [8] T. M. Apostol Itroductio to Aalytic Number Theory Spriger-Verlag 976. [9] G. Almkvist ad A. Meurma private commuicatio. [] V. Adamchik Itegral ad Series Represetatios for Catala's Costat i Web site [] R. W. Gosper: R m 6 log (z)dz Amer. Math. Soc. (997). [] S. Fich Glaisher-Kikeli Costat i Web site