A Faster Product for π and a New Integral for ln π 2

Transcription

1 A Fater Product for ad a New Itegral for l Joatha Sodow. INTRODUCTION. I [5] we derived a ifiite product repreetatio of e γ, where γ i Euler cotat: e γ = Here the th factor i the ( + )th root of the product ( + ) ( ) = I the proce we oticed a triigly imilar product repreetatio of : = () 4 6. () 3 5 I thi ote we give three proof of (). The third lead to a aalog for l( ) of itegral for l( 4 ) [4] ad γ [3], [4], [5]: l = [, ] x, ( + x)lx dx () 3 4 x l = ( + xy)lxy [, ] dx dy, ( 4) γ = [, ] x ( xy)lxy dx dy. Uig (3), we etch a derivatio of () ad () from the ame fuctio (a form of the polylogarithm [7]), accoutig for the reemblace betwee the two product. The fuctio alo lead to a product for e (due to J. Guillera [5]), e = , (5)

2 urpriigly cloe to product () for e γ.. THE ALTERNATING ZETA FUNCTION. The logarithm of product (), amely, + γ = + + = ( ) l( ), (6) = remided u of the erie (ee [6] ad []) ζ * () = ( ) + ( ) + = ( C), (7) which give the aalytic cotiuatio of the alteratig zeta fuctio ζ * ( ). The latter i defied by the Dirichlet erie (ee []) ζ * ( ) () = = ( R () > ). (8) (For example, uig the claic formula ζ * () = l for the alteratig harmoic erie for a ew proof ee [] oe ca derive the erie l = ( ) from (7) by coiderig it whe =.) Differetiatig (7) termwie ad ubtitutig the value of the derivative of ζ * at =, (ee []), yield the erie ζ * '( ) = l (9) + l = ( ) l( ) +, () ad expoetiatio produce product (). = 3. WALLIS'S PRODUCT AND EULER'S TRANSFORM. The pair of ifiite product () ad () call to mid aother pair, Walli' product for [7] ad Pippeger' product for e []:

3 =, ( ) e = ( ) It i iteretig to ote that product () ad (), whoe factor have expoet, coverge rapidly to umber ad e whoe irratioality ha bee proved (ee, for example, [9]), wherea product (), with expoet ( + ), coverge le rapidly to a umber e γ whoe (expected) irratioality ha ot yet bee proved. We give a ecod proof of (), uig () ad Euler' traformatio of erie ( ) a = ( ) + + a, (3) valid for ay coverget erie of complex umber [7, ec. 33B], []. Applyig (3) to the logarithm of Walli' product = + l = ( ) l (4) give l = ( ) l + +. (5) + = If we replace by, write the lat logarithm a l( + ) l( + ), ad the um o a the differece of two um i the firt of which we replace by, the the recurio + = lead to (), completig the ecod proof of (). The firt proof i baically the ame, becaue i [] we ue Walli' product to evaluate (9), ad we tae the Euler traform of (8) to get (7) for complex with R () >. Product () ad () are lied by Stirlig' aymptotic formula!~( e) : the formula i proved i [] uig () ad i ued i [] to etablih (). Product () ad () are lied by traformatio: a hypergeometric oe [5] for () ad Euler' for (). (To tregthe the li, we ca write erie (4) ad (5) a itegral of hypergeometric fuctio compare [5, Proof ] ad the obtai (5) from (4) by a hypergeometric traformatio equivalet to (3).) However, thi li doe ot explai the remarable reemblace betwee () ad (). Euler' traformatio accelerate the rate of covergece of a lowly covergig erie uch a (4) (ee [7, ec. 35B]). Thu, product () coverge fater tha product (), a Figure how.

4 4 Figure. Partial product of Walli' product ad it Euler traform 4. AVOIDING EULER. A third proof of () (due i part to S. Zlobi [8]) avoid uig Euler' traformatio altogether (compare the proof avoidig hypergeometric fuctio i [5]). We how that + y x I: = x dx dy = l. (6) Thi implie (), becaue if we factor ( x ) from the itegrad ad ue the biomial theorem, the termwie itegratio (jutified ice the itegrad i majorized by the erie ) yield (5) ad, therefore, (). To prove (6), we ue the geometric erie ummatio + = + = x x ( x) x x = ( x) x (7) to write y+ I = ( x) x dxdy. The itegrad i majorized by ( + ) (becaue max ( ) = x x x + + < ( + ) ad x y ), o we may perform the itegratio term by term, which by ivoig () give

5 5 ( + ) 4 6 I = l = l = l. ( + )( + 3) Thi prove (6) ad complete the third proof of (). Proof of (3). Equatio (6) ad the firt equality i (7) yield y x x dx dy = l. + x Reverig the order of itegratio (permitted ice the itegrad i oegative), we itegrate with repect to y ad arrive at formula (3). Alteratively, oe ca derive (3) from (4) by maig the chage of variable u = xy, v = x ad itegratig with repect to v: the reult i l miu itegral (3) (with u i place of x), ad equality (3) follow. 5. RELATING THE PRODUCTS FOR AND e γ. Recall that we derived product () for from the alteratig zeta fuctio ζ * ( ). Omittig detail, we etch a derivatio of product () for e γ from a geeralizatio of ζ * ( ). Thi accout for the reemblace betwee the two product. (Formula (8) ad (9) are due to J. Guillera [5].) We geeralize erie (7) for ζ * ( ) by defiig the fuctio + f t t (, ) = ( ) ( ) + ( < t <, C), (8) * = o that f(, ) = ζ ( ). Uig itegral (6) but replacig ( x ) with t( x), we ca how that the formula obtaied from (3) ad (9), exted to ζ * '( ) = f'( t, ) = t x, ( + x)lx dx x, ( t( x)) l x dx ( 9) where the prime ' i horthad for. We ow derive product () by evaluatig the itegral t f '( t, ) dt i two differet way. O the oe had, a glace at (8) reveal that thi itegral equal the right ide of (6). O the other had, ubtitutig (9) ito the itegral ad reverig the order of itegratio give f'( t, ) t( x) dt t ( t( x)) l x dtdx = l x x dx = +.

6 6 The lat i a claical itegral for Euler' cotat [, ec..3], [5], ad (6) follow, implyig (). Other product ca be derived i the ame way. For example, expoetiatig the itegral t f '( t, ) dt = give product (5) for e, which coverge more lowly tha Pippeger' product for e, becaue of the expoet i (5), veru i (). I order to idetify the fuctio f(, t ), we revere the order of ummatio i (8) ad um the reultig erie o. We the replace with, obtaiig f(, t ) = ( ) ( + ) t t ( ) + = t = = ( ) () for t atifyig < t ad for uitable. Therefore, f(, t ) ad ζ * ( ) are related to the fuctio Ft (, )= = t ( t <, R () > ) by the formula f(, t ) = F( t ( t ), ), ( ) * ζ () = F(,), for appropriate t ad. (With t =, equatio (8) ad () verify that formula (7) ad (8) for ζ * ( ) agree.) The fuctio Ft (, ), a pecial cae of the Lerch zeta fuctio Φ( zv,,)(ee [3, ec..], [6, Sec. 64]), i the polylogarithm Li ( t ) whe i a iteger [7, p. 89], [6, ec. 5, 64]. Relatio (8) ad () lead to a aalytic cotiuatio of Ft (, ), ad thu of the polylogarithm.

7 7 REFERENCES. G. Boro ad V. Moll, Irreitible Itegral: Symbolic, Aalyi, ad Experimet i the Evaluatio of Itegral, Cambridge Uiverity Pre, Cambridge, 4.. A. J. Colema, A imple proof of Stirlig' formula, Amer. Math. Mothly 58 (95) A. Erdélyi et al., Higher Tracedetal Fuctio, The Batema Maucript Project, vol., McGraw-Hill, New Yor, R. Graham, D. Kuth, O. Patahi, Cocrete Mathematic, d ed., Addio- Weley, Boto, J. Guillera, peroal commuicatio, 5 July H. Hae, Ei Summierugverfahre für die Riemache ζ -Reihe, Math. Z. 3 (93) K. Kopp, Theory ad Applicatio of Ifiite Serie, Dover, New Yor, L. Lewi, Polylogarithm ad Aociated Fuctio, Elevier North-Hollad, New Yor, A. E. Par,, e, ad other irratioal umber, Amer. Math. Mothly 93 (986) N. Pippeger, A ifiite product for e, Amer. Math. Mothly 87 (98) 39.. J. Sodow, Aalytic cotiuatio of Riema' zeta fuctio ad value at egative iteger via Euler' traformatio of erie, Proc. Amer. Math. Soc. (994) , Zero of the alteratig zeta fuctio o the lie R () =, Amer. Math. Mothly (3) , Criteria for irratioality of Euler' cotat, Proc. Amer. Math. Soc. 3 (3) , Double itegral for Euler' cotat ad l( 4 ) (preprit); available at 5., A ifiite product for e γ via hypergeometric formula for Euler' cotat γ (preprit); available at 6. J. Spaier ad K. B. Oldham, A Atla of Fuctio, Hemiphere, New Yor, J. Walli, Computatio of by ucceive iterpolatio, i A Source Boo i Mathematic, -8, D. J. Strui, ed., Priceto Uiverity Pre, Priceto, 986, pp ; reprited i Pi: A Source Boo, d ed., L. Berggre, J. Borwei, ad P. Borwei, ed., Spriger-Verlag, New Yor,, pp S. Zlobi, peroal commuicatio, 6 May 3. 9 Wet 97th Street, New Yor, NY 5 jodow@alumi.priceto.edu