UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Power series expansion connected with Riemann's zeta function Allard, Gabriel Louis Adolphe 1969

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1969_A6_7 A44.pdf [ 2.5MB ]
Metadata
JSON: 831-1.0052026.json
JSON-LD: 831-1.0052026-ld.json
RDF/XML (Pretty): 831-1.0052026-rdf.xml
RDF/JSON: 831-1.0052026-rdf.json
Turtle: 831-1.0052026-turtle.txt
N-Triples: 831-1.0052026-rdf-ntriples.txt
Original Record: 831-1.0052026-source.json
Full Text
831-1.0052026-fulltext.txt
Citation
831-1.0052026.ris

Full Text

A POWER SERIES EXPANSION CONNECTED WITH RIEMANN'S ZETA FUNCTION by GABRIEL LOUIS ADOLPHE ALLARD B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Computer Science We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1969 In presenting this thesis' in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and Study. I further agree that permission for extensive copying of this thesis for s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or p u b l i c a t i o n of this thes.is for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Computer Science The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date 11 August 1969. ABSTRACT i We consider the e n t i r e function g(s) = (1 - 2 1 _ S ) c ( s ) = Z ( - l ) n _ 1 n " S 1 whose set of zeros includes the zeros of c(s)', expand i t i n an everywhere converging Maclauring s e r i e s g(s) = Z a__ s n/n! ; a^ = g ( n ) (0) , |s| 0 Then we determine a n a l y t i c expressions for the c o e f f i c i e n t s a n which w i l l enable us to proceed with the numerical evaluation of some of these c o e f f i c i e n t s . To achieve t h i s , we define an operator D g acting on a r e -s t r i c t e d class of power s e r i e s and which we c a l l the zeta operator. Using the operator D g, we are able to express the c o e f f i c i e n t s a^ as i n f i n i t e n-dimensional i n t e g r a l s . Numerical values f o r the c o e f f i c i e n t s a^ and a^ are e a s i l y deter-mined. For a^ and a^j'we transform the multidimensional i n t e g r a l s into products of s i n g l e i n t e g r a l s and obtain i n f i n i t e s e r i e s expressions f o r these c o e f f i c i e n t s . Although our method can also be used on the following c o e f f i c i e n t s , i t turns out that the work involved to obtain an expression leading to a p r a c t i c a l numerical evaluation of a^, a^, seems p r o h i -b i t i v e at t h i s stage. We then proceed with the numerical computation of a^ and a^ and we use these c o e f f i c i e n t s to c a l c u l a t e the sums of r e c i p r o c a l s of the zeros of £(s) i n the c r i t i c a l s t r i p . F i n a l l y , assuming Riemann hypothesis, we c a l c u l a t e a few other quantities which may prove to be of i n t e r e s t . TABLE OF CONTENTS I Introduction I t The Zeta Operator I I I C o e f f i c i e n t s IV Analysis of Convergence V Numerical Evaluation VI Truncation and Round Off Errors VII Sums of Reciprocals of the Zeros of e(s) VIII Results 1. I INTRODUCTION i_ y Let c(s) be the Riemann Zeta function given by i t s D i r i c h l e t s e r i e s : (1) ?(s) = £n~ s ; s = a + i t , a>l. 1 S(s) has been the subject of numerous inv e s t i g a t i o n s f o r the past two centuries, mainly because of the fundamental r e l a t i o n , due to Euler: (2) c(s) = n d - p " 8 ) " 1 ' - n (i+p" s+p" 2 s+...) , o>i where p>l and runs over a l l primes. (2) may be regarded as an a n a l y t i c a l equivalent of the fundamental theorem of arithm e t i c . I t s importance derives from the f a c t that i t r e l a t e s the integers i n (1) to the primes i n (2). Moreover, (2) shows c(s) £ 0 when o>l or equivalently c ( l - s ) ^ 0 when a<0. A general reference on the Zeta function i s found i n Titchmarsh[l]. C(s) has a simple pole at s = 1 with residue 1 and can be extended to the l e f t of the l i n e c = 1. The r e s u l t i n g function i s n.-^lytic everywhere except f or the pole at s = 1. Hence: g ( s ) : (l-2 1*" s)5(s) and h(s) - £(s) - ( s - 1 ) " 1 are e n t i r e functions and have a Maclaurin ser i e s expansion i n a neighbourhood of s = 0. Thus: ,(3) g(s) = I a. s n/n! , a n = g ( n ) ( 0 ) . 0 . Another w e l l known r e l a t i o n i s the f u n c t i o n a l equation: (4) ?(s) = .x( s ) c ( l - s ) -s 1—s 1 where x(s) = 2 ir sin(-^7rs)r(l-s) and x ( s ) x ( l - s ) = 1 . When c<0, r ( l - s ) ? 0 and the only zeros of ?(s) are the zeros of s i n ( 2 _ 1 i r s ) at s = -2, -4, -6, .... These are the s o - c a l l e d t r i -v i a l zeros of C(s). I t can be shown, as i n Titchmarshfl], that £(s) has an i n f i n i t y of zo.fos i n the s t r i p 0<c<l, c a l l e d the c r i t i c a l s t r i p . A l l these zeros are complex and e i t h e r 11G on the l i n e a = 1/2, the c r i t i c a l l i n e , or occur i n pa i r s symmetrical about t h i s l i n e , with another p a i r of conjugate values since c(s) takes r e a l values when s i s r e a l . 2_ " • The Riemann hypothesis states that a l l these zeros l i e on the l i n e a =» 1/2,. The p r i n c i p a l importance of t h i s conjecture r e l a t e s to A n a l y t i c Number Theory: the Prime Number Theorem, proven by Hadamard i n 1896, asserts that i f ^ - ( x ) i s the number of primes <x then ir(x) - x/log x. At present no er r o r estimate of the type E(x) - 0(x a) , a<l i s known. I f Riemann hypothesis i s true then E(x) = 0 ( x ( 1 / 2 ) + G ) for every e>0. In 1914 G, Hi Hardy [2] proved there i s an i n f i n i t y of zeros on the c r i t i c a l l i n e . More recently R. Spira [3] has shown that under c e r t a i n r e s t r i c t i o n s Riemann hypothesis implies k ( s ) l < - k ( l - s ) | ., 1/2 < a < 1. Besides the t h e o r e t i c a l work there has been a great deal of numerical work, p a r t l y to v e r i f y that a l l zeros investigated were on the c r i t i c a l l i n e , p a r t l y to f i n d out more about the nature of the zeta function i n general. Let f ( t ) = c ( 2 ~ 1 + i t ) . Then any zero of f ( t ) i s a zero of c(s) on the c r i t i c a l l i n e . To c a l c u l a t e some of these zeros J , F. Gram [4] i n 1903 devised a method appl i c a b l e when t<30. Later E. C. Titchraarsh [5] used a method based on Riemann's work and s u i t a b l e f o r l a r g e r values of t, while A, M, Turing [6] proposed a method f o r a range of t. i n t e r -mediate between tho. two previous ones. In 1956 D. H. Lehmer [7] with the help c f a d i g i t a l computer investigated the f i r s t 25,000 zeros of ?(s) and a l l were found to l i e on the c r i t i c a l l i n e d 4 Moreover* i n the range t - 0(10 ) some roots were found to l i e very close to one another. Thus for la r g e r values of t, a computation of the zeros requires ar. improvement i n the accuracy of the e x i s t i n g methods. i t -Together with c(s) we also have the L - s e r i e s , Let m be a p o s i t i v e integer and l e t x be a residue c l a s s character modulo m. X i s then a complex valued function defined over a group G. The p r i n c i p a l character x o s a t i s f i e s : X ( x) = 1 f o r a l l x i n G. Then -8 L ( s | X ) - I' xfcOn" 1 i s c a l l e d the L-series associated with the residue cl a s s character x> The Zeta function i s formed with a p r i n c i p a l character... The f i r s t n o n - p r i n c i p a l L - s e f i e s i s g(s) = Z ( - l ) * 1 " 1 * . " 8 <= ( l - 2 X " s ) c ( s ) . 1 Like a l l s e r i e s formed with non-principal characters, g(s) i s an e n t i r e function. 4_ The aim of t h i s thesis i s to introduce and i n v e s t i g a t e a pos-s i b l y new method of numerical analysis f or ? ( s ) . Following (3) we w i l l develop a power se r i e s f o r the function g(s) whose set of zeros include a l l the zeros of C(s). Before proceeding any further, we w i l l mention a power se r i e s f o r c(s) developed i n 1954 by W. E. Briggs and S. Chowla [3]. Writing 5(s) - + Z A ( s - l ) n , s-1 Q n they found A - i ^ v n ni n where Y i s definad by +1 (5) ? ^ - i 2 g _ i + Y n . + 0(l) ^ k n+x n i . e . the sum on the l e f t of (5) i s approximated by an i n t e g r a l and -the Y n turn out to be higher analogues of Euler's constant. Un-fortunately these constants do not appear to be e a s i l y amenable to numerical computation. Now: ( s - l ) n = Z ( > n " k ( - l ) k 0 K - i - = - E s n , 0<|s|<l 0 i nl-s i ( l - s ) l o g 2 » n_ .n, , 1-2 = 1-e ° = - Z l o g 2 (1-s) /n! 1 = Z b s 0 n where b Q - - ( l o g 2 +1/2! l o g 2 2 +1/3! log 32 +...) b x = l o g 2 +2/2! l o g 2 2 +3/3! l o g 32 +... b 2 » -(1/2! l o g 22 +3/31 l o g 3 2 +6/4! log 42 +...) and so on. Then we f i n d : n . , / \ ~ n , °? . r_ ,nx n-K , , vK-, ?(s) - - Z s + Z A [Z (,) s (-1) ] 0 0 0 • - Z c s n 0 n where c Q = -1 + A Q - A 1 + A 2 - ... "~ ^ = -1 + A x - 2A 2 + 3A 3 - ... c 2 = -1 + A 2 - 3A.j + 6A^ - ... and so on. Hence we have: ( l - ^ - S U C s ) - [I b n s n ] [ Z c n s n ] 0 n 0 n = cO bO + ( c O b l + b 0 C l } 8 + .(c Qb 2 + c 1 b 1 + b Q c 2 ) s 2 + ... with b^ and c^ defined above. Using (3) and equating the c o e f f i c i e n t s of the powers of s, we f i n d : ao a co bo a i = c o b i + b o c i a 2 - 2 , ( c 0 b 2 + C i b l + b 0 C 2 ) and so on, with a^ defined i n (3). Next we wish to show that for a s u i t a b l e operator L we can w r i t e : a = L n f (x) I . with f (x) = ~ - . — n 'x=l 1+x I I THE ZETA OPERATOR Let s = a + i t , a > 0 , and define a l i n e a r operator D as s follows: D g acts on power serie s without the constant term. Since these form a vector space, say S, i t s u f f i c e s to define the a c t i o n of D, on the basis elements of S, namely x 1 1, n=l,2,3 7. Let D x = x n s T.- _ _ n _ n -p n -(s+p) _ n Ttien D D x = D x n r = x n ^ = D , x s p s s+p a -p -s _ _ n = x n - n = D D x P s and i n general n n (1) ( H D ) x n = D x n , s • Z s. . 1 S i 3 1 1 Hence the set {D }, s>_0, i s a semi group of operators,. s LEMMA 1 The operator Dg has the i n t e g r a l representation D gf(x) = p s^T f0 2 8 - 1 f(xe~Z) d z • 0 > °' Proof: n Using the subs t i t u t i o n s f(x) = x , t = nz , we obtain: 00 s - 1 n -nz , n, 0 0 ^ .s-1 - t -s _ n -s _, N J^z x e dz = x /Q t e n at = x n r(s) . Thus 00 s _ i n -nz n 1 fn z x e dz s D x . U s r(s) Since lemma 1 holds for n = 1 , 2 , . . . the operator D has an i n t e -s g r a l representation and under the proper.convergence conditions (2) D gf(x) = ^0 2 3 - 1 f ( x e " Z ) d 2 » f ( x ) e 3 Q - E - D -In terms of (1) t h i s implies that s . - l s -1 -z.,-. . ,-z (3) D f(x) =-=7—r =y—v /-..../-z.' ...z .. t(xe ) s r(s..)...r(s ) 0 0 1 m 1 m dz....dz 1 m subject to the r e s t r i c t i o n s ensuring convergence. (2) i n f a c t , shows D g i s e s s e n t i a l l y the M e l l i n transform, while „ n n -s D x = x n s makes D g a sort of f r a c t i o n a l i n t e g r a t i o n . R e c a l l that for |x|<l x ?? n , x » , n n - l n 1 ^ = I x a n d T+x" = I x where both s e r i e s converge uniformly. Let f(x) = x(l-x) X ; then: (4) D f(x) = Z x n n' S , |x| < 1. , S 1 When a > 1, (4) converges uniformly at x = 1. Hence: D s f ( x ) | x = 1 = Z n~ S = c(s). Using (2), we obtain the formula of Riemann: (5) C(s)r<s) =/Q z S " I ( e Z - l ) " 1 d z . In terms of (3), we obtain what appears to be a new ge n e r a l i z a t i o n of (5): (6) c(s) r ( s ) . . . r ( s ). 1 n f 0 0 s , - l s -1 Z , + . , i + Z " ' Z ^ ) (e -1) 0 dz- ... dz 1 n with n > 1, a. > 0 S Za. = a > 1 . = x 9 i Now l e t f(x) = x C x + i ) - 1 , |x| < 1; then (7) D f(x) - Z ( - l ) n - 1 x n n" S. 1 When x = 1, (7) becomes an a l t e r n a t i n g s e r i e s s a t i s f y i n g L e i b n i z ' s r u l e whenever a > 0 and thus converges to a l i m i t . Kence: D f ( x ) | _, - ? (-D 1 1" 1 n" S - g(s) , a > 0. Using our i n t e g r a l representation,and (3) section I, we f i n d : . (8) D f (x) 'x=i r(s) 'o 1 - » S - l , Z,-v-1 j /_ z (e +1) dz = g(s) = Z a s u/n! . 0 n We w i l l now i l l u s t r a t e some a p p l i c a t i o n s . Consider (6) with n=2 and transform the double i n t e g r a l by introducing the new v a r i a b l u=z 1+z 9, v=z . A f t e r a b r i e f manipulation there r e s u l t s : , s —1 s -1 / J ( i - y ) 1 " y 2 dy - r ( S l ) r ( s 2 > [ r ^ + S j ) 3 ~ \ that i s the representation of Euler's Beta function, and we avoid the usual i n t e g r a t i o n around the branch point. If instead we introduce polar coordinates s we obtain at once • T T / 2 S l " 1 Sr>~1 -(",+s-) fQ (cos 8 s i n " 6 ) (cos 8+ s i n 9) A * = B ( s 1 , s 2 > . Other transformations of various s p e c i a l cases of (3) lead to s i m i l a r r e s u l t s . For obvious reasons the operator D x ^ i l l be s c a l l e d the Zeta operator. Suppose that for a s u i t a b l e operator L, we can write: - ( 9 ) D __x_ = e 3 ^ _x_ , |x| <1. S 1+x 1+x Then using the exponential form of the Zeta operator, we may obtain d i r e c t l y the Maclaurin s e r i e s of g ( s ) : (10) = D s f ( x ) | x = 1 - e s L f(x)| x = 1 = E a s n/n! ; a « L n f (x) | , , f(x) -• x 0 n n X _ 1 1+x" provided we can define the L operator. 10. Using our i n t e g r a l representation f o r D g, we obtain: L f (x) = l i m D f(x) - f(x) . __ __ - l i m [ 1 Z s " x f ( x e " z ) dz - f(x) ] 3-K) T ( s ) s ' - s = l i m [ 1 /" z3"1 f ( x e " Z ) dz - 1 /" z 8 " 1 e~ Z f(x) dz] s->0 r(s+l) J r(s+l) (11) = ^ f ( ^ " Z ) - e" 2 f(x) d z > f ( x ) . s # R e c a l l i n g that f o r any p o s i t i v e r e a l numbers a sb co _ax - b x , , , /Q e - e dx = l o g _b x a and using (11) with f(x) = x 1 1 , we f i n d : -r.z -z , , _ n n ,<» e - e , n, . v (12) L x = x /„ dz = x (-log n) . 0 z In f a c t (12) defines the ac t i o n of the L operator on the basis of the space S. Hence we could formally define the L operator by L = l i m D - I -where I i s the i d e n t i t y operator. 5_ To prove (9), we s t a r t with a lemma. LEMMA 2 With m ^ 1 , n >. 1 j we have: ,m • n , . * m n L x - (-log n) x . 11. Proof; We use induction or. m ; s t a r t i n g with (12), we assume x 1 1 = (-log n ) ^ x n , k >_ 2. Then Tk+1 n T / T k n N _ r / . *k n, L x = L(L x ) = L [ ( - l o g n) x J , -nz -z / i v k n . ° < > e - e , = (-log n) x fQ dz , , Nk+1 n = (-log n) x . Thus L m x 1 1 = (-log n ) m x n for ia = 1,2,3, .... Q.E.D. Observe that we can also write: _ n -nz -z n T 2 n T , T n N T r >°° x e - e x , , L x = L(L x ) = L[/_ • — dz] U- z n - n < z i + z 2 ) " z l " n z 2 -zl. - n z l " z =» x f ^ f ^ [ a -e e -e (e -e d z 1 / z 1 d z 2 / z 2 3 4 with s i m i l a r formulae f o r L , L , .... We are now ready to prove (9) THEOREM 1 Let s = a + i t , a > 0. Then e s L f ( x ) - D sf(x) 5 f ( x ) . J ! ! L , | x | < 1. Proof: . x « , , x n - l n £ , l S i - l i , °? , , xi-1 i = E (-1) x = E (-1) x + E (-1) x 1 + x 1 1 n+1 P (x) + R (x). n n 12 Using lemma 2, ua have: T X T i sL n r a s L , n e x • [E • — j — ] x - [! s l ( " ^ ? g n ) 1 ] x n o l ! 53 (-log n) n n -s n e ^ ft x •» x n - D x s 0 0 s ( — 'op n.) where E — - * ( " — — converges f o r every f i n i t e value of n s whence 0 1 . e s I j x n i s w e l l defined. Thus e J A J P (x) - D c P_(x). n i • s n Since the se r i e s E (-1) x converges uniformly, we can take n large enough so that: | E (-1) 1" 1 x 1! - | R (x) | < e n+1 for any given e > 0. Thus f o r n large enough: | D R n(x) | - | f (-1) 1" 1 x 1 i " S | . n+1 and since a > 0 | E (-1) 1" 1 x 1 i " S | < | E ( - I ) 1 " 1 x 1 | - | R_(x)| < n+1 n+1 Hence I e s L P (x) - D f(x) I - I D R (x)I < e ' n s 1 s n r and . e s L f(x) = D f ( x ) . - Q.E.D. s n 13. Obviously the same kind c f proof would hold f o r any f(x) i n S s where f(x) i s uniformly convergent. We w i l l now apply our L operator to the L-series defined by g(s) and obtain formulae f o r • the c o e f f i c i e n t s defined i n (3) section I. I l l COEFFICIENTS Using (10) and theorem 1 i n section I I , we can write: (1) g(s) = E a n s n/n! = D g f ( x > l x o l = c s L f ( x ) | x = 1 , 0 ' * ' f (x) - -JL- | x | < i , . where L f(x) i s defined by (11) se c t i o n II ; L n f(x) i s given i n -d u c t i v e l y by L° f(x) •-•f(x) , L n + 1 f(x) = L [ L n f ( x ) ] and x i s put equal to 1 a f t e r L has been applied n times. Hence ; a. 'o L O i f e | x = 1 " 1 / 2 • T rt M .°°f(xe U ) -e u f(x) , a l " L f ( x ) | x - l " ' o — d u =/ 0°(l-e" u)(l+e" u)" 1 e" u du/u At u » 0, the zero of 1-e U cancels the pole of 1/u and the i n t e -grand i s regular i n the range of i n t e g r a t i o n . Vie f i n d : (2) (1-hTV 1 = E ( - l ) n a ~ n u , u e (0,=°) 1/2 , u = 0. 14. Comparison with the geometric series shows the series in (2) converges uniformly. Thus for u e (O,05), we have: , -u , -(n+l)u -(n+2)u . = E (.-I) du 1 + e ue 0 ~(n+l)u -(n+2)u co no « e ' _ _> ^ ' and a 1 = _ / „ E (-1) —1 du 2 0 v U - oo "(n+l)u ,„-(n+2)u = 1 E (-1) du 2 0 ° u where the interchange of the integral and summation signs io j u s t i -fied by the uniform convergence of the series involved. . Recalling the definition of Wallis'a product: jr _ 2.2.4.4.6.6.... 2 1.3.3.5.-).7«o.. we have: a = 1 [log 2 - log 3_ + log 3_ - . . . ] = 1 log v. 2 1 ' 2 4 2 2 X For a„, f(x) = •—— ' and 2' 1+x a 9 - L 2 f ( x ) | . 2 1 x=l r M r°° f/-/ -U-VN -U ,, -V, -V r^, -U v = /Q/Q {f(xa ) - e f(xe ) - e [f(xa ) '•-u , n da dvl . ' - e f (x) ]} 1 x=l u v r co rm r / i , - U - V . - l ... -V x-1 . -Us-1 , , / o 1 -U - / Q / 0 [(1+e ) - (1+e ) - (1+e ) + 1/2] e du/u e V dv/v. 15. Define Q(u9v) = ( l + e - ^ ) " 1 - (1+e"^" 1 - (1+e"")"1 + 1/2. Rewriting Q(u,v) as p(u,v)/q(u,v) , we find: (4) Q(u,v) =-1/2 ( l - e " U ) ( l + e " U ) " 1 ( l - e " V ) ( l + e " v ) " 1 ( l - e ' u " v ) ( l + e " u " v ) " 1 . Thus Q(u,v) has a zero of order 3 at u=v=0, thereby cancelling the poles of 1_ _1 , u v Hcnc:; the intogrand i s regular over the range cf integration; using (4) we can transform the doubl:-: integral i n (3) into products ofr.inglo integrals. ' T i t h u,v e (0, 0 0), wc have: , -u , -(n+l)u -(n+2)u ( 5) iz5L_ f ( - i ) n S± d u 1+e"11 ue u 0 , -v , -(n+l)v -(n+2)v ( 6 ) dy_ = ? ( - l ) n 2— =^ dv l+e- V ve V 0 ( 7 ) iz£!II , i . 2 I ( - l ) n G - ( r * + 1 ) u e ' ( n + 1 ) v l+e" u~ v 0 Comparison with the geometric series shows the series i n (5) s (6), and (7) converge uniformly. Hence: a 2 - -1 { / Q T x ( U ) du / Q T 1 (v) dv - 2 [ / J T 2 (U) du T 2 ( V ) dv - / " T 3 (U) du T 3 (v) dv +...]} - - | { [ ! ( - l ) n V T n + 1 ( u ) ] [ J ( - l ) n W n + 1(v)] - 2 ( [ E ( - l ) n W n + 2(u)] 0 0 o [ ? < - l ) n P n + 2 ( v ) ] - [ E ( - D n W n + 3 ( u ) ] [ E ( - l > n \ + 3 ( v ) ] +...)} 16. -(k-rj)x -(k+j+l)x where T. (x) = ? (-1) - — ; j=l,2,3,... 2 k=0 x -(k+j)x _ -(k+j+l)x and (x) « / Q ^ — dx ; j=l,2,3,..., k=0,l,2,... The interchange of the i n t e g r a l and summation signs i s j u s t i -f i e d by the uniform convergence of the serie s involved. Using the d e f i n i t i o n of Walli3'.s product, we f i n d : 2 2 2 2 (8) -2a_ = log TT_ - 2[log TT_._1 - l o g TT_.1_._3 + log _TT.1_._3._3 - ...] 2 2 2 2 2 2 , 2 2 2 4 3_ Before i n v e s t i g a t i n g the se r i e s defined by (8), we w i l l develop an a n a l y t i c expression f o r a,. Again f(x) = x and: J 1+x a 3 = L 3 f C x ) ! ^ -e W [ f ( x e U V ) - e Uf(xe v)-e. v{f(xe U-e Uf(x)}]} du dv dw]^ u v w V7here = / Q / Q / Q Q ( U , V S W ) e du/u e dv/v e dw/w Q(u,v,w) = 1 - 1 - 1 + 1 -, -u-v-w -, -v-w ,, -u-w ., -w 1+e 1+e 1+e 1+e 1 + 1 + 1 - 1 . ,. -u-v ., -v ., -u • .2 1+e 1+e 1+e 17. Again we rewrite Q(u,v,w) under the form p(u,v,w) ; we finds q(u,v,w) Q(u,v,w) - ( l - e ~ U ) ( l - e ~ V ) ( l - e ~ W ) POLY/DEN , _ T . M V -3u-3v-3w -3u-2v-2w -2u-3v-2w -2u-2v-3w where POLY = e -e -e -a - -2u-2v-2w 0 -2u-2v-w -2u-v-2w , -2u-v-w -De -3e -3e -3e 0 -u-2v-2w „ -u-2v-V7 ,. -u-v-2w c -u-v-w -u-v -3e -3e -3e -5e -e -u-v? -v-w.. -e -e +1 and DEN - ( l + e ^ - ™ ) ( 1 + e " ™ ) ( 1 + e " ^ ) (l+cT w) ( l + o ^ ) (l+c" v) (mf") . Hence Q(u,v,w) has a zero of order 3 at u=v=w=0 and the integrand i s regular over the range of i n t e g r a t i o n . When u,v,w e (0,°°), we have: (10) 1 « , l X n -nu-nv-nw = Z (-1) e , , -u-v-w _ 1+e 0 (11) 1 » , l N n -nv-nw = Z (-1) e , , -v-w 1+e 0 (12) 1 °? n -nu-nw = Z (-1) e l + e ~ u - w 0 ~~~ (13). 1 = « ( _ 1 } n e-nu-nv l + c " u - V 0 Comparison with the geometric s e r i e s shows the l a s t four s e r i e s converge uniformly. Our aim now'is to transform the t r i p l e i n t e g r a l i n (9) into products of s i n g l e i n t e g r a l s . Forming the product of POLY with the 4 serie s defined by (10, (11), (12), (13), c a n c e l l i n g and regrouping whenever a p p l i c a b l e , and c a l l i n g the 18. r e s u l t PROD, we obtain the uniformly convergent seri e s •ann-r, 1 t t ~ U ~ W -2u~2w. ~3u-3w N PROD = 1-6(e -e +e -...) -u-v-w M „, -u-w -2u-2w. -3u-3w - 6e [1-2(e -e +e - . . . ) ] . , -2u-2v-2w M 0 / -u-2 -2u-2w^ -3u-3w N 1 + 6e [1-2(e -e +e - . . . ) ] -For the remaining terms of the integrand i n (9), we have; 1-e U du 1-e v dv 1-e W dw , , -u u . -v v . . 1+e ue i+e ve 1+e we with Tj defined above. —.•==- = T. (u) du T. (v) dv T. (w) - W W 1 1 1 Comparison with the geometric seri e s shows the 3 se r i e s involved i n the l a s t product converge uniformly. Hence: 2a 3 » / Q T ^ U ) du / Q T x ( V ) dv / Q T x ( W ) dw ^ 6 [ / Q T 2(u) du fQ T ^ v ) dv fQ T 2 (w) dw - /J T3(u) du /J T x (V ) dv / Q T 3(W) dw + ...] {fQ T 2(u) du / Q T 2 ( v ) dv fQ T 2(w) dw OO CO CO - 2 [/Q T 3(u) du fQ T 2(v) dv / T 3(w) dw - fQ T 4(u) du fQ T 2(v) dv fQ T 4(w) dw + ...]} + 6 {f™ T 3(u) du / J T 3 ( V ) dv T 3(w) dw 2 . [ / 0 T 4(u) du fQ T 3 ( v ) dv / 0 T 4(w) dw / Q T 5(u) du fQ T 3 ( v ) dv fQ T 5(w) dw + ...]} 19. = [ Z ( - l ) n W . . ( u ) ] [ E ( - l ) n W ( v ) ] [ ? ( - l ) n W_,(w)] o n + 1 o n + 1 o n + 1 - 6 { [ E ( - l ) n • W n + 2 ( u ) ] l ^ ( " 1 ) n W n + l < v ) ] [ | ( " 1 ) n W n + 2 ( w ) ] - [ E ( - l ) n W n , , ( u ) ] [ ! ( - l ) n U.Av)][Z(-l) n W (w)] + ...} o n + 3 o n + 1 o n + 3 - 6 { [ E ( - l ) n W n + 2 ( u ) ] [ E ( - l ) n W n + 2 ( v ) ] [ E ( - l ) n W n + 2(w>] 0 0 0 - 2 ( [ E ( - l ) n W n + 3 ( u ) ] [ E ( - l ) n W n + 2 ( v ) ] [ Z ( - l ) n W n + 3(w ) 3 ~ [ ? ( - l ) n W . , ( u ) ] [ E ( - l ) n W . . ( v ) ] [ E ( - l ) n W M ( w ) ] +...)} 0 0 . 0 +6 { [ E ( - l ) n W ,„(u)][E(-l) n W ( v ) ] [ E ( - l ) n W ,(w)] 0 0 0 -2( [ Z ( - l ) n W n + A(u) ] [ Z ( - l ) r i W n + 3(v) ] [ Z ( - l ) n W n + 4(w) ] 0 0 0 - [ E ( - l ) n W ( u ) ] [ E ( - l ) n W , 0 ( v ) ] [ E ( - l ) n W (w)] +...)} o 0 0 with the symbols and W^+j defined as above. The interchange of the i n t e g r a l and summation signs i s j u s t i f i e d by the uniform con-vergence of the se r i e s involved. Again using Wall-is's product and rearranging, we f i n d : 2 2 2 2 (14) 2a, = logjnJloR £ -6(log TT_._1 - l o g JT._1.3_ +log _T__._1.3_._3 -...)] J 2 2 2 2 2 2 2 2 2 2 4 . 2 2 2 +6{logn._l[log TT._1 -2(log TT._1._3 - l o g £..1..3\1 +•••)] 2 2 2 2 2 2 2 2 2 2 4 2 2 2 +logTwl._3[log __T._1._3 -2(log TT_._1._3.3_ - l o g __T._1.3_.3_._5 2 2 2 2 2 2 2 2 2 4 2 2 2 4 4 +...)]+...}. 20. Before going to the next step in the numerical evaluation of &2 and a^j we w i l l outline some of the d i f f i c u l t i e s encountered i n the application of our method to the investigation of the next co-efficients a^a,.,.... For each new coefficient i t turns out that the number of terms in the Q function defined in (4) and (9) double with the addition of each new variable. Thus for , Q(u,v,w,x) w i l l have 16 terms. Actually we find;, (15) Q(u,v,w,x) = 1 - 1 - 1 - , -U-V-W-X . , -V-V7-X , , -u-w-x 1+e 1+e 1+e ,. -w-x -, -u-v-x .,, -v-x ,, -u-x 1+e 1+e 1+e 1+e 1 -, , -X - , -u-v-w n , -v-w .. -u-w 1+e 1+e 1+e 1+e + 1 • , -w i . -u-v -, -v .. -u 2 1+e 1+e 1+e 1+e Following the technique used for a.^ anc* w e n o w w a n t t o transform a quadruple integral into products of single integrals. To do so we must extract from Q(u,v,w,x) the factor ( l - e " u ) ( l - e " V ) ( l - e " W ) ( l - e ~ x ) in order to cancel the poles of 1/u, 1/v, 1/w, 1/x at 0. The resulting polynomial corresponding to PCLY in w i l l have of the order of 1000 terms while the product corresponding to DEN w i l l now have 16 factors, namely the denominators of the 16 terms in 21, (15). 11 of these factors w i l l be involved i n the se r i e s cor-responding to PROD and when the s e r i e s are m u l t i p l i e d out, the author has found h a u r i s t i c a l l y that no c a n c e l l a t i o n occurs u n t i l a s e r i e s not i n v o l v i n g the l a s t v a r i a b l e x has been m u l t i p l i e d into the product. In p r a c t i c a l terms t h i s implies that to f i n d a c a n c e l l a t i o n pattern i n the r e s u l t i n g s e r i e s , using a sort of brute force method as was done f o r a^, one has to look at several b i l l i o n terms i n the s e r i e s . Even with the help of a computer the work involved seems p r o h i b i t i v e at present, p a r t i c u l a r l y since often terms can be cancelled i n several possible ways. I t i s however possible that a better study of the c a n c e l l a t i o n pattern f o r a^ may give clues to the c a n c e l l a t i o n pattern f o r some of the following c o e f f i c i e n t s , thus rendering possible t h e i r numeri-c a l evaluation with our method. Next we s h a l l i n v e s t i g a t e the convergence of the se r i e s defined i n (3) and (14). IV ANALYSIS OF CONVERGENCE 1_ Before proceeding with the analysis of convergence, we w i l l develop some a n a l y t i c a l tools to be used i n the analysis and l a t e r for the truncation e r r o r c o r r e c t i o n s . R e c a l l that f o r jx| < 1 : 9 o { %n+l log (1-x) = - x - x V 2 -x "73 - ? (-D n — . v 0 n + 1 22. Hence f o r n >_ 2, and keeping i n mind the form of W a l l i s ' s product, we have; + log[l-(n+4)~ 2] + ... = - [ n ~ 2 + (n+2)" 2 + (n+4)~ 2 +-..] - 1/2 [ n ~ 4 + (n+2)~ 4 + (n+4)~ 4 +...] - 1/3 [ n " 6 + (n+2)""6 + (n+4)~ 6 +...] We are thus led to consider approximations to sums of the form E (n+2i)~ k ; n,k >_2. 0 Wc w i l l use the second Euler-l'aclaurin bum formula T:hich, according to Stcffcnsen [9], i s t h a mor" accurate one; t h i s formula i s w r i t t e n : 1 - 1 1 x 0 + l h m l-2k 2k-l (2) E f[x_ + (j+l/2)hj.= £/ U f ( y ) dy - E (1-2 1 ^ ) B_, h 0 ° h X0 1 ' x [ f ( 2 k - l ) ( x ^ + . h ) _ f ( 2 k - l ) ( X o ) ] + ^ . „-l-2nu . 2io+2 ; *2nri-2n c (2m+2) „ r ^ where E m = - i ( 2 m f 2 ) ; * CO . x Q < C < x Q +m . For our a p p l i c a t i o n , f(x) - x ', X Q = n-1, h • 2, and i » °=. F o l -lowing Ralston [10], we note that f ^ 2 l Q + 2 \ x ) and f ^ 2 a + 4 ^ (x) do not change sign and are of the same sign f or x A < x < i h . Thus our •J error i n the approximation i s l e s s than the magnitude of the f i r s t neglected term i n the summation of the r i g h t hand side of (2). Using the approximation and noting that f ^ 2^""X^ (x^+ih) = 0, we f i n d : and rv» - - - <n"1> ( n + 1> ( n+ 3> ' • • = 1 - 1 , 1 -3 1 -5 ( 3 ) l 0 g — n (n+2) (n+2)... 2 m + 12 m " 10 m _^  17 -7 31 -9 . 691 -1 + 56 m " 18 * + 44" m 38227 -13 . n , -15. j g 2 ~ m + 0(m ) , m f i \ H 2 (n-1)(n+1)... 1 - 2 1 -4 . 77 -6 269 -8 ( 4 ) l 0 g "5 —— = 4 m " I I 1 1 1 + 7 2 0 m - 8 4 0 m 44927 -10 1334909 -12 25200 Q •" 83160 m . 4300240459 -14 . -16* + 20130160 m + 0 ( m >' 2_ Define: (5) J m / - l o g 2 ! ... l o g 2 | ... ^ + l o g 2 * ... ^ 0 / J0 1 J 0 2 • J l 2 J2 where ia^j,j^ are p o s i t i v e integers with m^  odd, even and: m± m± m 1 + 1 m. — = r- , *» — - 7 - f o r 1 . < m. , 3 i V 1 j i + i m i + 1 1 1 m. m. m.,. m.+2 i 1 l+l 1 r • J ^ — •= — T - = - , = — — f o r j . > m, J i V 1 ' ji+i V 1 1 1 24. With the same constraints on in,, j ^ , we also defines TT m 0 2 (6) K m , = l o g -7T . . . - 7 — (log -77 . . . -. 2J j ). m 0/J 0 z j Q 2 j Q . m 1 / j 1 Observe that l o g 2 (n-D(n-fl)... . l Q g 2 _n n _ n n ... * w * (n-1) (n+1)... * Thus f o r J the two cases j A<ra A p.nd jrt>ra_ lead to the same E I Q / J Q y J u o r e s u l t s . On the other hand when J Q < C I Q and using Wallis's product, we haves i«» * m ° - i , 0 (n-D (n+1)... ' 1 0 8 2 *•• j ^ " " l 0 g ~n" — » n = ao + 1 while f o r jg>iiig, we finds ' TT m 0 , (n-1) (n+1)... _ l o g 2 • • • 3 ^ = 1 ° 8 ~ n T77 ' n = m 0 + 2 ' Hence for K ,. , the case i„<mrt gives the negative of the r e s u l t s f o r the case 3 Q > T O Q . Using (3) and (4), we f i n d f o r >_ 2, n = m^  - 1: (7) J - - k t T 1 - (n+1)' 1] - -ijr [ n - 3 - (n+1)" 3] m 0 / j Q 8 l a . 697 , -5 , 25267 . -7 , ,^-7, + 7200 [ n " ( n + 1 ) ] " 70560 t n " ( n + 1 ) 3 , 1040693 , -9 ' , .^-9, 27737929 , -11 , .,.-11, + 453600 [ n " ( n + 1 ) 3 ~ 1219630 [ n " ( n + 1 ) 3 , 2547222169677 , -13 . ^ s - 1 3 . , . o o r -15 , + 7870262400 [ n " ( n + 1 ) 3 " 3 1 2 0 [ n " ( n + 1 ) + 0 ( r " 1 6 ) 25. where the 8th term i s an "extrapolated" term whose meaning w i l l be made c l e a r i n se c t i o n VI l a t e r on. Using the same technique, we f i n d fa\ v 1 - 4 1 1 -6 ' 1157 -8 33361 -10 ^ 134531 ( 8 ) K m 0 / j Q = 8 M - 4 8 m + 1440 * " T720 m + "3024" m 553631263 -14 3315326155159 -16 n f -18. ~ 831600 S + 259459200 m + V ( m > j Q < n 0 5 m = mQ. When J Q > m^ , we reverse the signs i n (8) and set m = B I Q + I . We are now ready to study'the convergence of the serie s d e f i n i n g a 0 and a^. CONVERGENCE OF a 2 Using our preceeding d e f i n i t i o n s and (8) i n se c t i o n I I I , we have: (9) -2a 2 - l o g 2 |.- 2 J 1 / 2 - t Q - 2 ( t x - t 2 + t 3 -...) where obviously t ^ = l o g 2 j , t 1 = l o g 2 and so on. Thus t;j. > 0 f o r every i . Using the d e f i n i t i o n of Wall i s ' s product, we find." l i m J T ... m . = l i m jrr ... m = TT . _2 = 1 . m-*» 2 m-1 m-*° 2 m+1 2 TT 26. Thus: (10) l i m t = l o g 2 ! - 0 n and since the logarithm i s s t r i c t l y monotonic, we have t ^ + ^ < t ^ f o r every i . Thus (9) i s an a l t e r n a t i n g s e r i e s s a t i s f y i n g L e i b n i z ' s r u l e and converges to a l i m i t ; the error made when stopping the summation i s smaller i n magnitude than the magnitude of the f i r s t term omitted. To determine the rate of convergence, we use (4) and with a f i r s t order approximation we f i n d : (11) J 1 / 2 » |- ( 2 " 2 - 3 - 2 + 4" 2 - . . . ) . 2 2 3 Now n - (n+1) - 2/n + 0(n ). When n i s large enough we group the terms 2 by 2 i n (11) and with another f i r s t order approxi-mation, we f i n d : J l / 2 * i [ 2 " 2 ~ 3 " 2 + ••••+ 2 { n " * 3 + ( n + 2 >~ 3 +...>]•• -3 Therefore the rate of convergence for a 2 i s approximately n — CONVERGENCE 0? &3 Using our preceding d e f i n i t i o n s and (14) i n section I I I , we have: 2a 3 = lo g f ( l o g 2 f - 6 J 1 / 2 ) + 6 ( K 1 / 2 + K 3 / 2 + K 3 / 4 +...) = u Q + 6 ( u 1 + u 2 + u 3 +...) IT 2 IT where » lo g -^(log y " ^  Jl/2^9 u l = K l / 2 * a n d s 0 ° n * 27. Using the convergence r e s u l t s f o r a^a i t follows that each of the se r i e s defined by the u.'s converges to a l i m i t . Using x (10) above and the d e f i n i t i o n of K W P deduce t h ^ t : w . lira. K ,, .. = l i m K 0 m/(m-l) m/(m+i) and because the logarithm i s s t r i c t l y monotonic, ! u i + - j _ l < l u _ _ _ l -Furthermore the u.'s alternate i n sign i n accordance with the l p r operties of K , Hence Z u. i s an a l t e r n a t i n g s e r i e s s a t i s -n i J i . ~ f y i n g Leibniz's r u l e and thus converges to a l i m i t ; again the er r o r made when stopping the summation i s smaller i n magnitude than the magnitude of the f i r s t neglected term. To determine the rate of convergence, we use (S) above and with a f i r s t order approximation, we f i n d ; (12) 2 a 3 * c - 3/4(2~ 4 - 3~ 4 + 4~ 4 -...) -4 -4 -5 -6 for some constant c. B u t - (n+1) = 4n + 0(n ). Thus when n i s large enough, we w i l l group the terms 2 by 2 i n (12) and with another f i r s t order approximation, we f i n d ; 2 a 3 « c - 3/4{2"4 -3~ 4 +4~4 + 4[n" 5+ (n+2)' 5+ (n+4)~5+...]}, Hence the rate &f convergence f o r a 3 i s approximately n ~*. 28. V NUMERICAL EVALUATION EVALUATION OF a 2 R e c a l l that: - 2a 2 - l o g 2 f - 2 J 1 / 2 - t Q - 2 ( t r t 2 + t 3 - . . . ) . Following Hartree [11], we could at t h i s stage use an a c c e l e r a t i n g technique f o r a l t e r n a t i n g s e r i e s , due to Euler, which makes use of forward d i f f e r e n c e s and has the considerable advantage, from the point of view of round-off error, of transforming the a l t e r -a i nating s e r i e s i n t o a sum of p o s i t i v e terms. The method however has the inconvenience of leading to a much complicated formula f o r truncation error c o r r e c t i o n . Now f o r any r e a l numbers a,b, c,.,., we have: 2 • 2 2 2 l o g ab = (log a + l o g b) =» log a + 21og a log b + l o g b 2 2 2 l o g abc = l o g a + 21og a ( l o g b + l o g c) + l o g b 2 + 21og b l o g c + l o g c , e t c . In the s e r i e s defined by w e firoup t n e t e r m s 2 by 2, apply the above r e s u l t s , and cancel to f i n d : (1) J 1 / 2 = - [ l o g | (21og | + 21og \ + log |) + l o g | (21og | + 21og \ + 21og | + 21og I + l o g |-) +...] 00 - - I b. 1 0 ' 29, where b Q = l o g | (21og \ + 21og \ + l o g |) ~ b x = l o g | ( 2 1 o g | + 21og | + 21og | + 21og | + l o g | ) , and so on. Obviously here also we only have to sum up a s e r i e s of p o s i t i v e terms and, as we s h a l l see l a t e r , t h i s s e r i e s leads to a simple formula f o r truncation error c o r r e c t i o n . For each n, l e t F l be the f i r s t f a c t o r and F2 the second f a c t o r of b . n n n Following a notation now generally accepted, we present i n the form of a flow chart an algorithm f o r generating the sequence (b ) and evaluating a_. T i s truncation e r r o r c o r r e c t i o n to n 2 N . be developed l a t e r and M determines the end of the summation process. To reduce round o f f error, a l l terms of a sum are added s t a r t i n g at the term with the smallest magnitude. J 30. ALGORITHM FOR THE COMPUTATION OF a. INITIALIZATION lo g 3/2 + F l 3TT 2/32 •> x l o g x -*• F2 F1.F2 -»• b 1 n 4 m N Sum log(l+l/n) F l x[l+(2m-l)/(m A-2m 3)] log x F2 F1.F2 * b n+1 n m+2 -> m n -(1/2 l o g TT/2 + Sum) •*- a, ( n-1 -> n Sum + b -»• Sum n EVALUATION OF a. Rec a l l that: (2) 2 a 3 - l o g j ( l o g 2 j - 6 J 1 / 2 ) +6<K1/2+- K 3 / 2 + K 3 / 4 + . ..) xxn + 6 Z u 0 1 n 31. Each term decomposes into a f a c t o r , say F3, mu l t i p l y i n g an i n f i n i t e s e r i e s of the same type as that d e f i n i n g a 2 and where the same analysis a p p l i e s . Following the technique used f o r a 2 , we rearrange the se r i e s and obtain: 00 J, / 0 ° -E b as i n (1) above , 1/2 Q n J 3 / 2 = - [ l o g | (21og f + 21og \ + 21og | + l o g |) + l o g | (21og | + 21og | + 21og | + 21og | + 21og j.+ l o g \ ) + . . . ] = - Z c. 0 1 where c Q = l o g | (21og J + 21og \ + 21og | + l o g | ) C ], = l o g | (21og | + 21og j + 21og | + 21og | + 21og | + l o g | ), and so on. Then we f i n d : CO CO CO CO J 0 / , = -Z b , J c ,. = -Z c , J c ,, = -Z b , J , ,, = -Z c , 3/4 . 1 n 5/4 i n 5 ' 6 2 ' ' 2 n and so on. Hence to compute the terms u^ i n ( 2 ) , we have to generate only 2 s e r i e s , namely: B = -Z b and C = -Z c 0 n 0 n 32. Here also we could use Euler a c c e l e r a t i n g method for a l t e r -nating s e r i e s but again i t appears we would not derive any bene-f i t s from i t . Our truncation error c o r r e c t i o n , which w i l l be determined l a t e r , w i l l now involve 3 d i f f e r e n t c o r r e c t i o n s , one, say T ^ , to correct f o r the terms neglected i n evaluating B; another, say T^^, to correct f o r the terms neglected i n evaluating C; and f i n a l l y a t h i r d c o r r e c t i o n , say to correct for the terms neglected i n evaluating Z u^. Note that since (2) involves also J-J./2' w e c a n u s e t n e s a m e program to compute both c o e f f i c i e n t s a 2 and a^. To reduce round o f f error we w i l l again sum up i n backward order. For purposes of c l a r i t y and using our previous d e f i n i t i o n s , we w i l l present i n the form of a flow chart an algorithm for the computation of a^ alone since an algorithm f o r the computation of a_ has already been given. 33. ALGORITHM FOR a. INITIALIZATION lo g 3/2 + F l 3TT 2/32 -»• x log x F2 F1.F2 b Q 0 n 4 m Nl Surnl TN2 "* S u m 2 T.T0 Sum3 N3 log TT/2 .-»- F3 TT/2 •->• y -1 y(l-m )__-»• y log y F3 F3 (F3 2+2d ) •*• u n . n n+1 ->- n y(l+m -> y log y -> F3 F3(F3 2+2d ) u n n n+1 -»• n m+2 m log(l-l/m) -> F l x[l+(m2-2m) log x •+ F2 F1.F2 ->- c n n+1 n log (1+1/m) F l x(l-m' 2) -> x lo g x F2 F1.F2 b n m+2 -> m 2n •*• N n-1 •> n Sum3+u -y Sum3 n Suml+b n-> Suml Suml -> d2n n-1 -> n Sum2+c -> Sum2 n Sum2 •+ d2n+l > > >^ \ n ^ \ : 0 > Suml + b 2 m 0 F3(F3N-6d Q) - u Q 1 n (3 Sum3 + 1/2 u Q) a 3 34. EVALUATION OF TRUNCATION ERROR CORRECTION While the s p e c i f i c formulae w i l l be developed l a t e r , i t turns out that a l l our truncation e r r o r corrections are polynomials T„. of the form: Ni T N i = * cJ(n-2)-(2^-(n-l)-(2r^\ ; j - 1,3 ; L<7, »>2 . m=0 Hence we must f i n d a method to evaluate an expression of the form: n ~ k - ( n + l ) " k = [ ( n + l ) k - n k ] / [ n k ( n + l ) k ] j n,k>2 i n a way which w i l l minimize the l o s s of s i g n i f i c a n t figures while remaining e f f i c i e n t . One possible but somewhat i n e f f i c i e n t k k method i s to apply the binomial formula to (n+1) - n . On the other hand for any constants a,b and any integer k>2, we have: k k+1 ,k+l . , s . r _ k - i , i , a - b = (a-b) [E a b ] , 0 Se t t i n g a = n+1, b = n, then a-b = 1 and. k+1 .k+1 k - k - i , i a - b = E a b 0 The cases of i n t e r e s t to us are k = 3,5,7,9,11,13,15. With the above r e s t r i c t i o n s on a„b, we f i n d : (3) a 3 - b 3 = a(a+b) + b 2 (4) a 5 - b 5 = a(a 2+b 2)(a+b) + b 4 (5) a 7 - b 7 = a ( a 3 + b 3 ) ( a 3 - b 3 ) + b 6 3 3 where we use (3) to evaluate a - b ' 9 ,9 , 4,,4 W 2,, 2 W N , ,8 a - b = a(a +b ) (a +b ) (a+b) + b 11 ,11 , 5 ^ , 5 W 5 ,5 N , . 10 a - b = a (a +b ) (a -b ) + b 35. where we use (4) to evaluate a^-b^ 13 ,13 / 6,,6 W 3.,3*/ 3 ,3* . ,12 a - b = a (a +b ) (a +b ) (a -b ) + b 3 3 where we use (3) to evaluate a - b 15 ,15 , 7__, 7 W 7 .7. , , 14 a -b = a (a +b ) (a -b ) + b where we use (5) to evaluate a 7-b 7. The algorithm to implement the above formulae i s rather straightforward and w i l l be omitted. VI TRUNCATION AND ROUND OFF ERRORS TRUNCATION ERROR R e c a l l that: -2a 2 - log 2Tf/2 - 2 J l y , 2 . When evaluating the o r i g i n a l s e r i e s d e f i n i n g J2./2' w e 8 r o u P the terms 2 by 2. Suppose we stop our summation a f t e r the mth term i n the o r i g i n a l s e r i e s 9 where m i s even. Then our truncation e r r o r w i l l be: ' •' _ _ 2 (m+2)(m+4)(m+4)(m+6)... 1 2 (m+3)(m+5)(m+5)(m+7).. • . n l " 8 (m+3)(m+3)(m+5)(m+5)... " ° 8 (m+4) (m+4) (m+6) (m+6) .. Using (7) i n sec t i o n IV with n = m+1, we f i n d : (1) T n l - I [ „ - 1 - ( n + l ) - 1 3 - ^ [ n " 3 - ( n + l ) " 3 ] + | | ^ [ n " 5 - ( n + l ) " 5 ] 25267, -7 , ,, x-7 1 ,1040693f -9 , -70560 [ n " ( n + 1 ) ] l 453600 [ n " ( n + 1 ) 3 27737929 r -11 , ,., * - l l , , 2547222169677 r -13 . rfn -(n+1) h—-,0^0/.^ l n "(n+1) 3 1219680 1 v ' J 7870262400 -3120[n" 1 5-(n+1) - 1 5 3 + 0 ( n " 1 6 ) 36. Hence f o r our truncation e r r o r c o r r e c t i o n we simply use the f i r s t 8 terms i n ( 1 ) . The new truncation e r r o r i s of order n and following a remark made i n section IV, i t can be shown that the new er r o r should be.less in magnitude than the magnitude of the 8th term i n ( 1 ) . For r e c a l l that CO 2 a 0 = u n + 6 Z u 3 0 n As shown i n se c t i o n V, the sequence (u^) i s generated from the two se r i e s d e f i n i n g a n d ^3/2 a n d w e n a v e a l r e a c * y deter-mined a truncation e r r o r c o r r e c t i o n T ^ f o r Jj_/2* F o r ^3/2 w e group again the terms 2 by 2 i n the o r i g i n a l s e r i e s . I f we stop our summation at the mth term, where m i s even, our truncation e r r o r w i l l be: -2 (m+3)(m+5)... 2 (m+4)(m+6)... n2 ~ X ° 8 (m+4) (m+4)... ' 1 0 8 (m+5) (m+5) ... " " Thus f o r our truncation e r r o r c o r r e c t i o n w e u s e a g a i n t n e f i r s t 8 terms i n (1) and set n = m+2. The new truncation error w i l l again be of order n with a magnitude smaller than the magnitude of the 8th term i n ( 1 ) . CO When summing the s e r i e s Z u., we take the terms 2 by 2. 1 1 Suppose we stop the summation a f t e r the mth term, where m i s even. •37. Then the truncation error w i l l be: ^ 3 = V - ~ dnrf4 + dm+5 -, , . (m+2) (m+4) ... M 2 (m+2) (m+4)... „ _ , W h e r e dm+3 = l 0 g(^3")(m+3)„.. [ 1° S (m+3) (m+3) ... " 2 J(m+3)/(m+2)3 A - ^  (m+3) (m+5)... f 1 2 (m+3) (m+5)... - _ , am+4 " • L° 8(m+4)(m+4)... 1- L 0 8 (m+4) (m+4)... " Z J(m+3)/(m+4)J and so on. Then using our various approximations developed i n section IV, and more specifically (8), we find: (2) T = -l/8{[(n+l)"4+(n+3)"4+...] - [(n+2~4+(n+4)~4+...]} nJ +ll/48{[(n+l)"6+(n+3)"6+...] - [(n+2)"6+(n+4)"6+...]} where n = m+1. After approximating again, there results: < 3 ) T n 3 = - y*-3-^)~3i + m[»5-^~5i - forlo [ n " 7 - ( n + 1 ) " 7 3 , 58573r -9 , N-9, 35692501r -11 , ,.,-11, + 24l72 C n - ( n + 1 ) ] " 1330560[n " ( n + 1 ) 3 , 11952037439r -13 , ^ - 1 3 , , 0 Q n r -15 , ,^-15, + 28828800 [ n - ( n + 1 ) 3 ~ 4 2 9 0 [ n ~ ( n + 1 ) 3 +0(n" 1 6) where again the f i r s t 6 terms are exact and the 7th term i s an "extrapolated" term which should reduce the coefficient of our new truncation error, after we take for our correction T.._ the . • N3 f i r s t 7 terms of (3). The f i n a l truncation error w i l l again be —16 of order n and i t s magnitude should be less than the magnitude of the 7th term i n (3). 38. Hence f or a l l our truncation e r r o r c o r r e c t i o n s , the new erro r —16 i s of order n . A n t i c i p a t i n g l a t e r r e s u l t s , we w i l l f i n d that n=25, n=24, n=23, for T ^ , T 2> r e s p e c t i v e l y . Thus we can wri t e : T n l = E c . [ n - ( 1 + 2 i ) - ( n + l ) - " a + 2 i ) ] + 0 ( n " 1 6 ) , n-25. i=0 For T^^ we use the same equation with n=24, and f o r T ^ we have: m , r -(3+2i) . , l N - ( 3 + 2 i ) , . A / -16 N 0 « T _ = Z d.[n -(n+1) J + 0(n ) , n=23. n J i=0 1 We now wish to show that f o r these values of n, the l a s t "extrapolated" term i s s t i l l i n the region of convergence of the Euler-Maclaurin asymptotic expansion. Let c^ and d£ stand f o r the f u l l value of the extrapolated term. Then f o r T , we f i n d n l that: For T .: Til For T _: n3 c 6 * 2 5 " 1 4 = 8.75 x I O - 1 8 ; c} x 2 5 - 1 6 « 2,8 x I O - 1 9 c 6 x 2 4 " 1 4 = 1.54 x I O - 1 7 ; c} x 2 4 - 1 6 = 5.2 x I O - 1 9 d 5 x 2 3 - 1 4 « 3.6 x I O - 1 7 ; d£ x 2 3 ~ 1 6 = 1.5 x 1 0 ~ 1 8 and when we extrapolate a second time f o r each c o r r e c t i o n , we f i n d that the e r r o r bound goes down by another order of magnitude. Thus f o r our truncation e r r o r corrections the l a s t extrapolated term i s w e l l w i t h i n the region of. convergence of our approximation. Furthermore since the c o e f f i c i e n t s are fi x e d and the magnitude of the new e r r o r f o r each s e r i e s i s l e s s than the magnitude of the 39. f i r s t exact term neglected i n the c o r r e c t i o n , we can i n p r i n c i p l e , by taking n large enough, make the new truncation e r r o r as small as we please, leaving us to contend only with round o f f e r r o r . 3_ ROUND OFF ERROR The algorithms for the numerical evaluation of a^ and a^ aim to minimize the round o f f error occuring during a c t u a l com-putation. Since round o f f error grows as the number of terms to be evaluated i n each s e r i e s increases, we are l e d , w i t h i n the constraints of truncation e r r o r considerations, to minimize the number of terms to be taken for the evaluation of each s e r i e s . Meanwhile i t was found that: - -.06 and a^ - -.02 and owing to the l i m i t a t i o n s of the number representation i n s i d e the computer used to carry out the c a l c u l a t i o n s , a maximum of 16 s i g n i f i c a n t figures accuracy was allowed. Hence we are l e d to choose a value of n i n (1) and (3) such that the new —18 truncation error bound should be of order 10 or smaller. Suppose that for our 3 truncation e r r o r corrections we use the f i r s t exact terms to extrapolate and p r e d i c t a value for the next term. I f we now1 add an extra term with h a l f the predicted value, we have i n e f f e c t halved the error bound on the new truncation e r r o r . We wish next to determine a more r e a l i s t i c bound, K, f o r 40. the new truncation e r r o r . Computing experiments were conducted using equation (3) section IV, with 1, then 2, then 3, and up to 8 terms, the l a s t -one being extrapolated. We c a l c u l a t e d l o g |.. l o g j.. .-||-, l o g . .||-s which give .values of n i n the range we seek. We compared the d i f f e r e n t values obtained as more terms are added i n the approximation with the values given by the l o g function supplied by the Computing Center L i b r a r y , which i s u s u a l l y accurate to 15 decimal f i g u r e s . I t was concluded we can obtain a r e a l i s t i c bound on the new truncation error by taking fo r each s e r i e s 30% of the predicted magnitude f o r the extrapolated term. This i s equivalent to p o s t u l a t i n g that for T ^ and T ^ the value of the 8th term, and f o r T ^ the value of the 7th term, giving an exact c o r r e c t i o n i s between 20% and 80% of the predicted value, which seems to cover adequately the probable range. Hence we have: K = 1870 x 12 x 2 5 - 1 6 + 1870 x 11 x 2 4 - 1 6 + 2576 x 2 3 _ 1 6 < 3.1 x 1 0 ~ 1 8 which Is w e l l within the range we seek. Then according to the algorithm f o r the computation of. a^, i f we take 12 terms to evaluate the s e r i e s defined by B, we w i l l take 11 terms to evaluate the s e r i e s defined by C and 23 terms for the s e r i e s Eu.. This leads to the values of n f o r the order of the e r r o r x bound given above and to the value M=24 to determine the stopping point i n the algorithm. The remaining sources of round o f f error w i l l come mainly from c a l c u l a t i n g the argument of the logarithm 41. and from the logarithmic values themselves. I t was found that the l o g function used i n the c a l c u l a t i o n s gave an accuracy of 15 figures with a p o s s i b l e e r r o r of 2/10th i n the 16th f i g u r e . VII SUMS OF RECIPROCALS OF THE ZEROS OF g(s) We w i l l now use our calculated c o e f f i c i e n t s to obtain numerical values f o r quantities which may be of i n t e r e s t f o r future i n v e s t i g a t i o n s of ? ( s ) . Following Titchmarsh [1] and using Hadamard f a c t o r i z a t i o n theorem, we f i n d that: ? ( s ) " 2(s-i)rfil(i / 2 ) s 3 I (1-s/p) eS/P • b • 108 2* - 1 "2 n -1 Y - lim(I m - l o g n) n-*» 1 where p runs over a l l the zeros of £(s) i n the c r i t i c a l s t r i p . Hence we have: / 1 X „ l - s w , v ( 1 - 2 1 s ) e b s IT . , . s/p (1) (1-2 )c(8) • 2 ( s _ 1 ) r ( 1 + x s ) p (1-s/p) e . 2 We w i l l now develop power s e r i e s for the r i g h t side of (1). In the c r i t i c a l s t r i p , |sj < 1 and thus f ( s ) = ( 1 - 2 1 S ) ( s - 1 ) 1 i s a regular function and has a Maclaurin s e r i e s expansion v a l i d i n the s t r i p : f ( s ) = f(0) + f'(0) s + f " ( 0 ) s 2 / 2 + f ' " ( 0 ) s 3/6 + 42. We f i n d that: f ( 0 ) = l , f'(0)=l-21og 2 , f»'(0)=2(l-21bg 2 + lo g 2 2 ) , f"'(0)=2(3-61og 2 + 31og 22 - l o g 3 2 ) . Thus (2) , f(s)=l+(l-21og 2)s + 2(l-21og 2 + log 22) s 2/2 + 2(3-61og 2 + 31og 22 - log 32) o 3/6 .+ .... Next f o r a l l s i n the complex plane, we have: (3) e b s = l + b s + b 2 s 2 / 2 + b 3 s 3 / 6 + The function [T(l+2 x.s) ] 1 i s also an e n t i r e function and we can wr i t e : -1 2 3 [r(l+x)] =YQ + Y 1x + Y 2 X + Y 3 X + . . . where, following Nielsen [12], we have the recursion formula: n (n+D Y n + 1 - E y n _ r y n _ r ; y x Y ; > n - C(n) , n=2,3,4,.... r=0 Since T ( l ) =0! = 1, we s t a r t with YQ = 1 and we f i n d : Y l - Y, Y 2 = 2 " 1 ( Y 2 - 7 r 2 / 6 ) , Y 3 - 3' 1[ ( Y 3/2 - 7 r 2 Y/4) +C(3)]. Hence: (4) [ r ( l + 2 " 1 s ) ] " 1 = 1 + 2 _ 1 Y s +. 4 " 1 ( Y 2 - T T 2 / 6 ) s 2/2 - 4 ~ 1 [ Y 3 / 2 - 7 T 2 Y / 4 + c.(3)] s 3/6. + ... with the expansion v a l i d f o r the e n t i r e complex plane. S i m i l a r l y , we have: s /p (5) n(l-s/p ) e n = n(l-s/p )(l+s/p +• p"2 s 2/2 + p~3 s 3/6 +...) n n n n n n n = n(l-p" 2 s 2/2 - p"3 s 3/6 -...) n n n = 1 -(S p"2) s 2/2 - 2(1 p - 3) s 3/6 - .... n n 43, Let S. = Z p Observe that i n (5) the terms i n P ^ cancel and i n v n n therefore does not appear i n the f i n a l r e s u l t . Now we multiply out the s e r i e s defined i n (2), (3), (4) and (5), truncating each 3 s e r i e s and the product a f t e r terms i n s j we obtain: ( l - 2 1 " S ) ? ( s ) = 2 _ 1 [ l + l o g TT/2 S +(log 2 2Tr +21og 22 -41og 2TI l o g 2 -T T 2/24 -S 2) s 2/2 +{log 32rr +31og 2TT +2-61og 22Tr log 2 -61og 2 +61og 2TT l o g 2 2 -21og 32 -rr 2/8 l o g 2TT + T T 2 / 4 l o g 2 + c(3)/4 -31og 2TT S 2 +61og 2 S 2 - 2S 3> s 3/6 +...] 2 3 = a Q + a^ s + a 2 s /2 + a^ s /6 +.... Hence we can solve f o r S 2 and to f i n d : (6) S 2 - l o g 2 2 T r + 1 - 41og 2TT l o g 2 + 21og 22 - Tf 2/24 - 2a 2 (7) S 3 = 2 - 1 l o g 3 2 T r + (3/2)log 2TT + 1 - 31og 22iT log 2 + 31og 2rr l o g 2 2 - 3 l o g 2 - l o g 3 2 - ir 2/16 l o g 2TT + T T 2 / 8 l o g 2 + C(3)/8 -(3/2)log 2TT S 2 + 31og 2 S 2 -I t should be noted that the values obtained for S„ and S„ are 2 3 not contingent i n any way upon the Riemann hypothesis. Assuming that the Rienann hypothesis holds, S. Chowla [13] has shown that according to a c a l c u l a t i o n of Riemann SX = .02309570896612103381. 44. With the same assumption, we can also write: .•S, - Z 2a (al + t 2 ) " 1 - Z (1/4 + t V 1 1 n n n n n n S, = I 2 ( a 2 - t 2 ) ( a 2 + t V 2 = Z (1/2 - 2 t 2 ) (1/4 + t V 2 2 n n n n n n n n S = Z 2 ( a 3 - 3ant2)(a2 + t 2 ) - 3 - Z (1/4 - 3t 2)(1/4 + t 2 ) " 3 3 n n n n n n n n n where we sum over p a i r s of conjugate values. Thus using our previous r e s u l t s , we can c a l c u l a t e some furth e r quantities which may prove to be of i n t e r e s t . For example we f i n d that: R, = 2S, + S„ - Z (1/4 + t 2 ) " 2 1 1 2 n n R = 4~ 1(2S 1 - S 0) = Z t 2 (1/4 + t 2 ) - 2 z. ± J. n n n R_ = 2S 9 - S, = Z t 2 ( 3 - 4 t 2 ) ( l / 4 + t 2 ) - 3 3 I 3 n n n n but i t should be noted that these l a s t r e s u l t s now depend on the v a l i d i t y of the Riemann hypothesis. VIII RESULTS Most c a l c u l a t i o n s were c a r r i e d out at the U n i v e r s i t y Com-puting Center with an IBM System/360 Model 67 computer, using double p r e c i s i o n arithmetic with 64 b i t s representation of numbers. 45. The following results were obtained: a Q = .5 a± = .225791352644727433263098 a 2 = -.06102921183445199 a 3 = -.02347454420644166 5 2 = -.04615418116301331 5 3 = -.0001113877296476766 Rx = .00003723676922874619 R2 = .02308639977381384 R3 = -.09219697459637894 . The mathematical constants needed to carry out the numerical computations were taken from the Handbook of Mathematical Functions [14] where most constants are given with 25 figures accuracy. For a n the result i s exact and for a^ we have a minimum of 23 decimal digits accuracy. For a 2 we have to take into account the probable round off errors in the calculation of the argument of the log and i n the log values themselves. Hence we are led to expect a probable accuracy of 11 decimal digits for a 2 and thus a probable accuracy of only 8 decimal digits for a 3 since the sources of errors are similar to those of a 2, but the process i s repeated. This leads us to expect an accuracy of 10 decimal digits for S 2 > R^  and R 2 > and 6 decimal digits for S 3 and R 3« 46. BIBLIOGRAPHY [1] Titchmarsh, E. C. (1951). The Theory of the Riemann Zeta Function. Oxford University Press s Fair Lawn, N. J. [2] Hardy, G. H. (1914). Sur les Zeros de l a Fonction c(s) de Riemann. Comptes Rendus de l'Academie des Sciences (Paris), vol. 158, pp. 1012-1014. [3] Spira, R. (1965). An Inequality for the Riemann Zeta Function. Duke Mat. J . s vol. 32, pp. 247-250. [4] Gram, J. P. (1903). Note sur les Zeros de l a Fonction g(s) de Riemann. Acta Math., vol. 27, pp. 289-304. [5] Titchmarsh, E. C. (1935). The Zeros of the Riemann Zeta Function.- Proc. Royal Soc. (A), vol. 151, pp. 234-255 and vol. 157, pp. 261-263. [6] Turing, A. M. (1943).- A Method for the Calculation of the Zeta Function. Proc. London Math. Soc. (2), vol. 48, pp. 180r-197. [7] Lehmer, D. H. (1956). On the roots of the Riemann Zeta  Function.. Acta Math., vol. 95, pp. 291-298. [7a] Lehmer, D. H. (1956). Extended Computation of the Riemann  Zeta Function. Mathematika, vol. 3, pp. 102-108. [8] Briggs, W. E. and S. Chowla (1954). The Power Series Co- efficients of g(s). University of Colorado, Boulder, Colorado, C. U. Report No. C-4. 47. [9] Steffensen, J. F. (1950). Interpolation. 2nd ed., Chelsea Publishing Company, New York, N.Y. [10] Ralston, A. (1965). A First Course i n Numerical Analysis. McGraw-Hill Book Company, New York, N.Y. [11] Hartree, D. R. (1958). Numerical Analysis. 2nd ed., Oxford-University Press, Fair Lawn, N.J, [12] Nielsen, N. (1965). Handbuch der Theorie der Gammafunction. Chelsea Publishing Company, New York, N.Y. [13] Chowla, S. (1952). The Riemann Zeta and A l l i e d Functions. Bulletin of the A. M. S., Vol. 58, No. 3, pp. 287-305. [14] Abramowitz, M. and I. Stegun (1965). Handbook of Mathematical Functions. Dover Publishing Company, New York, N.Y. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0052026/manifest

Comment

Related Items