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AN ABSTRACT OF THE THESIS OF DOROTHY MARGUERITE RIVERA for the DOCTOR OF PHILOSOPHY (Degree) (Name) MATHEMATICS in presented on May 24, 1972 (Date) (Major) Title: ON A GENERALIZATION OF THE LERCH ZETA FUNCTION Signature redacted for privacy. Abstract approved: Fritz 0 erhettinger Application of a Mellin transform to a series which represents a generalization of the Lerch zeta function yields a transformation series. One obtains the asymp- totic behavior of the series, together with some associated expansions and limit relations and moreover, specialization of parameters yields several classical results. ® 1972 DOROTHY MARGUERITE RIVERA ALL RIGHTS RESERVED ON A GENERALIZATION OF THE LERCH ZETA FUNCTION by DOROTHY MARGUERITE RIVERA A THESIS submitted to OREGON STATE UNIVERSITY in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY June 1973 APPROVED: Signature redacted for privacy. C))/L/e\J? Professor of Mathematics in charge of major Signature redacted for privacy. Acta. hairman of Department of Mathematics Signature redacted for privacy. Dean of Graduate School Date thesis is presented May 24, 1972 Typed by Linda Knuth for Dorothy Marguerite Rivera ACKNOWLEDGMENT I wish to express my sincere gratitude to Professor Fritz Oberhettinger for his advice, encouragement, and guidance in the preparation of this thesis. Research for this thesis was supported in part by a National Science Foundation Science Faculty Fellowship. TABLE OF CONTENTS Page Chapter INTRODUCTION 1 A SUMMATION FORMULA INVOLVING THE MELLIN TRANSFORM 7 A GENERALIZATION OF LERCH'S ZETA FUNCTION 15 HURWITZ'S SERIES AND LERCHI TRANSFORMATION FORMULA 24 BIBLIOGRAPHY 29 APPENDICES Appendix I 32 Appendix II Appendix III 36 39 ON A GENERALIZATION OF THE LERCH ZETA FUNCTION CHAPTER I INTRODUCTION The transcendental function defined by the power series 00 a) = 0(z, s (1) (a + n) z n n= valid for IzI 1 0, -1, -2, ... a and arbitrary s has been the subject of a large number of investigations. For z = 1 Re s > 1 the restriction is necessary and CO (2) (n + a)-s = 0(1, s, a) = (s, a) Res > 1 n= is the Hurwitz zeta function. (3) 0(z, s, a) = zm 0(z, s The difference equation (a + n -szn, m + a) + n= m =1, 2, which is a consequence of (1) restrict the parameter main properties of (1) a 3, ... makes it sufficient t such that 0 < Re a 1 1. The (for a short condensation see [7, 2 vol. 1, p. 27]) were investigated by Lipschitz [16], Lerch [13] [la] Hardy and Barnes [3]. Important results of the above mentioned contributions are: admits the transfor- (set z =e-T) The series (1) mation into a Fourier series [13] Co (4)0(eT , s, a) = > (a + n)se -nT n=0 CO i2ffna i2Trn)s-1 T or Res < 0, 0 < a < 1 Re s < 1, 0< a < 1 which leads to the Lerch transformation formula (5) (e_T, s, a) - eaTF(l - s)T s-1 = -±(210s-lr (1 - s)eaT i(27ra + s) . 0(el2Tra - - s, 1-+ 2Tra. -e or, if we set + T= It ) log 0 (e-i2Tra, 1 (7) 1 _ s, 1 - 2;1 (6) 0(z, s, a) = i(27)5- r(1 (e-i2Tra , -e iff(2 + 2a) s -a z st log 27Ti , 0(ei2Tra, z - Sp 1 - log 2Tri Hardy's relation (7) lim[0(z, s a) - z-ar(l - s)(log j z+1 Os a) at z = 1, gives the character of the singularity of which is the only singular point in the finite part of the z-plane. 4) Barnes's result. The function 0(z, s, a) - z1-a cD(z, s, 1) has no singularities in the finite part of the z-plane, except the singularity due to z 1-a at the origin, and admits the expansion (6) (1)(z, s, a 1-a z)k = z -a > ' (log lc! 0(z, s, [(s - k, 1)1, k, a) k=0 convergent for 'log zi < 2ff. 4 In all these results the analysis employed is based on the conversion of the infinite series (1) into a loop integral, a subsequent deformation of the loop, and finally the evaluation of the integral by the residue theorem. In more recent contributions a different Apostol [1] proved (6) by a approach has been chosen. method based on the transformation theory of the theta functions and in a further paper [2] it was shown that Hurwitz's series for C(s, a), CO ru - Os, a) =-2(270 (9) s-1sin(2ffna + 2 s), n=1 0 <a <1, Re s <1 or 0 < a < 1, Res <0 This procedure, however, made can be obtained from (6). a rearrangement of (1) necessary, since (9) cannot be obtained from (6) by putting z = 1. It was shown later in a paper by Oberhettinger [20] that the transition z 1 in (6), or T 0 in (5), Hardy's limit relation (7). In can be made utilizing addition, a new proof of (5) and (7) was given applying the summation formulas of Poisson and Plana respectively. in (1) leads to a function The special case a = 1 5 n F(z, s) = z0(z, s, 1) = -s n z , 1 zl < 1, n=1 which is also known as Jonqufere's function [17 p. 33], whose properties have been extensively investigated [11, 12, 19, 21, 23, 26, 27]. It has been widely overlooked, however, that as a special case of cerning F(z s) 0 many results con- are simple consequences of Lerch's transformation formula (6). Furthermore, a large number of functions occurring in various branches of physics, for example the Fermi-Dirac functions [4, 18, 22], the Bose-Einstein functions [5], the Hahn functions [9], and a number of others [14] are likewise special cases of (1). In view of the fact that the various results 0 concerning have been obtained by rather different methods, an attempt is made in this thesis to present a More direct and unified approach to this important transcendental function. In addition a generalization of (1) is obtained. This is achieved by the use of a particular summation formula which is based on the Mellin transform and its asymptotic properties. These general considerations are the subject of Chapter II. In Chapter IlIwe apply these results to a simple special case. This leads to the transformation of a function defined by a generalization of (1) containing four parameters. The result is that 6 CO (a + T(T, v, a, 6) = n)-v e-T(n+a)6 n=0 cS >0 Re T> can be transformed into the convergent or asymptotic (depending on 6) series 1 r/1 - V v-1 - 6k, a)Tk (-1)k . k=0 Since for 6 = 1, Co (12) T(T, v, a, 1) = (a + n) -v eT (n+a) n=0 -Ta-T (1)(e. , v, a), the series (11) includes a generalization of Barnes's series (8) and Hardy's relation (7). In fact all the relations (4) - (9) can be derived from (11) upon specializing parameters. This is carried out in Chapter IV. 7 CHAPTER II A SUMMATION FORMULA INVOLVING THE MELLIN TRANSFORM Consider the two-sided Laplace transform of a function and its inversion formula F(t) 00 g(s) = - fF(t).est dt -co cf+ico F(t) - With x = e -t and g(s)esds 47T1 one has the equiva- F(-log x) = f(x) lent Mellin transform of a function . f(x) and its inver- sion formula co g(s) jr f(x)xS-1 x 0 cr+1.03 f(x) = 1 jr g(s)x s a-ico The theory of this type of an integral transform is well developed [6]. We note here a few simple properties: 8 A sufficient condition for the existence of (15), is that the integral (13) be absolutely convergent. function of and S represents an analytic g(s) The transform function in a strip al < Re s < a2, al are the abscissas of ordinary convergence of a2 The strip of analyticity may extend the integral (15). Re s. to infinity in one or both directions of latter case, al = a2, only. where represents an entire function. g(s) then In the If is defined on the line Re s = al = a2 g(s) We denote the abscissas of absolute conver- gence of the integral (15) by constant such that 01, 82 (al < 82). The in the inversion integral (16) is chosen a (31 <q < a 2 that is, the path of integra- . ' tion in (16), consisting of a straight line parallel to the imaginary s-axis at a distance a, lies in the strip of absolute convergence. We consider now a function x > 0 f(x) such that (15) and (16) hold. we form the series s= f(n + a). n= Then by (16) of a real variable With this function 9 1 S = g(s)(n + a sds, 2'rri 0-i00 Max(1, < <2 ' co where and Again, jrf(x)xs-1 x. g(s) are 0 the abscissas of absolute convergence. Interchanging the order of summation and integration, we have the n + a series which is Hurwitz's zeta function [7, vol. 1, p. 24] 00 (n + a C(s, a) = (18) This series is absolutely convergent for Moreover, c(s -s Re s > 1. 1 a) - ----- is an entire function of f(n + a) - 1 s- 1 We have then c+ico (19) S = C(s, a)g(s)ds c-ico c+1.00 = 3. h(s)ds 77 717 c-i00 where s. 10 h(s) = (20) s, a)g(s). Formula (19) represents the transformation of the left side series into a complex integral, the path of integration being of the type in (16). It may be noted that (19) is a generalization of a formula given by Ferrar [8] (see also [25, p. 63]). Of special interest is the case when the Mellin transform of the function g(s) involved in the summation is such that it is analy- f(x) tic in a half-plane to the left of the path Re s < c of integration except for a finite or infinite number of singularities of one-valued character. holds then for the function h(s). The same behavior This suggests the possibility of completing the path of integration in (19) to a closed path by adding a suitable simple contour such that all or part of the singularities of Re s < c lie inside the closed contour. h(s) C in The application of the residue theorem combined with a suitable behavior of h(s) may then effect the transformation of the left side series in (19) into a so-called residue series for the integral at the right side. This residue series may turn out to be finite, in which case a summation of the series f(n 4- a) has been effected. In the case of a convergent infinite series, this procedure leads to a 11 transformation of two infinite series into each other. Of special importance is the case in which a parameter z, h(s) say, such that contains h(s) as given by (20) is of the form h(s) = where cp(s) -scp(s), is independent of z. Then (under certain conditions) an asymptotic series of the sum for small positive $ in (19) can be given and this series, if z convergent, represents the exact result. This is expressed in the following theorem [6, vol. 2, p. 115]: Theorem. Let c+i= (21) (z) - 1 = fzs cp(s)ds, c-i= such that (a) cp(s) is analytic in a left half-plane Re s < c except for singular points of one-valued character (poles or essential singularities) X0,X1,X2,...;C > Re X0 > Re Xi > - The principal part of the Laurent expansion of =P(s) at Xv has the form 12 (22) - b(v) 1 X + b -1 (v) (v)-r (s X -2 (s - + b2 ) + + + ) rv 14)(x + iy) as 0 1Y1 03 Between two singular points xv+1 Re X there is a v+1 < (3. v < c, co < In each strip of finite width uniformly in Xv and (real) with (3v (v = 0, 1, 2, ...) < Re Xv, such that the integral 00 ji + iy)dy z-i17(4)(13 converges uniformly for 0 < z < Zv . [This is the case, for example, if cp(s) = cp(x + iY) = x 0(1 11 for fixed with Rea < O.] Then c+ioo 1 771 Jr z-s:P(s)ds = (1)(z) c-iconverges for 0 < z < Z and x. 13 (v) (23) b1 = (1)(z) + 1! + + (-log z) -A r.-1 + (-log z -.4 + f 21-1 as +ico -s z y(s)ds (through positive values), 0 where the last term is o(z We have then the ). following asymptotic expansion for 00 (24) (D(z): b(v) b(v) + (D(z) 1 2 1. (-log z) + + v= b(v) r,-1 rv 1 r as z 0 (-log z) + -xv (through positive values). 14 s-plane Figure 1. The first term in the series (24) is (v = 0) due to a shift to the left of the line of integration from c - jco pass between residue of to c + ico X and X 0 z-scP(s) at parallel to itself such as to under consideration of the 1 s = X0. The following terms result from a repeated operation of the same kind. A similar theorem holds for the asymptotics of an integral of the same kind as (21) for the case the singularities of Re s > c cp(s) z + 00 provided are in a right half-plane [6, vol. 2, p. 115]. 15 CHAPTER III A GENERALIZATION OF LERCH'S ZETA FUNCTION The analysis outlined in Chapter II is now applied to a special case. This procedure will lead to expansion formulas concerning a function which represents a generalSpecialization of ization of the Lerch zeta function. certain parameters will yield previously known results. We apply now (19) to the function 6 -V -TX f(x) =x e (25) Here 6 > 0 real and positive. T t and we choose, for the time being The analytic continuation to complex will be obvious later [See formula (34)]. op 1 tT g(s) s-v)-1 e T Then from x6 = t (15) and (25), with the substitution -it dt . > 0 . This is the gamma function integral and 1 7(v-s) (26) to be g(s) = 1 u 6 Res > Re V, r (s T 6 v------- > 0, 16 = Re v, The abscissas of absolute convergence are 12, 2 = 00. (25) and (26) we have By (19), -T(n+a) 71(n + a (27) 0 c+ico v s s 2fficS by s Ss, a)ds, Ciw c > Replacing V)( (l, Re v ) . one obtains .00 -T(n+a) (n + a (28) n=0 v T T = 77TE c+ic. icig. T5 c(6s, c > 1 a)r(s T) s, Max(1, Re v) This integral is of the form (21). uate the integrals (27) and (28). We proceed now to evalFirst we make use of (24) in order to establish the asymptotic behavior of (28) for small T. Having done this, we use (27) for a more de- tailed investigation of the possible convergence of the asymptotic expansion obtained by the previously mentioned theorem concerning an integral of the form (21). (28) and (21), we have z=T and Comparing 17 (29) = TT cp(s) a)r(s - (6s, . 6 It is easily verified that all conditions for [see (3), (9), (4), arities of hold The only singular- (10) Appendix I]. in the finite s-plane are poles of the cp(s) first order. cp(s) Hence only the coefficients (v) are bI different from zero; that is, in the expansion (24) only the first term inside the summation sign gives a contribution. c(z Since and since r(z) (k = 0, 1, 2, ...), a) - is an entire function of (-1)k 1 k! + k z is analytic near z = it follows that for the set of sin- gularities (all of them simple poles) r( T b(0) T 1 V k, blk) = = 7(v - v) OsS, a, [due to 1,k Sk, a)'-'' lc! [due to - -1-')] Therefore, by (24) and (28) CO (31) -ve-T(1+a)6 (n n=0 v-1 T r1 - v -6 )T c(v + k=0 Sk, a)Tk 18 Now the series on the left side of (31) represents a function of analytic in T Re T The possible > 0. convergence of the power series in on the right side T depends on the behavior of its coefficients for large indices We find from (5) and (1), Appendix I, that k. for large k and fixed 1 k! (32) - 6k, a) = v, 6, a k 0[(27')-6ke-k(1-6)log the following result: We obtain then The function 00 (33) T(T, v, a, 6) = (n + a) -v e -T(n+a)6 n= 6 0, -1, 2, > 0, a is an analytic function of T in . Re T By (32) > 0. one has V-1 (34) T(T, V, a, 6) - 1r(1 00 (-1)k - 6k, a Tk k! k=0 wherewhere the series is convergent for all convergent for ITI < 2, if 6 = 1. T If if 6 6 and < 1 > 1, the series is divergent and in this case the series on the right in (34) represents an asymptotic expansion of the 19 function of T Putting 0. T we can write (34) as follows. = log(.), Z(n (35) on the left as T v-1 6 + a)-vz(n+a) 1(log 11) r(1 - = 6 n=0 ov - (log z kl 6k, a) k=0 The series on the right is valid for all 0< 6 For < 1. tically as 6 > 1, For z + 1. in (35) converges for if 0 z the relation (35) holds asympto6 = 1, 'log z < the series on the right 27r Thus, we obtain . the limit relations v-l(36) (n + a)-ve-T(n+a)6 lim T+0 OV, (37) 1 6 a) n + a -vz(n+a) lim - 1 (v , -T- r 16 z+1 = ), vaLW,og a) we have For 6 = 1, 'P(-r, v, a, 1) = (n + a -ve-T(n+a) = e -T4 e -T v, a) , 20 CO T(log 1 , >,(n v, a, 1) + a)-vzn+a = za 4)(z, V, a) . Consequently, (38) (I)(e-T r(l - v, a) - e4T v)Tv-1 = 00 (-1)x Ta a) Tk -- --FTk.0 iTI (I)(z, (39) 27T < a v, a) - - ou 1og v1= 03 (log z) = z a) k! k=0 flog zl < 2Tr (The domain of convergence for this series is discussed in Appendix III.) and (40) lim n + a) -ve-nT rti - v, a) , T+0 (41) lim Z(n (n + a)-vzn - z-ar(1 - \)) (log 1)v-1 = v, s). The last two equations represent Hardy's limit relation (7), while (38) and (39) lead to Barnes's expansion (8). 21 It is worthwhile to note that the integral occurring in (27) can also be evaluated directly without recourse to the general theorem, as reflected by equations (21) (24). The integrand in (27) _s h(s) (42) = t6 s = 1 has poles of the first order at (k = 0, 1, 2, ...). a) (s F and s = v - Sk, Their respective residues are equal to T (-1) k! 1 t-----) S k- k 6T at s=l ov _ kS and a) s = v . We consider now the integral (43) jrh(s)ds taken over the closed contour as indicated in Figure 2, Cc > Max(1, Re v)]. 22 s -plane c+iRN v-(N+1)6 v-26 v-6 V -N6 v Figure 2. The radius of the half-circle in the left half-plane RN is chosen such that it separates the poles and s = v to infinity. (N 1)6 of h(s). s = v - N6 We then let RN tend The contributions to the integral along the horizontal lines vanish as This follows 23 from the behavior of due to (3), h(s) (10), (4), Appendix I. On the circumference of the half-circle S = Re iT We have for large R by (1), (7), (8), Appendix I, _v expER( ih(s)1 = 0 {R (44) 1)cos T log R + 1 1 + R cos T(log 2rr - T log T) /1 1)sin T - RT(T T log 6)]} - R cos Tk- - 1 7T < CP < 2 cos T < 0, In this interval exp[11(- - 1)cos T log 11] 6 sin T > 0 is the dominant one provided That is to say, the contribution to the integral 1. (43) along the half-circle vanishes as of T and the term if 6 < 1. If half-circle vanishes if 6 = 1, T < 2ff RN 4- > 1 (this is the condition leads to an exponential increase of half-circle as R increases. regardless the contribution along the for the convergence of the residue series). 6 co Thus The case h(s) on the the residue theorem leads exactly to our formerly obtained result (34), where in addition the asymptotic character of the residue series in the case 6 > 1 was stated. 24 CHAPTER IV HURWITZ'S SERIES AND LERCH'S TRANSFORMATION FORMULA Let us apply now the results obtained in Chapter III to two special cases. series (9). a = 1, and First we wish to derive Hurwitz's In (34) we set (5 = 1. T = < 1), (0 < i27 Then 11-7N eT2TriS(n+1)(n + 1)-v = + - n=0 ,n (-1)n e ----- (2113) 2 - n) n! . n= This relation is valid for 0 < Re v > 1, 0 < 1. Multi- ++ plying the above relation first by 2 e , then by and adding the results, we obtain CO (n + 1)-vcos 2[(n + 1)S - i v] n=0 (21-a)v-1sinorami-v) + Re v > 1, (-1) (2,T0)11(v-n)cos[ (v+h)], 0< < 1 nt- . 25 If the series on the right is split into even and odd terms, it follows (with sin(nv)F(1 - v) = r7rv co n vcos(271-n8 - 7n v) - ) that v-1 ff(27113) F(v) n=1 00 + cos( v) (-un (2n)! - (2")2n C(V 2n) + n= 00 + sin( 2 (-1)n (2n + I)! v) (2nfi)2n+1C(v - 2n - 1), n= Re v > 1, If v is replaced by 0 < 1 - s, < 1 . we obtain by means of (5), Appendix II, 00 s-1 nsin(21110 (45) 1 (2101-S C(S, r(1 - 2 s) which is Hurwitz's series (9). We proceed now to derive Lerch's transformation formula (5). mula (38). For this purpose we use the expansion forWe divide (38) by r(l - v) and replace the Hurwitz zeta function in the series at the right by the just derived expression (45). We have 26 -k (-1)k a) ru - v + k) (-1) mr(1 - v) 2(2ff)v-1(27T)-k r(1- v) kl 00 nv-1n-k sin(2Trna + 2 v- 2 k) . We observe that (binomial coefficients) (-1) k r(1 - v + k) k!F(1 - v) Then (interchanging the order of summation), 7 k, a) Tk ..2(21)v-1 (- ru - v) k=0 - 1) [k=0 n=1 2Trn . sin(27rna + v- k) Collecting the even and odd terms of this sum, 00 (-1)k k! k=0- - k a) k ru v) T 00 = v-1 sin(2Trna + 7IT v) . 2(21.)v- k=0 n=1 (\27<l ) ) 2Trn (T 00 n v-1 cos(2Trna + 7 Tr n=1 (2k:1)(-1)k(7k)2k4.1 2k 27 We use now the elementary formulas 00 f_ilkta \z2k 1 k2k) 2 [(1 + iz)a + (1- iz)a] k=0 co , (-1)k(2k+1)z2k+1 i[(1 + iz)a - (1 - iz)a] = k=0 from which it follows that 1 2 . 1(1 - )a e i6 + 1 2 i(1 + iz)ae-i6 = 00 = sin 2 (k) (-1) k a z2k - cos 2k+1 a. k=0 k=0 so that Co cc (-1) k! k=0 = Therefore, t (.\) - k, a) r(1 - v) k T (-1)kz2k+1 28 (I)(e -T , v, a) - r(1-V)e Ta Tv-1 , 00 e_2) i(2ffnairv) e 1-v n=1 (n + -i(21)v-1eTar(1-v) (n ,.... lir) lr-v which is (5) written in explicit form (see also [20, equation 11]). Re v < 0, The conditions given in [20], 0 < a < 1, 0 < Im T < 2ff are sufficient to justify interchanging the order of summation carried out previously, 29 BIBLIOGRAPHY On the Lerch zeta function. Pacific Apostol, T.M. Journal of Mathematics 1:161-167. 1951. Remark on the Hurwitz zeta function. Apostol, T.M. Proceedings of the American Mathematical Soc1951. iety 2:690-693. On certain functions defined by Taylor Barnes, E.W. series of finite radius of convergence. Proceedings of the London Mathematical Society 1906. 4:284-316. Applied The Fermi-Dirac integral. Dingle, R.B. Scientific Research 6:225-239. 1957. Dingle, R.B. The Bose-Einstein integral. Applied Scientific Research 6:240-244. 1957. Handbuch der Laplace Transformation. Doetsch, G. Basel, Birkhauser, 1960-1965. 3 vols. Erdelyi, A. et al. Higher transcendental functions. 3 vols. New York, McGraw-Hill, 1953-1954. Summation formulae and their relations Ferrar, W.L. Compositio mathematica to Dirichlet series. 1935. 1:344-360. 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Introduction to the theory of Titchmarsh, E.C. Fourier integrals. 2nd ed., Oxford, 1948. 394 p. On a function which occurs in the Truesdell, C.A. theory of the structure of polymers. Annals of 1945. Mathematics 46:144-157. Wirtinger, W. Ober eine besondere Dirichletsche Journal fiir die Reine und Angewandte Reihe. 1905. Mathematik 129:214-219. APPENDICES 32 APPENDIX I Asymptotic Behavior of and r(s) The asymptotic behavior of certain analytic functions is needed for the evaluation of the integral in (27) or This concerns mainly the (28) as outlined in Chapter III. functions that if g(s) g(s), r(s) g(s) 0 as First we notice Os, a). and Im s Re s for fixed :boo Os, a) transition as Let R -4- 00 Isl (in the strip With regard to Lebesgue lemma for Fourier integrals. and then This follows by the Riemann- of absolute convergence). F(s) f(x), is the Mellin tramsform of we consider two possibilities of the c°- s = x +iy = Re', arg s, and let a < arg s < in a given sector Let s= x + iy and let y for fixed ±-= x. We have (Stirling's formula [7, vol. 1, p. 47]) 1 (1) r(s) = r(s + a) (2) 2n7 s e -s a B es log s[1 [1 + O(!.)], + (2s--.)] ; fixed) (a, . + (3) Both formulas are valid as ± 00 in -Tr < arg s <7 . 33 Also 1 (3) iy)I = Ir(x as 1T 0(1y1X-2- e-71y1\ y + 0° for fixed x. C(s, a) As for the asymptotic behavior of large s for (0 < a < 1), it is clear (for instance from the integral expression [17, p, 23] or from Hurwitz's series (9)) that the dependency on the parameter gible importance. of C(s, a) a is of negli- That is to say, the asymptotic behavior for large can be replaced by that of s CO C(s, 1) = a(s). From the definition of C(s) = In-s 1 1 (Re s > 1) as a Dirichlet series, it is obvious that Ic(s)1 = 0(1) Re s > 1 . Using (4) and the well-known functional equation C(s) = 2(27)s-i51n(i s)r(1 - s)(1 - s) ff(27r)s-lc(1 - s) cos(i s)r(s) with s = x + iy, 1r(s)C(s)1 the result is ilYI ex log(21) , Re s < 34 [The equal sign in (4) and consequently in (6) follows by continuity if one considers (10)]. Applying (1) to (5), we obtain 1 '-- R ev(R,cP) where, = - R v(R, log R cos cp + R cos CP log(27) - (i - cOR sin cp + R cos cp as R (the validity in 0. in Z < cp <7 and < '*-7 CP < the latter interval follows by Schwartz's reflection principle). Similarly, x)(9) (3) applied to (6) gives k(s) I as Finally, in 1(x ± iy)I = 0(IyI2 = 1 y 0. x <0. and the critical strip 0 < x < 1 p. 82], we have / (10) (x as iy) ÷ 00 = 0 and 11 IY17-r) 0 < x < 1. [24, 35 It is not difficult to show (see the integral expression [7, vol. 1, p. 26] or use Hurwitz's series for (s, a)) that the same asymptotic relations hold also for Os, a). That is to say, in the asymptotic sense the terms involving a in Os, a) are of a lower order compared with the dominant part which is independent of a. 36 APPENDIX II Os, A Taylor Series Expansion for a) It follows from 00 Os, a) = Re s > 1 (n + a)-5 that n Os, (-1)n r(n + s) a) r (s) r(n + s) . -'a=1 Hence, by Taylor's theorem CO (1) (-1 n (a - 1)nr(n + s)(n + s) ) r(s)C(s, a) = n! 1 - if < 1, 1 s or, with Riemann's functional equation s - n) (210s+n c(1 r(n + s)On + s) 22= cos[(s + n)] 2 (2) r(s)(s, a) = Co 1.(21)s-1 y (-1)n [27(a - Inn n! cos[-(s + n)] 2 0 - 11 <1, s 1 . c(1 - s - n) 37 If (2) is split into even and odd terms in the summation, the result is r(s)c(s, (3) a) = (-1)n [2ff(a - 1) = Tr(21.) (2n)! COS CO (-1)n + , la - 11 (72- s) (- <1, 1 s S 2n) . we obtain sin(i s) cos(- s), c(s (4) s - 2n) + sin (-2 s) 0 Upon multiplying by (1 Tf 1)]2n+1 [27(a - (2n + 1)! ]2n a) = 03 2(27r)1r (-1)n sin (12- s 1-s 1 (2n)! [2ff(a-1)]2 C(1-s-2n) + 00 (-1)fl + cos (i s)/ 1 (2n+1)! [2ff(a-1)_] 2n+1 C(-s-2n) 0 la - 11 or, with a = 1 + a and < 1, c(s, S + a 1 ; = Os, a) - a-s 38 Os, a) (5) - a-s = Os, 1 + a) = CO = 2 (2Tr) s-1 (1-s sin(- s ((-2n1))11! (27m) 2n (1-s-2n) + 00 (-1)n + cos s (2n+1) ! a 2n+1 (2Tra) < 1, s 1 (-s-2 . 39 APPENDIX III Some Mapping Properties The transition from (34) to (35) represents a mapping of the-T.-plane into the z-plane by means of the mapping function z = e- Let T + iy, = a + it, x = e-a cos t, T then sin t Y = . in the T-plane is mapped into the The strip 0< t < 27 entire z-plane-and the half-strip 0 < a < is mapped into-the interior of the circle 0 < t < 27 co I = 1. z Of particular importance is the mapping of the T.-domain (ITI < 27) of the power series in (38) into that (Ilog zl < 27) of the corresponding series T= pe 1,4) z = eicp (39). Setting = rel.° one obtains (3) cos IzI = r = e-p =--p sin cp . It is sufficient to investigate the mapping of the upper half-circle 171 <2n into the z-plane, since the 40 same considerations apply to the lower half-circle. this purpose we observe the image point z as T For tra- verses the arc ABC of the upper half-circle (Figure 1). 'r-plane Figure 1. Then by (3), = r = e-271. cos eP 6 = arg z = -2ff sin eP . 41 It is obvious from (5) that from 6 = 0 to 0 = -2Tr decreases monotonically e as describes the arc AB, T as T Thus, for each ABC. ferent values of such that r> r' Tr -27 r. < 0 0 = 0 increases r e2ff along the entire arc < 0 there exist two dif- Denote these values by and r' r = r' if r (the smaller is encountered if the larger, if 0 < P < 7 ; P = to e back to while by (4), describes the arc BC; monotonically from 2ff e = and increases monotonically from 7 < cp < ir , and Therefore from (4) and (5), ). (6) r=e 47r2-02 (7) r' =e -1/4 < 1 cp< ff) Tr (0 < cp <). These are the polar equations of two curves in the z-plane. C We observe that r = r' dr, dr' de - au for = 0 e = -2ii; for 0 = 0 . Furthermore, dr de and de +' as -27 . and C' 42 C The last observation means that the slope of both and C' is zero at -2 It. e A few numerical examples will serve to illustrate the situation. In the following table, we give the cor- respondence of a point T on the arc ABC with its image z-point as expressed by (5), and r' to (6), (7) (r belongs to C C'). 4) sin cf) 0 (in degrees) 1 7.2 T 14.5 1 4 ---1T 1 --Tr 2 3 'Tiff T I/175 e2 e2 7T/ST 22.0 eT 30.0 -Tr e1TV 5 38,7 3 T 48.6 4 3 2 e7 eT T 61.1 7 --Tr 4 1 90 -21' 7 T iiTT e --g 'TA-5 eT --Tr 4 5 -21T eT 1 7 e e27T 0 T 1 r' = 17 r -27r sin cP ffiT ffirT 1 -21A-75 -LiTT e4 e--rri/T e iliTT e7 IT/TT e 1 43 We see that rmax r' max Thus, while = e 2ff % 530, = 1 r' min e= -27r = 1; 0.0019 . traverses the arc AB, the image point T traverses a spiral-like curve starting at tion rmin = 8 = 0 C' z in the negative direc- with rin and terminating at As T proceeds along the m ax = 1. arc BC the z-point travels along an "outer" Spiral C -2n with r' in the positive direction, starting at rmax = e2n The "inner" spiral . 0 = 0 and terminating at rmin = rax M = 1 8 =-2n with keeps (with the C' exception of the immediate neighborhood of close to the origin (for still only spiral C, rx, -315° 0 r = 1, a -2n) the value of 0.05), while the values of starting with with r very r' is for the "outer" increase rapidly. This fact is illustrated in Figure 2 (the numbers at certain points of C denote the distance from the origin). It is, of course, not possible to sketch the "inner" spiral C' because of its relatively very small linear dimensions and we therefore omit it in Figures 2 and 3. r' (Note that max rMin rmax rm. e 2n ix, 530 44 z -plane -21T < arg z < 0 Figure 2. Mapping of the upper half-circle of the T plane into the z-plane.