The American Mathematical Monthly, Vol. 92, No. 2. (Feb., 1985), pp

Transcription

1 The Gamma Function and the Hurwitz Zeta-Function Bruce C. Berndt The American Mathematical Monthly, Vol. 92, No. 2. (Feb., 1985), pp Stable URL: The American Mathematical Monthly is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. Thu Mar 6 22:28:

2 126 BRUCE C. BERNDT [February 4. I. M. Gelfand and G. E. Silov, Generalized Functions, vol. 3: Theory of Differential Equations, Academic Press, New York, J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Publications, New York, L. Harmander, Linear Partial Differential Operators, Springer Verlag, Berlin-Heidelberg-New York, F. John, Partial Differential Equations, 3rd ed., Springer Verlag, New York-Heidelberg-Berlin, K. Keller and A. Schneider, Ein funktionalanalytischer Beweis des Satzes von Cauchy-Kowalewsky, Manuscripta math., 39 (1982) S. v. Kowalevsky, Zur Theorie der partiellen Differentialgleichungen, J. Reine Angew. Math., 80 (1) (1875) S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, M. Nagumo, ~ ber das Anfangswertproblem Partieller Differentialgleichungen, Japan. J. Math., 18 ( ) L. Nirenberg, An abstract form of thg nonlinear Cauchy-Kowalewski theorem, J. Differential Geom., 6 (1972) T. Nishida, A note on a theorem of Nirenberg, J. Differential Geom., 12 (1977) L. V. Ovsjannikov, Singular operators in Banach scales, Dokl. Akad. Nauk SSSR, 163 (1965) ; Soviet Math. Dokl., 6 (1965) , A nonlinear Cauchy problem in a scale of Banach spaces, Dokl. Akad. Nauk SSSR, 200 (1971) ; Soviet Math. Dokl., 12 (1971) ,Abstract form of the Cauchy-Kowalewski theorem and its applications (Russian). Partial Differential Equations (Proc. Conf. Novosibirsk, 1978), 88-94, 250 "Nauka" Sibirsk, Otdel., Novosibirsk, T. H. Pate, A Direct Iterative Method for an Abstract Cauchy-Kowalewsky Theorem, Indiana Univ. Math. J., 30 (3) (1981) I. G. Petrovski, Lectures on Partial Differential Equations, Interscience Publishers 1961 (third printing). 19. R. Redheffer and W. Walter, Existence theorems for strongly coupled systems of partial differential equations over Bernstein classes, Bull. Amer. Math. Soc., 82 (1976) F. Treves, On the theory of linear partial differential operators with analytic coefficients, Trans. Amer. Math. SOC.,137 (1969) , An abstract nonlinear Cauchy-Kovalevska theorem, Trans. Amer. Math. Soc., 150 (1970) , Basic Linear Partial Differential Equations, Academic Press, New York-San Francisco-London, NOTES For instructions about submitting Notes for publication in this department see the inside front cover. THE GAMMA FUNCTION AND THE HURWITZ ZETA-FUNCTION BRUCEC. BERNDT* Department of Mathematics, University of Illinois, Urbana, IL The gamma function r(x)may be defined by n!(n+ 1)" r(x)= lim,+, x(x + 1)...(x+ n) ' where x is any complex number, while the Hunvitz zeta-function [(s, x)is defined by *Research partially supported by National Science Foundation Grant No. MCS

3 19851 NOTES 127 where a = Re s > 1 and x is any complex number. (Normally, it is assumed that 0 < x I1in the definition of {(s, x).) Observe that {(s, 1) = {(s), where {(s) denotes the Riemann zeta-function. The function {(s, x) can be analytically continued into the entire complex s-plane and is holomorphic except for a simple pole at s = 1. In 1894, Lerch [9, p. 131 established the following beautiful formula relating r(x) and {(s, x). THEOREM. If the prime ' denotes differentiation with respect to s, then (3) Lo~~(x).~(o,x) = - sf(o). This formula is not particularly well known. However, it has achieved some prominence in recent research. If L(s, X) denotes the Dirichlet L-function associated with the character X, then there is a classical formula for L'(0, X) that depends upon (3). Analogues of this formula have been established for other L-functions, e.g., p-adic L-functions, and analogues of (3) naturally arise. (See, e.g., [2], [3], [lo],and [Ill.) The main purpose of this note is to present a new, short proof of (3) that is more elementary than previously known proofs [9], [13, pp ], [14, p A second purpose is to show how (3) along with other properties of the Hurwitz zeta-function can be used to give short proofs of Gauss's multiplication theorem, Kummer's formula for Log r(x), and the reflection theorem. These proofs are new, but we do not claim that they are better or shorter than other proofs; we primarily wish to emphasize the little noticed connection between the two classical functions in the title of this paper. Proof of the Theorem. For a > 1 and x > 0, a straightforward application of the Euler- Maclaurin summation formula [12, p. 131 yields By analytic continuation, (4), in fact, is valid for a > -1. Differentiating both sides of (4) and then setting s = 0, we find that 1 =xlogx-x--logx+ lim dt 2 n-rm In particular, Thus, (5) and (6) give (7) {'(o,x) - {'(O) = lim {Log(k + 1) - Log(k + x)) n-ca

4 BRUCE C. BERNDT [February On the other hand, by (I), -Log(n + x) +xlog(n + 1) n LogI'(x)= lim Logk+xLog(n+l)- zlog(k+x) k=o A comparison of (7) and (8) completes the proof for x > 0. By analytic continuation, (3) holds for all complex x. In the sequel, we shall need one fact about r(x) and a few properties of [(s, x). Recall that Euler's constant y may be defined by Then, from (I), it is an easy exercise to show that (9) rf(i) = -y Recall that [14, p and Lastly, we shall need the most important property of [(s, x), Hunvitz's formula [12, p. 371, [14, p. 2691, where 0 < x < 1 and u < 1. (A simple proof of (12) can also be found in [I].) GAUSS'SMULTIPLICATION THEOREM ([14, p. 2401). For any complex number x and positive integer n, Proof. For x > 0 and u > 1, by (2),

5 = (1 - ns){'(s, x) - nslog n[(s, x), where in the penultimate line we set r =jn + k. By analytic continuation, the extremal sides of (14) are equal for all s. Setting s = 0, employing (3), and using (10) and (ll), we find that, for x > 0, which is clearly equivalent to (13). By analytic continuation, (13) holds for all complex x. The following elegant representation is due to Kummer [7], [ti,pp Hardy [4], [S], [6, pp , has given two proofs, and references to other proofs may be found in [4]. See also [14, p KUMMER'S FORMULA. If 0 < x < 1, then Log T(x) = I 1 Log( ncsc( nx)) + C (y + Log(2nk))sin(2nkx), mk k=l where y denotes Euler's constant. Proof. Differentiate both sides of Hunvitz's formula (12) with respect to s, set s = 0, and employ (3) to find that Logr(x) = ~'(0, X) - ~'(0) The second series on the right side of (16) is well known to equal $Log($csc(nx)), 0 < x < 1[14, p Using also (9) and (ll), we readily find that (16) reduces to (15). REFLECTION FORMULA ([14, p. 2391). Ifx is any complex number, then (17) r(~)r(i- X) = TCSC(~X). Proof. Suppose first that 0 < x < 1.Then by Kummer's formula (15), which is equivalent to (17). By analytic continuation, (17) holds for all x. References 1. B. C. Berndt, On the Hunvitz zeta-function, Rocky Mountain J. Math., 2 (1972) B. Ferrero and R. Greenberg, On the behavior of p-adic L-functions at s = 0, Invent. Math., 50 (1978) B. H. Gross and N. Koblitz, Gauss sums and the p-adic r-function, Ann. of Math., 109 (1979) G. H. Hardy, A new proof of Kummer's series for logi'(a), Mess. Math., 31 (1902) , On Kummer's series for logi'(a), Quart. J. Math., 37 (1906) , Collected papers, vol. IV, Clarendon Press, Oxford, E. E. Kummer, Beitrag zur Theorie der Function T(x) = j~e-"ux-'du, J. Reine Angew. Math., 35 (1847) , Collected papers, vol. 11, Springer-Verlag, Berlin, 1975.

6 130 BRUCE C. BERNDT [February 9. M. Lerch, DalSi studie v oboru MalmstCnovskych fad, Rozpravy Ceske Akad. 3, no. 28, 1894, 63 pp. 10. C. J. Moreno, The Chowla-Selberg formula, J. Number Theory, 17 (1983) T. Shintani, On special values of zeta-functions of totally real quadratic fields, Proc. Inter. Congr. Math., Helsinki, 1978, vol. 2, Acad. Sci. Fennica, Helsinki, 1980, pp E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Springer-Verlag, Berlin, E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., University Press, Cambridge, MISCELLANEA Just as surely as our understanding of Nature is really valid only to the extent that it is mathematical, so also our understanding of higher domains must be based on mathematical models. -R. Steiner ( ), quoted in Mathematical Reviews 82k : Each natural science contains genuine Science only to the extent that it contains Mathematics.... It may be that a formal philosophy of Nature in general, that is, a philosophy that deals only with general concepts, is possible without Mathematics, but a formal natural science dealing with definite objects (whether Physics or Psychology) is possible only with the use of Mathematics, and since each natural science contains only as much genuine Science as it contains a priori knowledge, it follows that a natural science is a genuine Science only to the extent that Mathematics can be applied to it. -1mmanuel Kant, Metaphysische Anfangsgrtmde der Natunvissenschaft, Perhaps we should sometimes listen to the other side. Goethe, for example, did not like Mathematics. Mathematicians are amazing people. In virtue of their accomplishments, they have set themselves up as a universal guild and will acknowledge only what suits their circle, what their method of organization can produce. A prominent mathematician once said, when someone strongly recommended a topic in Physics, "But can't it be reduced to calculation'.)" -Goethe, Maximen und Reflexionen, no It surely does not follow that the hunter who kills the game must also cook it. A cook might go hunting and shoot well; but he would be badly mistaken if he claimed that only a cook can be a good shot. It seems to me that this is the situation of mathematicians who claim that nobody can understand or discover physical phenomena without being a mathematician, since they should be pleased enough if the meat is brought to their kitchen for them to lard it with formulas and dress it as they like. -Goethe, Maximen und Reflexionen, no ANSWER TO PHOTO ON PAGE 93 The photo is of Louis de Branges, who proved in 1984 the conjecture Bieberbach made in 1916, by establishing a strong inequality proposed by the Russian function-theorist I. M. Milin.