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• W1990221590 abstract "The classical Eisenstein series are essentially of the form Em n((m + rl)z + n + r2) s, m, n ranging over integer values, Imz > O, r1, r2 rational and s an integer > 2. In this paper we show that if s is taken to be complex the series, with rl, r2 any real numbers, defines an analytic function of (z, s) for Imz > 0, Re s > 2. Furthermore this function has an analytic continuation over the entire s plane, exhibted explicitly by a convergent Fourier expansion. A formula for the transformation of the function when z is subjected to a modular transformation is obtained and the special case of s an integer is studied in detail. Introduction. In this paper we are concerned with the function G (z, s, r 1' r2) defined as the sum of the series Y, n 1/((m + r )z + n + r2Y), for z having positive imaginary part, s an arbitrary complex number and r1, r2 arbitrary real numbers. m, n range over all integer values. In ?I we define which branch of the 'multi-valued' complex power should be taken, and show that although the series converges only for Res > 2, G has an analytic continuation to all values of s. This is done by exhibiting a Fourier expansion for G convergent for all s. For r1, r2 rational numbers and s an integer > 3 one sees that G coincides essentially with the classical Eisenstein series; this then is the reason for the title. A few more details on this will be found in ?III. In ?II we determine how G transforms when z is subjected to a transformation z (az + b)/(cz + d) belonging to the modular group. To put this paper in proper perspective we remark that its significance is not just in the formulas in ? H but also in that considering G as a function of the variable s, several classical functions are obtained for special values of s and r1, r2, each of whose functional equations has always been derived separately while here they all appear as specializations of one general formula. Thus, besides the above mentioned Eisenstein series for s = k > 3, rl, r2, rational, we also obtain in ?III Hecke's generalized Eisenstein series for s = 2, while the case s = r = r2= 0 gives us the transformation formula for logqr, ri being Dedekind's 71 function. Received by the editors September 30, 1971. AMS 1970 subject classifications. Primary 10D05; Secondary 30A14, 30A16, 30A58." @default.
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• W1990221590 date "1972-01-01" @default.
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• W1990221590 title "Analytic continuation of Eisenstein series" @default.
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• W1990221590 doi "https://doi.org/10.1090/s0002-9947-1972-0306148-1" @default.
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