ARGUMENT INVERSION FOR MODIFIED THETA FUNCTIONS. Introduction

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1 ARGUMENT INVERION FOR MODIFIED THETA FUNCTION MAURICE CRAIG The standard theta function has the inversion property Introduction θ(x) = exp( πxn 2 ), (n Z, Re(x) > ) θ(1/x) = θ(x) x. This formula is proved in many places [1, p.11], [11, p.124, p.475]. It forms the basis for Riemann s second proof [1, p.32], [8, p.147] of the functional equation for the zeta function. As entry 8 in Chapter 15 of the econd Notebook, Ramanujan [7] gives a like formula (Proposition 1 below) concerning the modified theta function Here, f(x) = 1/(e x 1) and summation is for n 1. ϕ(x) = f(πxn 2 ). (1) A fairly circuitous proof appears in [2]. In this paper I deduce Ramanujan s formula from Riemann s relation, and also state the more general result, with n k (k = an even positive integer) replacing n 2 in equation (1). The key to my proof is the observation that f in equation (1) is the same as in equation (4) below. This remark brings to light a whole class of argument inversion formulae, of which the extension just mentioned is only a first instance. Reminders Let s be a complex variable with real part σ. For σ > the gamma function is defined by the integral Γ(s) = x s 1 e x dx. (2) Integration by parts recovers its basic recurrence property: sγ(s) = Γ(s + 1). This equation extends Γ(s) to the complex plane, except for simple poles at non positive integers, the pole at zero visibly having residue unity. The zeta function is defined for σ > 1 by ζ(s) = 1/n s, an absolutely convergent series. Here, and below, the summation index n runs through all positive integers. Writing [11, p.267] 1

2 2 MAURICE CRAIG L(s) = ( 1) n 1 /n s, we find that ζ(s) L(s) = 2 1 s ζ(s), which extends the definition of ζ(s) to < σ < 1, and shows that there is a simple pole of residue unity at s = 1. As is well known [1, pp.3 32], [8], [11, p.269], ζ(s) can be continued to a function, analytic in the whole complex plane (except this pole), that satisfies the functional equation π s cos(πs/2)γ(s)ζ(s) = 2 s 1 ζ(1 s). (3) Moreover, by letting s in equation (3), we see that ζ() = 1 2. The main result The proof of Proposition 1 below employs a standard procedure with perhaps four basic steps: (i) express ϕ(x) by an inverse Mellin transform [1, p.28], [4, p.196]; (ii) use Cauchy s residue theorem [11, p.112] to shift the contour of integration to its image under the mapping s 1 s, then restore the contour by a change of variable 1 s s; (iii) invoke the functional equation (3) to modify the integrand; (iv) make a second application of the residue theorem and so identify the answer obtained. Proposition 1. With R(x) = π/6x + ζ( 1 2 )/2 x and ω = exp(2πi/8) we have ϕ(1/x) R(1/x) = x/2 [ {ω 1 f 2πω ] [ 2nx + ωf 2πω 1 ]} 2nx / n. Proof. In equation (2) replace x by nx, divide by n s and sum to produce (for σ > 1) Mellin inversion gives Γ(s)ζ(s) = 2πif(x) = C x s 1 f(x)dx. (4) x s Γ(s)ζ(s)ds, (5) where the path of integration C may be taken upwards on any line σ = c with c > 1. I shall invoke just two such lines, : σ = 3/2 and T : σ = 2. For now, replace x within equation (4) by πxn 2 and proceed as before to obtain first thence π s Γ(s)ζ(s)ζ(2s) = x s 1 ϕ(x)dx, 2πiϕ(x) = (πx) s Γ(s)ζ(s)ζ(2s)ds. C

3 ARGUMENT INVERION FOR MODIFIED THETA FUNCTION 3 To apply Cauchy s theorem, choose c = 3/2 and close the contour to the left, so as to form a rectangle B descending on the line σ = 1/2. Now B envelopes the pole at s =, due to the gamma function, and the poles at s = 1/2 and s = 1, due to the zeta factors within the integrand. By our earlier evaluations, the sum of residues at these poles is precisely R(x). The asymptotic behavior of Γ(s), explained in [6, p. I 2], entails nil contributions from the horizontal sides (with imaginary parts ± ). Thus 2πiR(x) = (πx) z Γ(z)ζ(z)ζ(2z)dz B = (πx) s Γ(s)ζ(s)ζ(2s)ds (πx) s 1 Γ(1 s)ζ(1 s)ζ(2 2s)ds. For the first integral, z traverses B in the positive sense. For the other two integrals note that, as s ascends, the variable 1 s descends the line σ = 1/2. To the last integrand apply equation (3), first with 1 s, then with 2s 1, in place of s. After cross multiplying the resulting equations we find that 2π 2 i[ϕ(x) R(x)] = (x/8π 2 ) s 1 cos(πs/2)γ(2s 1)ζ(2s 1)ζ(s)ds. o (2π 2 i/x)[ϕ(8π 2 /x) R(8π 2 /x)] = x s cos(πs/2)γ(2s 1)ζ(2s 1)ζ(s)ds, where the right side can be written (by Mellin summation [4, p.214]) as g(nx), with xg(x 2 ) = x 1 2s cos(πs/2)γ(2s 1)ζ(2s 1)ds = x s cos[π(s + 1)/4]Γ(s)ζ(s)ds/2. T Reversing the order of integration and summation is validated by the uniform convergence of n s on. The subsequent change of variable shifts the contour (without crossing poles) to the line T : σ = 2. Writing 2 cos[π(s + 1)/4] = ω exp(πis/4) + ω 1 exp( πis/4) we can apply equation (5) with c = 2 (that is, T replacing C) to conclude that 4xg(x 2 ) = 2πi[ω 1 f(ωx) + ωf(ω 1 x)]. Of course, the expression in brackets is self conjugate and hence real valued. Passage from the real argument x of equations (4 5) to complex arguments ω ±1 x can be justified, as in [4, pp.24 5], by considering integrals whose contours are circular sectors of angle π/4. The function z s 1 f(z) has no poles for arg(z) π/4, while z s f(z) as z within this region. The same applies if π/4 arg(z). For brevity I omit the easy reduction, to an expression by trigonometric and hyperbolic functions, needed for checking agreement with [2], [7].

4 4 MAURICE CRAIG Analogues Further results are stated without proof, the purpose being merely to affirm that Ramanujan s formula is not an isolated one. To this end, let k denote an even positive integer and write (as generalises equation (1)) ϕ(x) = f(πxn k ). Proposition 2. With R(x) = ζ(k)/πx + Γ(1/k)ζ(1/k)/k(πx) 1/k + 1 4, ω = exp(πi/2k) and y n = 2π(2nx) 1/k we have ϕ(1/x) R(1/x) = [( 1) k/2 (2x) 1/k /ik] n 1/k 1 ( 1) j [ω 2j 1 f[ω 2j 1 y n ) ω 1 2j f(ω 1 2j y n )], the range for the inner sum being 1 j k/2. The proof is almost unchanged, but makes appeal (for α = π(1 s)/2k) to the elementary identity ( 1) k/2 1 sin kα/ cos α = 2 ( 1) j 1 sin(2j 1)α, 1 j k/2. When k = 2, the series on the right side reduces to sin[π(1 s)/4] = cos[π(1+s)/4]. A similar expansion of the ratio on the left is lacking if k is odd. To exemplify further results of the same kind, let K(s) = ( 1) n 1 /(2n 1) s, so that Γ(s)K(s) is the Mellin transform for h(x) = 1/(e x + e x ) = ( 1 2 )sech x. The partial fractions expansion πh(πx) = ( 1) n 1( n 1 )/[( n 1 ) ] 2 + x entails the functional equation K(1 s) = (2/π) s sin(πs/2)γ(s)k(s). This relationship is proved initially for < σ < 1, but provides thereafter analytic continuation of K(s) to the region σ <. Next, if k is a positive integer and ψ(x) = h(πxn k ), then ψ(x) has Mellin transform π s Γ(s)K(s)ζ(ks). Let ω = exp(πi/2k) as before. Proposition 3. With R(x) = Γ(1/k)K(1/k)/k(πx) 1/k 1 4 and y n = 2π[(n 1 2 )x]1/k, we have ψ(1/x) R(1/x) = (2/k)(x/2) 1/k [( 1) n 1 /(2n 1) 1 1/k ]F (y n ). Here, for odd values of k the function F(x) is given by F (x) = f(x) + [ω 2j f(ω 2j x) + ω 2j f(ω 2j x)], 1 j (k 1)/2 whereas, for even values, F (x) = [ω 2j 1 f(ω 2j 1 x) + ω 1 2j f(ω 1 2j x)], 1 j k/2.

5 ARGUMENT INVERION FOR MODIFIED THETA FUNCTION 5 When k = 1, Proposition 3 reads [ r n /(1 + r 2n ) = x ] ( 1) n 1 q 2n 1 /(1 q 2n 1 ), (6) with q = e πx and r = e π/x. Thus we recognise it [3, p. 285] as a form of the identity θ 2 (1/x) = xθ 2 (x). Incidentally, the second factor on the right is q n /(1 + q 2n ), as follows on expanding the general term in a geometric series and reversing the order of double summation. Conclusion If x 1 the series (1) converges rapidly. Ramanujan [7, l.c.] viewed Proposition 1 as determining the error, when y = 1/x is close to zero, entailed in approximating ϕ(y) by R(y), a remark that extends to the other results above. Clearly these formulae by no means exhaust the possibilities. For example, other Dirichlet series [5],[1, p. 57] can replace K, while Davies [4, pp ] and Ogg [6, pp.i 43 45] indicate applications, resembling equation (6), to inversion properties for classical modular functions [9]. Of course, the essential novelty in Ramanujan s identity is that, unlike these identities, it is not otherwise familiar from elliptic function theory. References [1] Bellman, R. A brief introduction to theta functions. Holt, Rinehart and Winston, New York, [2] Berndt, B. C. and Evans, R. J., Chapter 15 of Ramanujan s second Notebook: Part 2, Modular forms: Acta Arith. XLVII (1986), pp [3] Borwein, J. M. and Borwein, P. B. Pi and the AGM. John Wiley & ons, New York, [4] Davies, B., Integral transforms and their applications (2nd edn). pringer Verlag, New York, [5] Hurwitz, A., Einige Eigenschaften der Dirichlet schen Funktionen F (s) = (D n).1/n s, die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten. Zeitschr. für Math. u. Phys. 27 (1882), pp (= Math. Werke Bd. I, pp ). [6] Ogg, A., Modular forms and Dirichlet series. W. A. Benjamin Inc., New York, [7] Ramanujan,., Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay, [8] Riemann, B., Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse: Monats. Berlin Akad. (1859), pp (= Gesamelte Abhandlungen, Chelsea Publishing Co., New York, pp and pp ). [9] Weber, H., Lehrbuch der Algebra, Bd 3., Chelsea Publishing Co., New York. [1] Weil, A., Elliptic functions according to Eisenstein and Kronecker. pringer, New York, [11] Whittaker, E. T. and Watson, G. N. A course of modern analysis (4th edn). Cambridge, 1962.