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Kyushu J. Math. 65 (2011), 39–53 doi:10.2206/kyushujm.65.39

THE MODULAR RELATION AND THE FUNCTION

K. CHAKRABORTY, S. KANEMITSU and X.-H. WANG (Received 5 March 2010)

Dedicated to Professor Ken’ichi Sato on his sixtieth birthday with compliments and friendship

Abstract. In this paper we shall locate a class of fundamental identities for the function and trigonometric functions in the chart of functional equations for the - functions as a manifestation of the underlying modular relation. We use the -transform but not the inverse Heaviside . Instead we appeal to the reciprocal relation for the Euler which gives rise to the partial fraction expansion for the cotangent function. Through this we may incorporate basic results from the theory of the digamma (and gamma) function, thereby dispensing also with the beta-transform. Section 4 could serve as a foundation of the theory of the through the digamma function.

1. Introduction and the digamma function

We recall that Euler’s solution of the Basel problem, the evaluation of the value of the infinite ∞ 2 series n=1(1/n ), depends on the product representation for the sine function (cf. (20) below). Since the value is ζ(2) = π2/6, there must a relation between the sine function and the Riemann zeta-function, which has been established as a circle of equivalent identities in [13, Ch. 5]. In this paper we shall give another foundation of this circle of identities by making use of the Euler digamma function, thereby suggesting a new foundation of the theory of the gamma function through the digamma function. We use the following notation throughout this paper (cf. e.. [5]): (a) s = σ + it is the complex variable; is another complex variable, used interchangeably with s; N 1 () ψ(s) =− lim − log(N + s) N→∞ n + s n=0 is the digamma function (cf. (40)); s (c) log (s) = ψ(u) du 1 is the log-gamma function (in both the digamma and log-gamma functions, non- positive integer values of s are excluded);

2000 Subject Classification: Primary 11M35; Secondary 11M06, 33B15. Keywords: Euler digamma function; gamma function; Hurwitz zeta-function.

c 2011 Faculty of Mathematics, Kyushu University 40 K. Chakraborty et al

() γ =−ψ(1) is the Euler constant (cf. (44)); ∞ 1 (e) ζ(s, a)= (n + a)s n=0 is the Hurwitz zeta-function, where σ>1, a ∈ C,anda = non-positive integer; ∞ 1 (f) ζ(s)= ζ(s, 1) = ,σ>1, ns n=1 is the Riemann zeta-function; (g) for x ≥ 0, a ∈ C, a = non-positive integer, and u ∈ C, u Lu(x, a) = (n + a) 0≤n≤x is the partial sum of the Hurwitz zeta-function ζ(−u, a); ¯ (h) Bk(t) = Bk(t −[t]) is the kth periodic Bernoulli ; k k − (i) B (t) = B tk r k r r r=0

is the kth Bernoulli polynomial (in both of which Bk is the kth , and [t] is the integral part of t); and   m,n  aj (j) Gp,q x  bj is the Meijer G-function defined by (cf. [5, Ch. VI, p. 206] and [25])   m,n  a1,a2,...,ap Gp,q x  b1,b2,...,bq m − n − + 1 j= (bj s) i= (1 ai s) = 1 1 xs , (1) πi q − + p − 2 (γ ) j=m+1 (1 bj s) i=n+1 (ai s) where (γ ) is the vertical Bromwich path extending from γ − i∞ to γ + i∞ so that all poles of (bj − s), j = 1, 2,...,m, are to the right, and all poles of (1 − ai − s), i = 1, 2,...,n, are to the left of (γ ). The last separation of poles condition is essential for an integral to be the G-function. The notation is slightly different in main references [5], [19]and[25]. In [5]and[25], the notation is the same except for the sign of the abscissa of the integration path being opposite. In [13, Ch. 5] we have presented the theory of the gamma function through the Hurwitz zeta-function and located various fundamental identities. Here we shall appeal rather to the Euler digamma function, which is the Laurent constant of the Hurwitz zeta-function around s = 1, and locate the same class of identities, the highlight being the claim that the partial fraction expansion for the cotangent function is equivalent to the for the Modular relation and digamma function 41

Riemann zeta-function. The partial fraction expansion for the cotangent function ∞ ∞ 1 2z 1 1 1 1 1 cot πz= − = + − (2) πz π n2 − z2 πz π n + z n − z n=1 n=1 is usually proved as an application of the residue (cf. e.g. [22, p. 113]) and regarded as independent of the theory of zeta-functions. However, our recent theory of general modular = relations (cf. [8, 9, 14]), which gives identities equivalent to the functional equation ϕ(s) − = ∞ s = ∞ s R(s)ψ(r s),whereϕ(s) n=1(αn/λn), ψ(s) n=1(βn/μn) and R(s) is a general gamma factor, reveals that it is a specification of the following general modular relation:   ∞  ∞  αn 2,0  − βn 1,1 μn  1 − a G zλn  = − G  λs 0,2 s, a r s 1,1 z r − s n=1 n n=1 μn L s−w + Res((a − s + w)χ(w)z ,w= sk). (3) k=1 Indeed, the left-hand side of (3) reduces to the cotangent function in view of the inverse Heaviside integral   − 2,0 2  = z+1/2 G0,2 x  1 2x Kz−1/2(2x) (4) z, 2 √ −z and the reduction formula K1/2 = (π/2z)e . On the other hand, the right-hand side of (3) reduces to the partial fraction expansion of the coth in view of the beta-transform (in previous literature it used to be called the Mellin– Barnes integral, but this refers to a much wider class of functions, cf. [19])  (z + s)(−s)  + −z = 1 s = 1 1,1 −1  1 (1 x) x ds G1,1 x  , (5) 2πi (c) (z) (z) z for x>0and−ξ0, 0

ψ(z) − ψ(1 − z) =−π cot πz, (10) for z/∈ Z, which gives rise to the partial fraction expansion for the cotangent function; cf. Proposition 4. Note that using Legendre’s representation for the digamma function [13, (2.59), p. 45] we may deduce (8) (whence (9)), so that we could base the whole discussion on the theory of the digamma function and that (10) is equivalent to (15) under logarithmic differentiation. In Section 3, we make a hydromechanical intermission to the Karman vortex street and we shall give a rigorous foundation, with the aid of the functional equation for the Riemann zeta-function, of the complex potential for a vortex row in which there are infinitely many vortices with the same intensity and rotating in the positive direction at the same distance apart. In Section 4, we deduce all the results on the digamma function from the integral representation, showing that it can give a foundation of the theory of the gamma function through that of the digamma function.

2. The first circle

PROPOSITION 1. The Weierstrass product representation for the gamma function ∞ −1 − z (z + 1) = e γz ez/n 1 + , (11) n n=1 or the Gaussian representation n! (z) = lim nz (12) n→∞ z(z + 1) ···(z + n) is a consequence of (42). The difference equation

(z + 1) = z(z) (13) is a consequence of (11) and the Gaussian representation ∞ 1 1 γ = − log 1 + (14) n n n=1 for the Euler constant. Proof. We prove (13). We rewrite (11) for (z) as follows: ∞  n n (z) = eγ −γz e−1/n e−z/n n − 1 n − 1 + z n=2 ∞ 1  n + 1 n = eγ −γz e−1/(n+1) e−z/(n+1) . z n n + z n=1 Factoring out (z)/z and substituting (14), what remains is e · e−z · ez−1 = 1andthe result follows. 2 Modular relation and digamma function 43

PROPOSITION 2. The reciprocal relation π (s)(1 − s) = (15) sin πs for the gamma function is a consequence of the asymmetric form of the functional equation

− − πs ζ(1 − s) = 21 sπ s (s) cos ζ(s) (16) 2 for the Riemann zeta-function. Also, (15) in turn implies (10).

Proof. We change s into 1 − s in (16) and deduce its counterpart:

− πs ζ(s)= 2sπs 1(1 − s) sin ζ(1 − s). (17) 2 Now multiplying (16) and (17) and canceling the common factor ζ(s)ζ(1 − s), we arrive at (15). 2

Remark 1. The proof of Proposition 2 is modelled on Eisenstein’s 1849 proof (cf. [24]) of the functional equation for the Lipshitz–Lerch transcendent. A standard proof (cf. e.g. [17]) is via the beta-function and use is made of (7). The above proof suggests that the gamma function is a factor at infinity place of the adelic product for the zeta-function in question.

LEMMA 1. The asymmetric form (16) of the functional equation for the Riemann zeta- function is a consequence of that for the Hurwitz zeta-function (0

s−1 (π/2)is −(π/2)is ζ(s, a)=−i(2π) (1 − s)(e l1−s(a) − e l1−s (1 − a)), (18) = ∞ 2πina s where ls (a) n=1 e /n stands for the function, which in the long run is a consequence of (50).

A recent proof of (18) based on the Fourier expansion of the Dirac -function can be found in [1]or[10]. A more laborious but easier proof can be found in [20] (for the Riemann zeta) and [19] (for the general case). It amounts to completing the integral in 1 1 1 ζ(s, a)= a1−s + a−s + sa−s−1 s − 1 2 12 + ∞ s(s 1) ¯ −s−2 − B2(t)(t + a) dt, σ > −2, (19) 2 0 ∞ ¯ + −s−2 in the form −a B2(t)(t a) dt, then using the absolutely converging Fourier series for ¯ B2(t) and appealing to a formula for the incomplete gamma function. We refer to the above references.

PROPOSITION 3. The product representation for the sine function ∞ sin πz  z2 = 1 − (20) πz n2 n=1 is a consequence of (11) and (15). 44 K. Chakraborty et al

Proof. Writing −z for z in (11), we get ∞ −1 − z (1 − z) = eγz e z/n 1 − . (21) n n=1 Multiplying (11) and (21), we deduce that ∞ −  z2 1 z(z)(1 − z) = 1 − , (22) n2 n=1 where we used the formula (13). Plugging (15) in (22) gives (20). 2

PROPOSITION 4. The partial fraction expansion for the cotangent function ∞ 1 2z 1 cot πz= − πz π n2 − z2 n=1 ∞ 1 1 1 1 = + − , (23) πz π n + z n − z n=1 where the series is uniformly convergent in any domain not containing integer multiples of π, is equivalent to (20). It is a consequence of (10) and (42). Proof. Equations (20) and (23) are related by logarithmic differentiation. More details will appear in Remark 4 in Section 4. The second assertion immediately follows from those formulas. 2

PROPOSITION 5. The partial fraction expansion for the hyperbolic cotangent function and (23) are equivalent: ∞ 1 1 1 x 1 1 coth πx = + = + , Re x ≥ 0. (24) 2 e2πx − 2 2πx π n2 + x2 1 n=1 Proof. This follows by putting ix = z in (23) (i.e. we move from the right half-plane into the upper half-plane). 2

LEMMA 2. The partial fraction expansion for coth x and the (symmetric form) functional equation − s − − 1 − s π s/2 ζ(s)= π (1 s)/2 ζ(1 − s) (25) 2 2 are equivalent. Proof. This is a specification of (3). First proofs can be found in [11]and[15]. 2 Remark 2. Historically, (24) was first used to deduce (25) (the fifth proof of [23], where an appeal to (9) is needed). Then Koshlyakov [15] deduced (24) from (25) on the use of (8). Supplementarily, we state a result which allows us to skip the above propositions and deduce (23) directly from (18) or rather its equivalent under (15): + (1 s) πis/2 −πis/2  l−s (x) = i {e ζ(1 + s, x) − e ζ(1 + s, 1 − x)}.(33) (2π)1+s Modular relation and digamma function 45

PROPOSITION 6. The functional equation (18) for the Hurwitz zeta-function implies the partial fraction expansion (23) for the cot function.

Proof. We remark that the functional equation (18) for ζ(s, x)may be expressed on the basis of (15) as (33). First we assume that Im x>0. Then the sum for l0(x) converges for every s ∈ C,and the left-hand side is ∞ e2πix 1 l (x) = e2πinx = = (−1 + i cot πx). (26) 0 1 − e2πix 2 n=1 By , this holds true for every x ∈ R − Z. We consider the as s → 0 and with s>0 on the right-hand side of (33).Firstwe note that

− {eπis/2ζ(1 + s, x) − e πis/2ζ(1 + s, 1 − x)}−{ζ(1 + s, x) − ζ(1 + s, 1 − x)} − = (eπis/2 − 1)ζ(1 + s, x) − (e πis/2 − 1)ζ(1 + s, 1 − x) πs − = 2i sin {eπis/4ζ(1 + s, x) + e πis/4ζ(1 + s, 1 − x)} 4 sin(πs/4) π − = 2i {eπis/4sζ(1 + s, x) + e πis/4sζ(1 + s, 1 − x)}, (π/4)s 4 which tends to πi as s → 0 on account of lims→0 sζ(1 + s, x) = 1. Secondly, since ∞ 1 1 ζ(1 + s, x) − ζ(1 + s, 1 − x) = − ,σ>0, (n + x)1+s (n + 1 − x)1+s n=0 we get

lim (ζ(1 + s, x) − ζ(1 + s, 1 − x)) s→0,s>0 ∞ ∞ 1 1 1 2x = − = + . n + x n + 1 − x x x2 − n2 n=0 n=1 Hence the limit of the right-hand side of (33) as s → 0 through positive values is ∞ 1 2x i + . x x2 − n2 n=1  Combining this with l0(x) on the left-hand side of (33) , we conclude (23). 2

PROPOSITION 7. The functional equations in symmetric form (25) and in asymmetric form (16) are equivalent under (46) and (15).

The proof is immediate.

LEMMA 3. The functional equation (25) for the Riemann zeta-function and (18) for the Hurwitz zeta-function are equivalent. 46 K. Chakraborty et al

Proof. This can be found in [11] and is a manifestation of the far-reaching modular relation principle. 2 We are now in a position to state the main result of the paper.

THEOREM 1. By the theory of the digamma function, all formulas (15), (16), (18), (20), (23), (24) and (25) are equivalent in the sense of the following logical scheme.

(42) ⇒ (11) ⇒ ⇐⇒ ⇐⇒ (24) ⇐⇒ ⇐⇒ (18) ⇒ (20) (23) (25) (18) ⇒ (16) ⇒ (15) ⇐ (27) ⇐ (18) (16)

LEMMA 4. (Berndt) The functional equation (18) for the Hurwitz zeta-function implies Kummer’s Fourier series for log (x), which reads (x) 1 1 log √ =− log(2sinπx)+ (γ + log 2π)(1 − 2x) 2π 2 2 ∞ 1 log n + sin 2πnx, 0 < Re x<1, (27) π n n=1 which implies the reciprocal relation (11). The proof was due to Berndt [2]. The proof depends on Lerch’s formula (52) and the integral representation for ζ(s, a) givenby[13, (3.6), p. 55] with x = 0. We postpone the proof of the following theorem to Section 4. Note also our recent paper [7] which develops the theory of arithmetical Fourier series from that of the modular relation.

THEOREM 2. Kummer’s Fourier series for log (x) is equivalent to the functional equation (25) for the Riemann zeta-function.

3. The Karman vortex street

Consider a two-dimensional flow of a vortex with the complex potential given by  w =− log z, 2πi where >0 is a constant. This is the flow (in the positive direction) around the vortex at the origin. Indeed, putting w = φ + iψ and z = reiϑ,wehave   ϕ =− ϑ, ψ = log r. (28) 2π 2π The streamline ψ = const. is r = const., i.e. concentric circles around the origin. The radiation velocity v(r) and the tangential velocity v(ϑ) are given respectively by ∂ϕ 1 ∂ϕ  v(r) =− = 0,v(ϑ)=− = , ∂r r ∂ϑ 2πr so that the induced velocity caused by the vortex is only the circulation v(ϑ) around the origin. The constant  is called the intensity of the flow. Modular relation and digamma function 47

Apparently, if the vortex is at the point α, the complex potential is given by  w =− log(z − α). (29) 2πi In most textbooks on fluid mechanics, one finds the following argument. We consider the 2n + 1 vortices with the central one at the origin. Then by (29) and the principle of superposition, the complex potential is  n wn =− log z + (log(z − ka) + log(z + ka)) . (30) 2πi k=1 Since log(z − ka) + log(z + ka) = log(z2 − ka2), it follows, after slight transformation, that  πz n z2 wn =− log 1 − + const. 2πi a k2a2 k=1 Hence choosing coordinates that annihilate the constant and letting n →∞, we obtain ∞  πz  z2 w = lim wn =− log 1 − . (31) n→∞ 2πi a k2a2 k=1 Comparing (31) with (20), we conclude (33). We make this more rigorous by a zeta-regularization.

THEOREM 3. The complex potential of a vortex row in which there are infinitely many vortices with the same intensity and rotating in the positive direction at distance a apart is given as the formal sum   z/an w = lim wn =− log z + log(z − an)e , (32) n→∞ 2πi which can be expressed as the zeta-regularized limit [16, (2), p. 224]  πz w =− log sin . (33) 2πi a Proof. The complex potential w may be given as the limit of (30) in the following sense. Let us recall the partial fraction expansion of the cotangent function (23) valid for all z ∈ C save for integer multiples of π: ∞ 1  1 1 z = + + , cot − (34) z n=−∞ z πn πn where the prime on the summation sign means that the term with n = 0 is omitted. We may integrate (34) term by term from 1 to z to obtain the formal sum  log sin z = log z + log(z − πn)ez/πn, (35) where the sum is over all non-zero integers taken in a symmetric way. Since we may interpret (35) to mean ∞  z sin z = z 1 − ez/πn, (36) n=−∞ πn i.e. (20), we obtain (33) as the zeta-regularization of (35). 2 48 K. Chakraborty et al

Example 1. (The Karman vortex row) With (33) in mind, we may consider the Karman vortex street, which is the row of two rows of infinite vortices with the same intensity but with the opposite rotation. We express them as two rows above and below the real axis at height h and each vortex in one row lies in the middle of the other row. The complex potential of the above vortex row is given by  π ih w =− log sin z − , 2πi a 2 and that of the lower row is  π ih a w =− log sin z + + . 2πi a 2 2 Hence their composition potential is  sin(π/a)(z − ih/2) w =− log . (37) 2πi cos(π/a)(z + ih/2) The stream function ψ is the imaginary part of (37) and can be computed to be  cosh(2π/a)(y − h/2) − cos(2π/a)x w = log . (38) 4π cosh(2π/a)(y + h/2) + cos(2π/a)x This follows from

|sin(x + iy)|2 = 4(cosh 2x − cos 2y), |cos(x + iy)|2 = 4(cosh 2x + cos 2y).

It is known due to Karman that the vortex street is stable when sin(πh/a) = 1orh/a ≈ 0.281 [16, p. 228].

4. Integral representation for the partial sum and uniqueness theorem

In this section we shall deduce all the results that we used in Section 2 from the following theorem.

THEOREM 4. [13, Theorem 3.1, p. 55] For x ≥ 0, a ∈ C, and a =non-positive integer, we have l −1 1 ¯ −r L− (x, a) = (n + a) = log(x + a) − ψ(a) − B (x)(x + a) 1 r r 0≤n≤x r=1 ∞ ¯ −1−l + Bl(t)(t + a) dt, (39) x where the last integral may be estimated as O(x−l−1).

COROLLARY 1. (i) Equation (39) with x →∞implies the generic definition for the digamma function N 1 ψ(z) =− lim − log(N + z) , (40) N→∞ n + z n=0 Modular relation and digamma function 49 and (39) with a = 1 and x →∞implies N 1 γ = lim − log(N + z) . (41) N→∞ n n=1 (ii) We have that ψ(a) admits the Gaussian representation ∞ 1 1 ψ(z) + γ = − , (42) n z + n − 1 n=1 the series being uniformly convergent. (iii) Equation (39) with x = 0 implies the integral representation ∞ 1 −1 ¯ −2 ψ(z) = log z − z + B1(t)(t + z) dt. (43) 2 0 Proof. Only (ii) needs a proof. Formula (39) gives 1 1 ψ(a) = log(x + a) − + O , n + a x 0≤n≤x which we rewrite as N + a N 1 1 N 1 1 ψ(a) = log − − + log N − + O . N n − 1 + a n n N n=1 n=1 Taking the limit as N →∞and substituting (41) with z = 1, we deduce (42). In (41), z is usually taken to be 0, but it can be any number as in (41) because log(N + z) − log(N + w) → 0,N→∞. 2

COROLLARY 2. (i) We have γ =−ψ(1). (44) (ii) We have the duplication formula = 1 + 1 + 1 + ψ(2z) 2 ψ(z) 2 ψ(z 2 ) log 2, (45) and, a fortiori, = 2z−1 −1/2 + 1 (2z) 2 π (z)(z 2 ). (46) Proof. This can be found, for example, in [3]. Indeed, using (40) in the form 1 1 1 ψ(z) + ψ z + 2 2 2 1 1 1 = lim log(x + z) x + z + − x→∞ 2 2 2n + 2z 0≤2n≤2x − 1 2n + 1 + 2z 0≤2n+1≤2x+1 1 1 2 1 = lim log 4 x + z + − log 2 − , x→∞ 2 2 n + 2z 0≤n≤2x+1 50 K. Chakraborty et al we conclude that it is ψ(z) − log 2. Clearly, (46) follows from (45) if we use 1 = 1/2 2 (2 ) π . Remark 3. The property in Corollary 2 is a special case of the distribution property (or Kubert identity) shared by a wide class of functions (cf. [18]). Now we claim that formula (42) is a unique definition of ψ(z) in the sense of the Dufresnoy–Pisot type uniqueness theorem (cf. [4]). First we state a lemma.

LEMMA 5. If the function g : R+ → R (R+ meaning positive reals) satisfies lim (g(x + n) − g(n)) = 0, 0 0; (c) f is a solution of the difference equation

f(x+ 1) − f(x)= g(x), x ∈ R+. If such a function exists, it is given by the Gaussian representation n−1 f(x)= lim λ + xg(n) − g(x) − (g(x + k) − g(k)) . (48) n→∞ k=1 THEOREM 5. (i) Equation (42) is exactly (48) as in Lemma 5 and gives a proper definition of the digamma function. (ii) The digamma function ψ(x) defined by (40) is a unique solution (convex for large argument) of the difference equation 1 f(x+ 1) − f(x)= ,x∈ R+. (49) x Proof. We check the conditions in Lemma 5. With g(x) = 1/x, (47) is satisfied. Property (a) follows from (44) and (c) follows from (40); only (b) remains and this follows from

 2 ψ (a) =− − ζ(3,a+ 1)<0. 2 a3 Assertion (ii) is known as the Bohr–Mollerup theorem, and is a consequence of Lemma 5. We could cover√ (ii) also by our Corollary 2, part(ii), if only we assume we know the value ζ (0) = log(1/ 2π). We may also regard (ii) as Lerch’s formula (52) (see [4]).

LEMMA 6. Let z denote the complex variable not taking non-positive integer values. Then under the definition (41) the two definitions (42) and (40) for ψ are equivalent. Proof. Substituting (41) in (42) in the form N 1 1 ψ(z) + γ = lim − N→∞ n z + n + 1 n=1 N 1 N 1 = lim − log(N + z) − − log(N + z) , N→∞ n z + n − 1 n=1 n=1 Modular relation and digamma function 51 we deduce that N 1 ψ(z) + γ = γ − lim − log(N + z) , N→∞ z + n n=0 whence (40). On the other hand, (40) may be written as N 1 1 N 1 ψ(z) = lim − − − log(N + z) , N→∞ n z + n − 1 n n=1 n=1 ∞ 1 1 = − − γ, n z + n − 1 n=1 i.e. (42). 2 Finally, we state a theorem that establishes Lerch’s formula, the link between the gamma function and the zeta-function, which in turn proves Theorem 2.

THEOREM 6. If we suppose the integral representation for ζ(s, z), ∞ 1 1−s 1 −s ¯ −s−1 ζ(s, z)= z + z − s B1(t)(t + z) dt, σ > −1, (50) s − 1 2 0 and (43) for ψ(z), and also the value 1 = 1/2 (2 ) π (51) as known, then we have Lerch’s formula

 (z) ζ (0,z)=−log√ . (52) 2π Proof. Integrating (43) from 1 to z, we obtain z ∞ z 1 ¯ −2 log (z) = log z − dz + B1(t) (t + z) dz dt 1 2z 0 1 ∞ ∞ 1 ¯ −1 ¯ −1 = z log z − z − log z + 1 − B1(t)(t + z) dt + B1(t)t dt. (53) 2 0 1 On the other hand, differentiation of (50) gives ∞  1 ¯ −1 ζ (0,z)= z log z − z − log z − B1(t)(t + z) dt. (54) 2 0 Comparing (53) and (54), we see that ∞  ¯ −1 ζ (0,z)= log (z) − 1 − B1(t)t dt, (55) 1 and it remains to evaluate the last integral. 1 = s − For this we differentiate the formula ζ(s, 2 ) (2 1)ζ(s) to obtain  1 = s + s −  ζ (s, 2 ) 2 (log 2)ζ(s) (2 1)ζ (s). 52 K. Chakraborty et al

=−1 Hence in view of ζ(0) 2 , a consequence of (50),  1 = =−1 ζ (0, 2 ) (log 2)ζ(0) 2 log 2. = 1 1 Now put z 2 in (55) and use the value of (2 ) to obtain √ ∞ 1 ¯ −1 − log 2 = log π − 1 − B1(t)t dt. 2 1

Hence ∞ √ ¯ −1 1 + B1(t)t dt = log 2π, (56) 1 which is of some interest in its own right [21, p. 345, (3)]. Finally, (54) and (56) combine to give (52). This completes the proof. 2 Remark 4. We have had a feeling that ψ is a number-theoretic special function because of its heavy relevance to number-theoretic results: cf. [6](and[13, Ch. 8]) for the equivalence of the finite expression for L(1,χ)and the Gauss formula for ψ at rational points; cf. [12] for the discrete mean value of L(1,χ), etc. In this paper, we have made it clear how ψ is more intimately connected to the zeta-function than the gamma function, by establishing a circle of identities from its theory, in contrast to the gamma function, which has a mere catalytic effect in Proposition 2 although it appears as an essential factor in the functional equation. We have also investigated the logarithmic differentiation relation (log sin) = cot parallel to (log ) = ψ, revealing that the right-hand side is more intimately connected to by showing the first circle of identities centering around the partial fraction expansion for the cotangent function (cf. e.g. Proposition 5). As Proposition 6 shows, it is an expression in the corresponding modular form whose inversion formula translates in the Lember series through Proposition 5. Naturally, we are led to consider another approach through the polylogarithm function in view of (26), which we hope to conduct elsewhere.

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K. Chakraborty Harish-Chandra Research Institute Chhatnag Road Jhunsi Allahabad 211019 India (E-mail: [email protected])

S. Kanemitsu School of Humanity-Oriented Science and Engineering Kinki University Iizuka Fukuoka 820-8555 Japan (E-mail: [email protected])

X.-H. Wang Graduate School of Advanced Technology Kinki University Iizuka Fukuoka 820-8555 Japan (E-mail: [email protected])