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Arithmetic Properties of the Herglotz Function

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Arithmetic Properties of the Herglotz Function ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION DANYLO RADCHENKO AND DON ZAGIER Abstract. In this paper we study two functions F (x) and J(x), originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these func- tions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark's conjecture. We also discuss connections with 1-cocycles for the modular group PSL(2; Z). Contents 1. Introduction 1 2. Elementary properties 2 3. Functional equations related to Hecke operators 4 4. Special values at positive rationals 8 5. Kronecker limit formula for real quadratic fields 10 6. Special values at quadratic units 11 7. Cohomological aspects 15 References 18 1. Introduction Consider the function Z 1 log(1 + tx) (1) J(x) = dt ; 0 1 + t defined for x > 0. Some years ago, Henri Cohen 1 showed one of the authors the identity p p π2 log2(2) log(2) log(1 + 2) J(1 + 2) = − + + : 24 2 2 In this note we will give many more similar identities, like p π2 log2(2) p J(4 + 17) = − + + log(2) log(4 + 17) 6 2 arXiv:2012.15805v1 [math.NT] 31 Dec 2020 and p 2 11π2 3 log2(2) 5 + 1 J = + − 2 log2 : 5 240 4 2 We will also investigate the connection to several other topics, such as the Kronecker limit formula for real quadratic fields, Hecke operators, Stark's conjecture, and cohomology of the modular group PSL2(Z). The function J(x) can be expressed via the formula π2 (2) J(x) = F (2x) − 2F (x) + F (x=2) + 12x 1who learned it from H. Muzzafar (Montreal), see [2, Ex.60, p. 902{903] 1 2 DANYLO RADCHENKO AND DON ZAGIER in terms of the function 1 X (nx) − log(nx) (3) F (x) = x 2 0 := (−∞; 0] ; n C C r n=1 Γ0(x) which is therefore in some sense more fundamental. Here (x) = Γ(x) is the digamma function. The function F was introduced in [18], where it was shown, in particular, that it satisfies the two functional equations x 1 F (x) − F (x + 1) − F = −F (1) + Li ; x + 1 2 1 + x (4) 1 log2 x π2(x − 1)2 F (x) + F = 2F (1) + − ; x 2 6x 0 for all x 2 C = C r (−∞; 0], where Li2(x) is Euler's dilogarithm function. A function essen- tially equivalent to F (x) was also studied by Herglotz in [5] and for this reason we will call F the Herglotz function. In fact, Herglotz also introducedp thep function J(xp) and found explicit evaluations [5, Eq.(70a-c)] of J(x) for x = 4 + 15, 6 + 35, and 12 + 143. Several other identities of this kind were found by Muzzafar and Williamsp [13], together with some sufficient conditions on n under which one can evaluate J(n + n2 − 1). We give a more systematic treatment of identities of this type in Section 6. 2. Elementary properties Integral representations. Using the well-known integral formula for (x) (see [17, p. 247]) Z 1 e−t e−xt (x) = − −t dt ; 0 t 1 − e we see that the function F also has the integral representations Z 1 Z 1 1 1 −tx 1 1 x dy (5) F (x) = −t − log(1 − e ) dt = + log(1 − y ) : 0 1 − e t 0 1 − y log y y d x x By differentiating the last integral and using the obvious identity dx Lim(y ) = Lim−1(y ) log y we get also the following expression for the n-th derivative of F Z 1 (n) 1 1 n x dy (6) F (x) = − + log (y) Li1−n(y ) : 0 1 − y log y y We will give another useful integral representation of F in Section 7.3. Asymptotic expansions. The Herglotz function has the asymptotic expansion 1 X Bn ζ(n + 1) F (x) = F (1) − Li (1 − x) + (x − 1)n(1 − x−n) 2 n n=1 as x ! 1, the asymptotic expansion 2 1 π X Bn ζ(n + 1) F (x) ∼ − − x−n ; 12x n n=2 as x ! 1, and the asymptotic expansion 2 1 ζ(2) log x X Bn ζ(n + 1) F (x) ∼ − + + 2ζ(2) + 2F (1) + xn ; x 2 n n=1 as x ! 0. In terms of the formal power series 1 X n Z(x) = ζ(1 − n) ζ(1 + n) x 2 R[[x]] ; n=1 ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION 3 these can be rewritten as 8 2 > log x −1 >−Z(x) + + 2F (1) − ζ(2)(x − 2 + x ) ; x ! 0 ; > 2 < 1 − x (7) F (x) ∼ Z − Z(1 − x) − Li2(1 − x) + F (1) ; x ! 1 ; > x > 1 :>Z ; x ! 1 : x The expansion at x ! 1 follows from the asymptotic expansion for the digamma function P1 −n (z) ∼ log z + n=1 ζ(1 − n)z , while the two other expansions can be derived easily from the expansion at 1 using the functional equations (4). 0 Analytic continuation. The function F is analytic in the cut complex plane C , because 0 P1 −2 the series (3) converges locally uniformly there. From the formula (z) = m=0(z + m) we 0 0 get a convergent series representation for F in C 1 1 X X 1 1 (8) F 0(x) = − : (nx + m)2 nx n=1 m=0 1−y 2 Let us look at this from a different point of view. The change of variables y 7! x = ( 1+y ) 0 0 maps the unit disk fy : jyj < 1g bijectively and conformally onto C . The analyticity of F in C together with the second formula in (7) then implies that the power series 1 − y 2 4y 4y 4y (9) G(y) := F − F (1) + Li = Z − Z 1 + y 2 (1 + y)2 (1 − y)2 (1 + y)2 has radius of convergence ≥ 1. The power series Z(y) is factorially divergent, but we can show directly that the difference of two Z's appearing on the right-hand side of (9) is convergent in the P γ0(x=n) unit disk by the following argument. First, note that Z(x) + ζ(2)x is equal to − n≥1 n . P Bm m Here γ0(x) = n≥1 m x is the formal power series discussed in detail in [22, (A.13)], where it was shown that it satisfies x γ − γ (x) = log(1 − x) + x : 0 1 − x 0 From this we obtain 1 n−1 4 16y2 X 1 X y (10) G(y) + ζ(2) = ' n ;'(x) := − log(1 − x) − x : (1 − y2)2 n 2j 2 n=1 j=0 1 − 2( n − 1)y + y R 1 x P m 1 Using '(x) = 0 ( 1−tx − x)dt and the generating series m≥0 Um(t) x = 1−2tx+x2 for Cheby- shev polynomials of the second kind, we obtain 1 n−1 16y2 X 2 Z 1 2 X 2j [ym+1] G(y) + ζ(2) = U (t) dt − U − 1 : (1 − y2)2 n m n m n n=1 −1 j=0 Now breaking up the integral into n integrals over intervals of length 2=n and using the estimate 0 2 jUm(t)j ≤ 4m(m + 1), t 2 [−1; 1], we find that the right hand side is O(m ), and hence that the series G(y) indeed has radius of convergence at least 1. 0 Relation to the Dedekind eta function. The analytic extension of F to C satisfies another important functional equation. To state it, and for later purposes, it is convenient to introduce the slightly modified function π2 log2 x (11) F(x) = F (x) − F (1) + x − 2 + x−1 − : 12 4 Proposition 1. For any z 2 H = fz : Im(z) > 0g we have F(−z) − F(z) = 2πi log((z=i)1=4η(z)) ; πiz=12 Q 2πinz where η(z) = e n≥1(1 − e ) is the Dedekind eta function, and the branch of log((z=i)1=4η(z)) is chosen to be real-valued on the imaginary axis. 4 DANYLO RADCHENKO AND DON ZAGIER −1 Proof. This follows from (−x) − (x) = x + π cot(πx). Compare [16, p. 27]. As a corollary, F(x) cannot be extended analytically across the negative real axis either from above or from below. Relation to the Rogers dilogarithm. The functional equation (4) already shows that F is intimately related to the dilogarithm function Li2. If we use the modified function F from (11) instead, we get nicer identities involving the Rogers dilogarithm. Recall that the Rogers dilogarithm is defined by 1 (12) L(x) = Li (x) + log(x) log(1 − x) ; x 2 [0; 1] ; 2 2 and extended to R by ( π2 1 3 − L( x ) ; x > 1 ; (13) L(x) = x −L( x−1 ) ; x < 0 : 1 π2 The key property of L is that it defines a continuous map from P (R) to R= 2 Z that satisfies x(1 − y) y(1 − x) (14) L(x) + L(y) − L(xy) − L − L = 0 (mod π2=2) : 1 − xy 1 − xy Now, if we use F instead of F , then we get \clean" versions of the functional equation (4) with the Rogers dilogarithm in place of Li2: for x > 0 we have x x F(x) − F(x + 1) − F = L(1=2) − L ; x + 1 x + 1 (15) 1 F(x) + F = 0 : x 3.
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