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 √ √ 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 √ π2 log2(2) √ J(4 + 17) = − + + log(2) log(4 + 17) 6 2 arXiv:2012.15805v1 [math.NT] 31 Dec 2020 and √ 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 ∞ X ψ(nx) − log(nx) (3) F (x) = x ∈ 0 := (−∞, 0] , n C C r n=1 Γ0(x) which is therefore in some sense more fundamental. Here ψ(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 ∈ 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 introduced√ the√ function J(x√) 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 Williams√ [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 ψ(x) (see [17, p. 247]) Z ∞ e−t e−xt ψ(x) = − −t dt , 0 t 1 − e we see that the function F also has the integral representations Z ∞ 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 ∞ 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 ∞ π X Bn ζ(n + 1) F (x) ∼ − − x−n , 12x n n=2 as x → ∞, and the asymptotic expansion 2 ∞ 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 ∞ X n Z(x) = ζ(1 − n) ζ(1 + n) x ∈ R[[x]] , n=1 ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION 3 these can be rewritten as 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 → ∞ . x The expansion at x → ∞ follows from the asymptotic expansion for the digamma function P∞ −n ψ(z) ∼ log z + n=1 ζ(1 − n)z , while the two other expansions can be derived easily from the expansion at ∞ using the functional equations (4). 0 Analytic continuation. The function F is analytic in the cut complex plane C , because 0 P∞ −2 the series (3) converges locally uniformly there. From the formula ψ (z) = m=0(z + m) we 0 0 get a convergent series representation for F in C ∞ ∞ 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 {y : |y| < 1} 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 ∞ n−1 4 16y2 X 1 X y (10) G(y) + ζ(2) = ϕ n , ϕ(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 ϕ(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 ∞ 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 |Um(t)| ≤ 4m(m + 1), t ∈ [−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 ∈ H = {z : Im(z) > 0} 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 ψ(−x) − ψ(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 ∈ [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. Functional equations related to Hecke operators As it turns out, as well as the two functional equations (4) the Herglotz function satisfies infinitely many further 1-variable functional equations, one for each Hecke operator Tn. Before we make this statement precise, we look at a related elementary problem about relations satisfied by cotangent products.
3.1. Identities between cotangent products. Let ( cot πx , x ∈ , C(x) = C r Z 0 , x ∈ Z , so that C is periodic, vanishes on Z, and is holomorphic away from Z. Define C (x, y) = C(x)C(y) + 1 , which will be our main interest in this subsection. Note that it is even, symmetric, and satisfies (16) C (x, y) − C (x, x + y) − C (x + y, y) = 0 whenever xy(x + y) 6= 0. (See [20] for an interpretation of this type of functional equation in terms of the cohomology of SL(2, Z).) We are interested in more general relations of the form X λiC (aix + biy, cix + diy) = const , i where ai, bi, ci, di are integers with aidi − bici 6= 0. 0 −1 1 −1 Let Γ be the group PSL2(Z), which is generated by the matrices S = ( 1 0 ) and U = ( 1 0 ), 2 3 1 1 with S = U = 1, and let T = US = ( 0 1 ). For n ∈ N, let Mn be the set of 2 × 2 integer matrices of determinant n, modulo {±1}, and let Rn = Q[Mn]. We denote both the element a b of Mn corresponding to a matrix ( c d ) of determinant n and the corresponding basis element of R by [ a b ]. Define M = S M and set R = [M] = L R , which is a graded ring. n c d n∈N n Q n∈N n ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION 5
2 This ring acts on the right on the vector space of even meromorphic functions on C by the formula X X f ◦ λ [ ai bi ] x, y = λ f(a x + b y, c x + d y) . i ci di i i i i i i i The same formula defines the action of R on the space V of even complex-valued functions 2 2 on C /Z . We also define δ1, δ2 ∈ V by
δ1(x, y) = C(x)δ(y) , δ2(x, y) = δ(x)δ(y) , where δ = δZ : C → {0, 1} is the characteristic function of Z. The functional equations that we are interested in are elements of the set I = {ξ ∈ R | C ◦ ξ = const} , which is a right ideal in R. First, we give an algebraic description of I. Proposition 2. An element ξ of R lies in the ideal I if and only if
(1 − S)ξ ∈ ker(Φ1) ∩ ker(Φ2) , a b where Φi : R → Q(u, v) ⊗ V, i = 1, 2, are homomorphisms defined for γ = [ c d ] ∈ M by 1 c det(γ)−1 Φ (γ) = ⊗ δ ◦ γ . Φ (γ) = ⊗ δ ◦ γ , 1 cu + dv 1 2 cu + dv 2 Proof. From the definitions of C and δ it follows that δ(x) C(x + ε) = + C(x) + O(ε) , x ∈ / , ε → 0 , πε C Z where we recall that C|Z = 0. If we fix ξ0 = (x0, y0) and set (x, y) = (x0 + εu, y0 + εv), then 1 1 C (ax + by, cx + dy) = Φ2 (1 − S)γ (x0, y0) + Φ1 (1 − S)γ (x0, y0) + O(1) . π2vε2 πε
Thus the function f := C ◦ξ is continuous if and only if (1−S)ξ ∈ ker(Φ1)∩ker(Φ2). When this happens, the function f for fixed y is holomorphic, 1-periodic, and bounded when | Im(x)| → ∞, and hence constant by Liouville’s theorem. Applying the same argument to f as a function of y, we get that f(x, y) is constant. We now use the criterion in Propostion 2 to write down nontrivial elements in the ideal of relations I using Hecke operators. Following [1], we say that an element Ten ∈ Rn (n ∈ N) “acts like the n-th Hecke operator on periods” if it satisfies ∞ (17) (1 − S)Ten = Tn (1 − S) + (1 − T )Y, ∞ for some Y ∈ Rn, where Tn is the usual representative of the n-th Hecke operator X X a b T ∞ = ∈ [M ] . n 0 d Q n ad=n 0≤b It was shown in [1] that such elements exist for every n ∈ N and satisfy rf |2−kTen = rf|kTn for the period polynomial rf of any holomorphic modular form of weight k, whence the name. Theorem 1. Suppose that Ten ∈ Rn acts like the n-th Hecke operator on periods. Then X (18) (C ◦ Ten)(x, y) − l C (lx, ly) = c(Ten) , l|n where c: R → Z is the group homomorphism defined on generators by 1 − sgn(a + b) sgn(c + d) , a + b 6= 0 , c + d 6= 0 , a b 7→ 1 − sgn(a + b) sgn(d) , a + b 6= 0 , c + d = 0 , c d 1 − sgn(b) sgn(c + d) , a + b = 0 , c + d 6= 0 . 6 DANYLO RADCHENKO AND DON ZAGIER Proof. We apply the criterion from Proposition 2. Since δ1 ◦ (1 − T ) = δ2 ◦ (1 − T ) = 0, the ∞ subspace (1−T )Rn lies in both ker Φ1 and ker Φ2, and therefore Φi((1−S)Ten) = Φi(Tn (1−S)). Next, we calculate X X X n−1 δ(ax + by)δ(dy) = (ad0)−1den(y, d0)δ(ad0x) ad=n 0≤b Now, for x, y > 0 we calculate s+1 Z ∞ (2π) s −s−1 1 −s−1 −s−1 − C (ixt, iyt) t dt = y A(x/y, s) + 2 Γ(s + 1) ζ(s + 1) (x + y ) 4 0 (xy)−1 x π2 1 1 log x = + y−2F 0 + − − + O(s − 1) s − 1 y 12 y2 x2 xy (xy)−1 x log(xy) = + y−2F 0 − + O(s − 1) . s − 1 y 2xy (Note that C (ixt, iyt) is exponentially small for t large.) Since all entries of matrices in Ten are nonnegative, we have c(Ten) = 0 (where c is the homomorphism defined in Theorem 1), and equation (18) then implies that for x > 0 X log((ax + b)(cx + d)) X n log(l2x) ν (F 0| γ)(x) − n = F 0(x) − . γ 2 2(ax + b)(cx + d) l 2x γ=( a b )∈M l|n c d n 0 Next, we integrate the above identity. The primitive of (F |2γ)(x) is F(γx), while for the logarithmic term we use (assuming that ad − bc = n) Z log((ax + b)(cx + d)) x + b/a 1 1 n dx = L + log2(n(x + b/a)) − log2(n(x + d/c)) 2(ax + b)(cx + d) x + d/c 4 4 for c > 0 and its variant Z log((ax + b)d) 1 n dx = log2(n(x + b/a)) 2(ax + b)d 4 for c = 0 (both statements are easily checked by differentiation). Thus, modulo constants, the left-hand side of (19) is equal to 1 X X X n ν log2 n(x − γ−1(0)) − ν log2 n(x − γ−1(∞)) − log2(l2x) . 4 γ γ l γ γ−1(∞)6=∞ l|n 1 A simple calculation using (17) shows that the homomorphism ϕ: R → Z[P (Q)] defined on −1 −1 the generators by γ 7→ [γ (0)] − [γ (∞)] satisfies ϕ(Ten) = σ(n)([0] − [∞]). Thus, the above expression simplifies to 1 X n σ(n) log2(nx) − log2(l2x) = A log x + const , 4 l n l|n as claimed. The following proposition shows that a Ten consisting of matrices with nonnegative entries exist for all n ∈ N. Similar elements were constructed by Manin [10] and others. Proposition 3. The element Tbn ∈ Rn defined by X a b (20) T = bn c d 0≤c An essentially equivalent version of Tbn (with inequalities for rows instead of columns) was given by Merel in [11]. Here are the first three nontrivial examples 2 2 1 0 1 1 2 0 2 0 X 1 j X 3 0 2 1 T = + + + , T = + + , b2 0 2 0 2 0 1 1 1 b3 0 3 j 1 1 2 j=0 j=0 3 3 X 1 j X 4 0 2 0 2 0 2 1 2 2 3 1 T = + + + + + + . b4 0 4 j 1 0 2 1 2 0 2 1 3 2 2 j=0 j=0 For the special case Ten = Tbn, Theorem 2 gives the following explicit functional equation. Corollary. For all n ∈ N and x > 0 the modified Herglotz function F satisfies X x + b/a 1 + b/a (21) F Tbn (x) − σ(n)F(x) = L − L + An log x , 0 x + d/c 1 + d/c 0 The functional equations corresponding to the three Tbn’s given above are x x+1 2x x 1 1 (22a) F(2x) + F( 2 ) + F( 2 ) + F( x+1 ) − 3F(x) = L( x+1 ) − L( 2 ) + 2 log(2) log(x) , 3x 3x x x+1 x+2 2x+1 F(3x) + F( x+1 ) + F( 2x+1 ) + F( 3 ) + F( 3 ) + F( 3 ) + F( x+2 ) − 4F(x) (22b) 2x+1 x 2x 2 = L( 2x+4 ) + L( x+1 ) + L( 2x+1 ) − L(1) − L( 3 ) + log(3) log(x) , P3 x+j 4x 2x 2x+1 2x+2 3x+1 j=0 F( 4 ) + F( jx+1 ) + F( x+2 ) + F( 2 ) + F( x+3 ) + F( 2x+2 ) − 6F(x) (22c) P3 jx x x+1 3x+1 2 = j=1 L( jx+1 ) + L( x+2 ) + L( x+3 ) + L( 3x+3 ) − 4L(1) + L( 3 ) + 3 log(2) log(x) . We will make use of the first and the third of these in Section 6 to obtain explicit evaluations of the function J(x) for certain quadratic units x. We remark that there is no loss of generality in passing from the apparently more general functional equation (19) to the special case (21), because one can show that the functional equation coming from any other choice of Ten would be the same up to a linear combination of specializations of the functional equations (15). 4. Special values at positive rationals The special values of the Herglotz function at positive rationals turn out to always be ex- pressible in terms of dilogarithms. Here it is more convenient to use the original function F (x) instead of F(x). Theorem 3. For any positive rational x the value F (x) − F (1) is a rational linear combination 2πix 2πi/x of Li2 with arguments belonging to the cyclotomic field Q(e , e ). ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION 9 Proof. From the second integral representation in (5) we get Z 1 P P P Q P P dt F − F (1) = F − F = Q − P log(1 − t ) . Q Q P 0 1 − t 1 − t t Therefore, using the distribution relations X Q X 1 log(1 − tP ) = log(1 − αt) and = , 1 − tQ 1 − βt βP =1 αQ=1 we see that for all positive integers P , Q one has ! P X X X F − F (1) = f(α, β) − f(α, β) , Q αP =1 βQ=1, β6=1 βP =1, β6=1 where Z 1 log(1 − αt) f(α, β) = dt . 0 t(1 − βt) The statement of the theorem follows using the identity β α − β f(α, β) = Li − Li , 2 β − 1 2 1 − β which can be proved easily by differentiating both sides in α. We give three concrete examples of the theorem. x 1 2 1. Setting (P,Q) = (1, n) in the above proof and using Li2( x−1 ) + Li2(x) = − 2 log (1 − x) gives n−1 (n − 1)(n − 2)π2 log2 n 1 X πj (23) F (n) − F (1) = − + + log2 2 sin . 24n 2 2 n j=1 Notice that this statement is stronger than Theorem 3, since we only get products of logarithms rather than dilogarithms. Conjecturally, the only positive rationals x for which F (x) − F (1) can reduce to products of logarithms are x = n or x = 1/n, since for other values of x the corresponding formal combination of arguments of Li2 does not lie in the Bloch group of Q . n n 2. Combining (23) with the 3-term relation (4) gives F ( n+1 )−Li2( n+1 ) as a bilinear combination 2πi/n of logarithms of elements of Q(ζn, ζn+1), where ζn = e . 3. As a further example, which we will generalize in Section 7.2, we have 2 1 4 π2 π 2π 2 (24) F − F (1) = Li − + log2 2 sin + log2 2 sin + log(2) log . 5 2 2 5 5 5 5 5 An argument similar to the one used in the proof of Theorem 3 allows one to compute also the derivative of F at rational points. p 0 p Proposition 4. For any coprime p, q > 0 the difference q F ( q ) − (1 + log(p)) is a linear combination of Li2 at p-th or q-th roots of unity with coefficients in Q(ζp, ζq). Proof. Replacing the integral reprsentation (5) by (6) in the previous proof we obtain ! P P X X X F 0 − F 0(1) = g(α, β) − g(α, β) , Q Q αP =1 βQ=1, β6=1 βP =1, β6=1 where Li (α) − Li (β) Z 1 α log(y) α 2 2 , α 6= β , g(α, β) = − dy = α − β 0 (1 − αy)(1 − βy) Li1(α) , α = β . 0 P The result then follows by noting that F (1) = 1, and that αP =1,α6=1 Li1(α) = − log(P ). Similarly, one can show that the value (p/q)kF (k)(p/q) − F (k)(1) + (−1)k(k − 1)! log(p) is a combination of Lim, m = 2, . . . , k + 1 at p-th or q-th roots of unity with coefficients in Q(ζp, ζq). 10 DANYLO RADCHENKO AND DON ZAGIER 5. Kronecker limit formula for real quadratic fields The function F appeared in [18] in a formulation of a so-called Kronecker limit formula for real quadratic fields. (The original Kronecker limit formula was the corresponding statement for imaginary quadratic fields.) This is a formula expressing the value log ε %(B) = lim Ds/2ζ(B, s) − , s→1 s − 1 √ 2 D+ D where B is an element of the narrow class group of the quadratic order OD = Z + Z 2 of discriminant D > 0, ζ(B, s) is the corresponding partial zeta function (the sum of N(a)−s over all invertible OD-ideals a in the class B), and ε = εD is the smallest unit > 1 in OD of norm 1. The numbers %(B) are of interest because the value at s = 1 of the Dirichlet series P s −1/2 P LK (s, χ) = a χ(a)/N(a) equals D B χ(B)%(B) for any character χ on the narrow ideal class group. The formula proved in [18] is that X (25) %(B) = P (w, w0) , w∈Red(B) where the function P (x, y) for x > y > 0 is defined as y π2 x 1 1 x P (x, y) = F (x) − F (y) + Li − + log γ − log(x − y) + log , 2 x 6 y 2 4 y √ −b+ D and Red(B) is the set of larger roots w = 2a of all reduced primitive quadratic forms Q(X,Y ) = aX2 + bXY + cY 2 (a, c > 0, a + b + c < 0) of discriminant D which belong to the class B. Recall that narrow ideal classes correspond to PSL2(Z)-orbits on the set of w w0 −w w0 integral quadratic forms: if b = w + w ∈ B with 1 2√ 2 1 > 0, then the quadratic Z 1 Z 2 D N(Xw1+Y w2) form Q(X,Y ) = N(b) is in the corresponding orbit. The set Red(B) is finite and every element w ∈ Red(B) satisfies w > 1 > w0 > 0. Let us also denote l(B) = #Red(B). The set Red(B) can also be understood in terms of continued fractions. To any element w ∈ K 0 with w > w (here we assume that K is embedded into R) one can associate a continued fraction 1 w = b − , 1 1 b − 2 . .. which by the standard theory of continued fractions is eventually periodic. The sequence {bj}j≥1 is periodic (without a pre-period) if and only if the number w is reduced (i.e., w > 1 > w0 > 0), and then has period l = l(B). We call the l-tuple ((b1, . . . , bl)) the cycle associated to w. If we −1 fix a narrow ideal class B and take any b ∈ B such that b = Z + Zw with w reduced, then the equivalence class of ((b1, . . . , bl)) modulo cyclic permutations depends only on B. The set Red(B) is then simply {w1, . . . , wl}, where 1 w = b − , j j 1 b − j+1 . .. and both bj and wj depend only on j (mod l). Thus we can restate (25) as X 0 (26) %(B) = P (wj, wj) . j (mod l) 2 more precisely, the group of narrow ideal classes of invertible fractional OD-ideals, where two ideals belong to the same narrow class if their quotient is a principal ideal λOD with N(λ) > 0. For more details see [19] or [4] ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION 11 P P 0 0 Q Using the fact that j (mod l) wj + 1/wj = j (mod l) wj + 1/wj and j (mod l) wj = ε, we can rewrite (26) as π2 X (27) %(B) = 2γ log ε − l(B) + P(w , w0 ) , 6 j j j (mod l) where the function P(x, y), x > y > 0 is now defined by a simpler formula y (28) P(x, y) = F(x) − F(y) + L . x One can use the Kronecker limit formula (25) to prove various properties of %(B). For example, the functional equation (4) was used in [18] to prove Meyer’s formula π2 %(B) − %(B∗) = − l(B) − l(B∗) , 6 where B∗ = ΘB, and Θ is the narrow ideal class of principal ideals of negative norm. The key observation used in that proof is equivalent to the identity 1 1 (29) P(x, y) = Pe(x − 1, 1 − y) − Pe 1 − , − 1 , x > 1 > y > 0 , x y π2 where Pe(x, y) = F(x) − F(y) + L(−y/x), that allows one to rewrite %(B) − 2γ log ε + 6 l(B) in terms of the continued fraction associated to the wide ideal class containing B. For details we refer to [18], but we do state here a version of the Kronecker limit formula for wide ideal classes, since we will use it in Section 6 below. A number x ∈ K is called reduced in the wide sense if 0 x > 1, 0 > x > −1, and to any such number one associates a cycle [[a1, . . . , am]], which is simply the periodic part of the regular continued fraction of x, i.e., x = a1 + 1/(a2 + 1/(a3 + ... )) with W ai = ai (mod m) for i ∈ Z. We define Red (A) to be the set of reduced numbers x (in the wide W sense) such that Zx + Z is a fractional ideal in A. Then Red (A) = {x1, . . . , xm}, where 1 x = a + . i i 1 a + i+1 . .. Proposition 5 ([18, Corollary 2, p. 179]). Let A be a wide ideal class in a real quadratic field K, 0 ε0 > 1 the fundamental unit of K, x1, . . . , xm the elements of K satisfying xi > 1, 0 > xi > −1 and such that {1, xi} is a basis of some ideal in A, and [[a1, . . . , am]] the corresponding cycle of integers. Then m 2 m X 0 π X (30) %(A) = 2 Pe(xi, −x ) + 4γ log ε0 − (ai − 1) i 6 i=1 i=1 with Pe as above. 6. Special values at quadratic units While there does not seem to be a general formula, akin to that of Theorem 3, that would express in closed form the individual values F(x) for x in real quadratic fields, there are many rational linear combinations of these values that can be evaluated. One way to obtain such identities is to specialize functional equations satisfied by F(x), for example the 3-term rela- tion (15), or the more complicated functional equations from Section 3. A completely different source of identities, as was explained in [18, §9], stems from the fact that if χ is a genus char- acter on the narrow class group of a real quadratic field of discriminant D, then the special −1/2 P value LK (1, χ) = D B χ(B)%(B) is equal to a product LD1 (1)LD2 (1), where the splitting D = D1D2 corresponds to the genus character χ, and LD(s) is the Dirichlet series XD L (s) = n−s . D n n≥1 12 DANYLO RADCHENKO AND DON ZAGIER √ 2 2 π2 n n − 1 4 J (n + n − 1) − n 24 2 3 2S1,12 3 3 2 6S1,8 4 3 · 5 2S1,15 + 4S5,12 3 5 2 · 3 3S1,24 + 2S8,12 6 5 · 7 2S1,140 + 4S5,28 4 7 2 · 3 7S1,12 + 2S8,24 2 8 3 · 7 2S1,28 + 4S12,21 4 9 2 · 5 22S1,5 + 4S8,40 3 11 2 · 3 · 5 3S1,120 + 2S8,60 + 2S12,40 + 2S5,24 12 11 · 13 2S1,572 + 4S13,44 3 13 2 · 3 · 7 3S1,168 + 2S8,21 + S12,56 + S24,28 14 3 · 5 · 13 2S1,780 + 4S5,156 + 4S13,60 16 3 · 5 · 17 2S1,1020 + 4S12,85 + 4S5,204 3 2 19 2 · 3 · 5 6S1,40 + 12S5,8 + 2S12,120 + 2S24,60 3 21 2 · 5 · 11 3S1,440 + 2S8,220 + 2S5,88 + 2S40,44 22 3 · 7 · 23 2S1,1932 + 2S28,69 + 2S21,92 4 23 2 · 3 · 11 4S1,33 + 2S8,264 + S12,44 + S24,88 3 27 2 · 7 · 13 3S1,728 + 2S8,364 + 2S28,104 + 2S13,56 3 29 2 · 3 · 5 · 7 3S1,840 + S12,280 + 2S5,168 + S24,140 + S28,120 + 2S21,40 + S56,60 34 3 · 5 · 7 · 11 2S1,4620 + 4S5,924 + 2S28,165 + 2S60,77 + 2S21,220 36 5 · 7 · 37 2S1,5180 + 4S5,1036 + 4S37,140 4 41 2 · 3 · 5 · 7 4S1,105 + 2S8,840 + S12,140 + 2S5,21 + S24,280 + S28,60 + 2S40,168 + S56,120 3 43 2 · 3 · 7 · 11 3S1,1848 + 2S8,924 + S12,616 + S24,77 + S28,264 + S44,168 + S33,56 + S21,88 56 3 · 5 · 11 · 19 2S1,12540 + 4S12,1045 + 4S5,2508 + 2S44,285 + 2S76,165 3 61 2 · 3 · 5 · 31 3S1,3720 + S12,1240 + 2S5,744 + S24,620 + 2S40,93 + S60,1032 + S120,124 3 69 2 · 5 · 7 · 17 3S1,4760 + 2S8,2380 + 2S5,952 + 2S28,680 + 2S40,476 + 2S56,85 + 2S136,140 3 77 2 · 3 · 13 · 19 3S1,5928 + 2S8,741 + S12,1976 + S24,988 + 2S13,456 + S76,312 + S152,156 3 83 2 · 3 · 7 · 41 3S1,6888 + 2S8,861 + 3S12,2296 + S24,1148 + S28,984 + S56,492 + 4S21,328 3S + S + 2S + S + 2S + S + 2S 131 23 · 3 · 5 · 11 · 13 1,17160 12,5720 5,3432 24,2860 40,429 44,1560 13,1320 +S60,1144 + S88,780 + 2S104,165 + S120,572 + S156,440 + S220,312 √ Table 1. Values of J (n + n2 − 1) Since LD(1) can be evaluated explicitly as 2√h log ε , D > 0 , |D| LD(1) = √2πh , D < 0 , w |D| and %(B) can be evaluated in terms of F via the Kronecker limit formula (27), this leads to nontrivial identities for F(x). We will discuss different realizations of this idea in this section, looking first at the case where each genus consists of only one narrow ideal class. 6.1. Explicit formulas when there is one class per genus. In Table 1 (resp. Table 2) we collect all numbers n less than 100 000 (and conjecturally all n) such that each genus of binary quadratic forms of discriminant 4(n2 − 1) (resp. 4(n2 + 1)) contains exactly one narrow ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION 13 √ 2 2 π2 n n + 1 2 J (n + n + 1) − n 24 1 2 S1,8 2 5 4S1,5 3 2 · 5 S1,40 + 2S5,8 4 17 2S1,17 5 2 · 13 S1,104 + 2S8,13 2 7 2 · 5 3S1,8 + 4S5,40 8 5 · 13 2S1,65 + 4S5,13 11 2 · 61 S1,488 + 2S8,61 13 2 · 5 · 17 S1,680 + 2S8,85 + 2S5,136 17 2 · 5 · 29 S1,1160 + 2S5,232 + 2S29,40 19 2 · 181 S1,1448 + 2S8,181 23 2 · 5 · 53 S1,2120 + 2S5,424 + 2S40,53 31 2 · 13 · 37 S1,3848 + 2S13,296 + 2S37,104 37 2 · 5 · 137 S1,5480 + 2S8,685 + 4S5,1096 47 2 · 5 · 13 · 17 S1,8840 + 4S5,1768 + 2S40,221 + 2S13,680 + 2S85,104 73 2 · 5 · 13 · 41 S1,21320 + 2S5,4264 + 2S40,533 + 2S13,1640 + 2S104,205 √ Table 2. Values of J (n + n2 + 1) √ equivalence class. In each case we give an identity for J (n + n2 ± 1), where log2(2) π2 1 J (x) = J(x) − + x − 2 24 x is related to F(x) by J (x) = F(2x) − 2F(x) + F(x/2) , parallel to the relation (2) between J(x) and F (x). The numbers Sp,q are defined as Sp,q = log(εp) log(εq) , √ where εd for d a fundamental discriminant is the fundamental unit of Q( d) if d > 1, and εd is defined to be 2 if d = 1. √ Theorem 4. The values of J (n + n2 ± 1) for the 45 known one-class-per-genus cases are as given in Table 1 and Table 2. We observe that, with the exception of n = 7, 9, 23, and 41, the cases listed in Table 1 also follow from Theorem 6 (for n even) or from Theorems 8 and 9 (for n odd) of [13]. The identities listed in Table 2 appear to be new. As already√ indicated, we will deduce Theorem 4 from the following general expression for J (n + n2 ± 1) as a combination of Kronecker limits %(B). √ 2 Theorem 5. For√ all n ≥ 1 the value J (n + n ± 1) is a rational linear combination of ζ(2), log(2) log(n + n2 ± 1), and the values %(B) for at most four narrow classes B of quadratic forms of discriminant 4a(n2 ± 1), with a = 0, ±1. √ 2 Proof. We first consider the case of J (u) for u = n+ n − 1. Define B1 and B2 to be the narrow classes with cycles ((2n)) and ((n + 1, 2)), respectively, and if n is odd also define B3 and B4 to be n+3 n+1 the narrow classes corresponding to the cycles 2 , 2, 2, 2 and 2 , 4 respectively. Then we claim that (31) 2J (u) = %(B1) − %(B2) + log(2) log(u) , n even , (32) 4J (u) = %(B1) + %(B2) − %(B3) − %(B4) + 3 log(2) log(u) , n odd, n 6= 7 . 14 DANYLO RADCHENKO AND DON ZAGIER We have nu + 1 2u o Red(B ) = {u} , Red(B ) = , , 1 2 2 u + 1 and for n odd also n 4u 3u + 1 2u + 2 u + 3o n 4u u + 1o Red(B ) = , , , , Red(B ) = , . 3 3u + 1 2u + 2 u + 3 4 4 u + 1 4 Since u0 = 1/u, we can use the Kronecker limit formula (27) to rewrite equation (31) as 1 u+1 u+1 2u 2 2J (u) = P u, u − P 2 , 2u − P u+1 , u+1 + ζ(2) + log(2) log(u) , where we recall that P(x, y) = F(x) − F(y) + L(y/x). This identity, in fact, holds for all real u > 1, as can be derived easily using the functional equation (22a) and the following simple relation for the Rogers dilogarithm u 1 1 2L u+1 + 2L u − L u2 = 2L(1) . Similarly, (32) is equivalent to 1 u+1 u+1 u+1 u+1 4J (u) = P u, u + 2P 2 , 2u − 2P 4 , 4u 4u 4 3u+1 u+3 − 2P 3u+1 , u+3 − 2P 2u+2 , 2u+2 + 3ζ(2) + 3 log(2) log(u) . Again, this identity holds for all u > 1, and can be derived using the functional equations (22a) and (22c) for F(x). Finally, for n = 7 we have √ √ 0 7 4J (7 + 48) = %(B1) − %(B2) + 2ζ(2) + 2 log(2) log(7 + 48) , 0 where B2 corresponds to the cycle√ ((6, 2, 2)). This can again be derived using (22a) and (22c). 2 The case of J (v) for v = n + n + 1 is similar. Here we define A1 and A2 to be the wide ideal classes with cycles [[2n]] and [[n − 1, 1, 1]] respectively. We claim that 4J (v) = %(A1) − %(A2) + 2 log(2) log(v) + nζ(2) , n > 2 . Once again, using 2v v+1 v−1 Redw(A1) = {v} , Redw(A2) = v+1 , v−1 , 2 , and the Kronecker limit formula (30) one can rewrite the identity as 1 2v 2 v+1 v−1 v−1 v+1 1 2J (v) = Pe v, − Pe , − Pe , − Pe , − ζ(2) + log(2) log(v) , v v+1 v−1 v−1 v+1 2 2v 2 (with Pe as in (29)) which follows easily from (22a) and the 3-term relation (15). Finally, the cases n = 1,√2 do not fit the above scheme, but can be derived directly using√ functional equations 2 2 that J (n + n + 1) is a linear combination of ζ(2) and√ log(2) log(n + n + 1). For instance, the case n = 1 follows directly√ by substituting x = 1 + 2 into (22a) and applying the 3-term relation (15) with x = 2. The derivation for n = 2 is slightly more complicated, but again only involves the functional equations (15) and (22a). Proof of Theorem 4. Since in all the cases listed in the tables there is one class per genus of quadratic forms, we can rewrite the corresponding linear combination of %(Bi) as a rational 1/2 linear combination of D LOD (1, χ), where χ runs over genus characters of the corresponding class groups. In each case the identity then follows from the factorization of LO(s, χ) into a product of two Dirichlet L-functions (for general discriminants, see, for example, [7]). There is a small subtlety in some cases when we need to consider a combination %(B) − %(B0) 0 with B√of discriminant D and B of discriminant 4D and a priori the resulting expression for 2 J (n + n ± 1) may involve constant terms of ζOD (s) and ζO4D (s) at s = 1. However, a simple but somewhat tedious calculation using the explicit expression for ζO(s) (see, say [6]) and the relation between the class groups of discriminants D and 4D (see, e.g., [4, Cor. 5.9.9]) shows that in all these cases the nontrivial contributions coming from ζO(s) cancel out. √ Note that to say that ε is a number of the form n + n2 ± 1 is equivalent to saying that ε > 1 is a quadratic unit with even trace (or a unit in an order of even discriminant). As far as we can ascertain, there are no similar formulas for J (ε) when ε is a unit of odd trace. ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION 15 √ 2 6.2. General case. The proof of Theorem 4 shows that J (n+√ n ± 1) can always be expressed as an algebraic linear combination of ζ(2), log(2) log(n + n2 ± 1), and terms of the form 1/2 a 2 D LOD (1, χ), where D = 4 (n ± 1), a ∈ {0, ±1}, and χ is a narrow class group character. Recall that the Stark conjecture predicts that the special value at s = 1 of the Artin L-function of a Galois representation ρ for a Galois extension E/F is a simple multiple (a power of π times an algebraic number) of the so-called Stark regulator (the determinant of a certain matrix Q dim(ρ) of logarithms of units) in such a way that the factorization ζE(s) = ρ LE/F (s, ρ) of the Dedekind zeta function of E matches the factorization of the regulator of E obtained by decomposing the group of units of E (after extending scalars to Q) into irreducible Gal(E/F )- representations. For the general formulation of the Stark conjecture, see [15, p. 25–28]. In the case of an abelian extension H/K (in our cases, a ring class field) of a real quadratic field K, all irreducible representations of Gal(H/K) are one-dimensional, all irreducible rep- resentations of Gal(H/Q) are one- or two-dimensional, and for any character χ of Gal(H/K) we have LH/K (s, χ) = LH/Q(s, ρ), where ρ is the two-dimensional representation of Gal(H/Q) induced from χ. The Stark regulator for LH/Q(1, ρ) is then a 2 × 2, 1 × 1, or 0 × 0 determinant of logarithms of units in H, depending on whether tr(ρ(σ)) is 2, 0, or −2, where σ denotes the complex conjugation. If H is the ring class field of a real quadratic field K corresponding to an order OD ⊂ K and χ a character on the corresponding narrow ideal class group, then LK (s, χ) is the L-function of the corresponding character on Gal(H/K), and since H is totally real, the Stark conjecture predicts that LK (1, χ) should be an algebraic multiple of a 2 × 2 determinant of units in H (compare with [14, p. 61]). We therefore obtain the following result. Proposition 6. Let ε > 0 be a quadratic unit with even trace. Assume the abelian Stark conjecture for the real quadratic field Q(ε). Then J (ε) is a rational linear combination of ζ(2), log(2) log(ε), and of 2 × 2 determinants of algebraic units in the narrow ring class field of the quadratic order Z[2ε]. √ Here is an explicit numerical example. The field K = Q( 257) has class number 3 and its 3 2 Hilbert class field H is K(α), where α satisfies α − 2α − 3α + 1 = 0. If we let α1 < α2 < α3 be the roots of x3 − 2x2 − 3x + 1, then to very high precision we find √ √ ? log(−α1) log(3 − α1) J (16 + 257) = 4ζ(2) + log(2) log(16 + 257) + 2 . log(α3) log(3 − α3) As another example, this time with a non-fundamental discriminant, we find √ √ ? log(β1) log(γ1) 2 J (10 + 3 11) = 5ζ(2) + log(2) log(10 + 3 11) + , log(β2) log(γ2) β2−9β +4 where β > β are the two largest real roots of x4 − 11x3 + 24x2 − 11x + 1 and γ = j j . 1 2 j 1−βj 7. Cohomological aspects Note that the 3-term relation x x (33) F(x) − F(x + 1) − F = L(1/2) − L , x + 1 x + 1 when viewed modulo the more elementary right-hand side is exactly the period relation satisfied by the component ϕS of a 1-cocycle {ϕγ}γ∈PSL2(Z) such that ϕT = 0. (Cf. [20] and also [9] for more discussion of this relationship.) In this section we will discuss various ways—all of them still somewhat provisional—in which one can relate F to 1-cocycles for PSL2(Z). 7.1. An interplay between the 5-term relation and the 3-term relation. If we extend F to an even function on R r {0}, then the weaker form of relation (33) x − 1 (34) F(x) − F(x − 1) + F = L(x) − L(1/2) (mod ζ(2)) x 16 DANYLO RADCHENKO AND DON ZAGIER now holds for all x ∈ R r {0, 1}. The function P(x, y) defined by (28) then becomes an even 2 piecewise continuous function on all of R and satisfies the functional equation x − 1 y − 1 (35) P(x, y) − P(x − 1, y − 1) + P , = 0 (mod ζ(2)) x y for all x, y ∈ R r {0, 1}. Conversely, if F is any function and we define L and P (modulo constants) by equations (34) and (28), respectively, then the left hand side of (35) becomes a sum of 15 F’s that can be grouped into 5 L’s and then becomes the famous 5-term relation (equivalent to (14)) for L: y y − 1 1 − 1/y (36) L − L + L − L(y) + L(x) = 0 (mod ζ(2)) . x x − 1 1 − 1/x The anti-invariance of F and L under inversion implies that P is also anti-invariant and this together with (35) implies that P defines a cocycle for the group PSL2(Z) with values in the 2 space of functions from R to R/ζ(2)Z, by mapping T to 0 and S to P. The above discussion thus gives an interesting connection, which we have not yet understood completely, between the 5-term relation and 1-cocycles for the group PSL2(Z). 2 ab × 7.2. A cocycle with values in Λ (Q ). A different construction of a 1-cocycle can be obtained from the evaluation of F at positive rationals given by Theorem 3. The formula that we established in the proof has the form F (α) − F (1) = Li2(ξα) , 2πiα 2πi/α for a certain ξa ∈ Z[Q(e , e )]. It is therefore interesting to know when a linear combina- tion of ξα’s lands in the Bloch group of Q (see [21]). For this we have to calculate δ(ξα), where 2 × δ : Z[Q] → Λ (Q ) ⊗Z Q is the usual differential given by δ([x]) = x ∧ (1 − x). We find that if α = p/q,(p, q) = 1, then δ(ξα) = β(α) − β(1/α) − p ∧ q , 2 ab × where β : Q → Λ (Q ) ⊗Z Q is defined by X (37) β(p/q) = (1 − λp) ∧ (1 − λ) ((p, q) = 1) . λq=1 We further note that the element ξα is Galois invariant, so whenever δ(ξα) vanishes, by Galois descent we get an element in the Bloch group of Q, which is a torsion group. Since K2(Q) is 2 × torsion (see Theorem 11.6 in [12]) the group Λ (Q ) ⊗ Q is equal to δ(Q[Q]). Therefore, P Z whenever a linear combination ci(β(αi) − β(1/αi)) vanishes, there are rational numbers xj P P i P P and dj such that i ciξαi + dj[xj] is in ker(δ), and then the sum i ciF (αi) + j djLi2(xj) vanishes modulo products of logarithms of algebraic numbers. A rather simple example of this n n2+1 was given in (23). As a slightly less trivial example, one can show that β( n2+1 ) = β( n ), n 1 n2 which implies that F ( n2+1 ) − F (1) − 2 Li2( n2+1 ) is a bilinear combination of logarithms of algebraic numbers, generalizing (24). Two remarks about the function β appearing above are that it takes values in Λ2 of the group of cyclotomic units and that it satisfies the two identities β(−x) = β(x) and β(x + 1) = β(x). Because of this, sending T 7→ 0, S 7→ ϕ, where ϕ(p/q) = δ(ξp/q) + p ∧ q defines a PSL2(Z)- cocycle. This cocycle can also be lifted to a cocycle given by T 7→ 0, S 7→ ξ with values in 1 ab ab functions from P (Q) to Q[Q ]/(Q[Q] + C(Q )), where C(F ) denotes the subspace of Q[F ] spanned by all specializations of the 5-term relation. A further remark is that β itself can be used to construct infinitely many 1-cocycles, now 1 2 × P (Fp) 2 × with values in Λ (Q(ζp) ) for any prime p: the mapping βp : Z/pZ → Λ (Q(ζp) ) ⊗Z Q given by βp(n) = β(n/p) satisfies the functional equations βp(n) = βp(−n) = −βp(1/n) = βp(n + 1) + βp(n/(n + 1)), and thus sending T 7→ 0, S 7→ βp defines a PSL(2, Fp)-cocycle. A final remark is that there is a formal similarity between the formula (37) and the classical Dedekind sums arising in the modular transformation behavior of the Dedekind eta function. This similarity can be made precise using the notion of generalized Dedekind symbols due to Fukuhara [3]: the mapping (p, q) 7→ β(p/q) is an even generalized Dedekind symbol with values ARITHMETIC PROPERTIES OF THE HERGLOTZ FUNCTION 17 2 ab × in Λ (Q ), and the mapping (p, q) 7→ δ(ξp/q) + p ∧ q is its reciprocity function. Note that from this point of view the functional equations discussed in Section 3 (or, more precisely, the corresponding functional equations for β) are analogous to the functional equations discovered by Knopp [8] for the classical Dedekind sums. 7.3. The Herglotz function and the weight 2 Eisenstein series. In this final subsection, we will give an explanation of the cocycle nature of F by writing it as an Eichler-type integral. We slightly modify the definition of F to X ψ(nz) − log(nz) + (2nz)−1 π2 F ?(z) = = F (z) + . n 12 z n≥1 Using Binet’s integral formula [17, p. 250] for the digamma function, 1 Z ∞ 1 2t dt ψ(x) = log x − − 2πt 2 2 , 2x 0 e − 1 t + x we obtain the following integral representation, valid for Re(z) > 0: ∞ ! Z ∞ X 1 2t dt Z i∞ 1 1 F ?(z) = − = − H(τ) + dτ , n(e2πnt − 1) t2 + z2 τ + z τ − z 0 n=1 0 P∞ n 2πiτ where H(τ) = n=1 σ−1(n) q . (Here, as usual, we use q to denote e and σν(n) for the sum of the ν-th powers of the positive divisors of n.) Note that H(τ) = log(q1/24/η(τ)), where η 0 0 is the Dedekind eta function, and that its derivative is 2πi G2(τ), where G2(τ) is the weight 2 1 P∞ n Eisenstein series G2(τ) = − 24 + n=1 σ1(n) q without its constant term. Integrating by parts, we therefore get Z i∞ 2 ? 0 τ F (z) = 2πi G2(τ) log 1 − 2 dτ . 0 z This formula is valid as it stands for Re(z) > 0 and gives yet another proof of the analytic 0 continuation of F to C by deforming the path of integration to remain to the left of z if z is in the second quadrant and to the right of −z if z is in the third quadrant. The above integral for F ? can be rewritten as F ?(z) = F +(z) + F −(z), where Z i∞ ± 0 τ τ F (z) = 2πi G2(τ) log 1 ∓ ± dτ . 0 z z + − 0 + − Note that both F and F continue analytically to C . If we denote by H and H the upper and lower half-planes respectively, then by splitting the integral from 0 to i∞ at z or −z we get F ±(z) ≡ H±(z) − H±(−1/z) + log(z)H(±z) − U(z±1) z ∈ H± , R ∞ 0 where “≡” means “modulo elementary functions”, U(z) = 2πi z G2(τ) log(τ)dτ, and Z ±i∞ ± 0 ± H (z) = ±2πi G2(±τ) log(τ − z) dτ (z ∈ H ) . z Since H±(z) is obviously a 1-periodic function in H± and U(z) is essentially the primitive of ± ± ± G2(z) log(z), this computation implies that F (z) − F (z + 1) − F (z/(z + 1)) = 0 modulo elementary functions, and that T 7→ 0, S 7→ F ± defines a 1-cocycle with values in the space of suitably nice analytic functions modulo elementary functions. The above calculations go through much the same way for the higher Herglotz functions X ψ(nz) F (z) = (k > 2) , k nk−1 n≥1 P n with H replaced by n≥1 σ1−k(n)q , which for even k is essentially the Eichler integral of the Eisenstein series of weight k on the full modular group. The functions Fk were defined in [16] in connection with a higher Kronecker “limit” formula for ζK (B, s) at s = k/2 (for k even) rather than the limiting value at s = 1. It is likely that most of the properties we have given have analogues for higher Herglotz functions. This could be a topic for future research. 18 DANYLO RADCHENKO AND DON ZAGIER References [1] Y.-J. Choie, D. Zagier, Rational period functions for PSL(2, Z), in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math. 143, AMS, Providence, RI, 1993, pp. 89–108. [2] H. Cohen, Number Theory. Analytic and Modern Tools. Graduate Texts in Mathematics, vol. II. Springer, New York, 2007. [3] S. Fukuhara, Hecke operators on weighted Dedekind symbols, J. reine angew. Math. 593, pp. 1–29 (2006). [4] F. Halter-Koch, Quadratic Irrationals. An Introduction to Classical Number Theory, Pure and Applied Math- ematics, CRC Press, Boca Raton, 2013. [5] G. 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Zagier, Zetafunktionen und quadratische K¨orper. Eine Einf¨uhrungin die h¨ohere Zahlentheorie, Springer, Berlin-New York, 1981. [20] D. Zagier, Quelques cons´equences surprenantes de la cohomologie de SL(2, Z), Le¸consde math´ematiques d’aujourd’hui, Cassini, Paris, 2000, pp. 99–123. [21] D. Zagier, The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry, Vol. II, pp. 3–65. Springer, Berlin, 2007. [22] D. Zagier, Curious and exotic identities for Bernoulli numbers, in: T. Arakawa et al., Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics, 2014, pp. 239–267. ETH Zurich, Mathematics Department, Ramistrasse¨ 101, 8092 Zurich,¨ Switzerland Email address: [email protected] Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy Email address: [email protected]