On Kronecker limit formulas for real quadratic fields

Shuji Yamamoto

REVIEW

On Kronecker limit formulas for real quadratic fields

Shuji Yamamoto

Number Theory · Submitted 27 Feb 2006 · 15 references

Pith's verdict: Solid generalization of Zagier's Kronecker limit formula to ray classes, but one load-bearing computation is omitted in the proof of Theorem 5.1.1.

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arXiv IDmath/0602615 (version 1)

Pith IDGNV62Q4Q, Pith's permanent record of this paper and its review

CitationShuji Yamamoto. "On Kronecker limit formulas for real quadratic fields." arXiv:math/0602615 (2006). https://pith.science/paper/GNV62Q4Q

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Editor's note

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Colleague,

Here's the take on Yamamoto's paper (math/0602615). It does what the title says: extends Zagier's continued-fraction Kronecker limit formula from narrow ideal classes to arbitrary ray classes of a real quadratic field. Theorems 2.2.1 and 4.1.1 give clean formulas for the Laurent constant at s=1 and for ζ(0,C), and Theorem 5.2.3 gives a neat relation for Shintani's X(C) under the C2 twist. The example in §5.3 with X2(C)=1 is a nice concrete check. This is a genuine advance in the Stark–Shintani direction.

The exposition is mostly careful, and the main analytic machinery (the cone decomposition, the use of Egami's method) is standard and reproduced in enough detail. The paper is honest about what it does not do; it explicitly excludes the (+1,+1) signature case.

The soft spot is exactly where the stress-test note lands. In the proof of Theorem 5.1.1, after reducing to the case x_k=1, y_{k+1}=0, x_{k+1}=y_k in (0,1), the paper states an auxiliary identity for G(ω_k, ...)+G(ω_{k+1}, ...) and then says 'We omit the detailed computation.' That is not a routine check; it is a Barnes double-gamma translation identity, and it carries the cancellation of the Bernoulli term. Since this theorem feeds directly into Theorem 5.2.3 and Corollary 5.2.5, a sign or factor error here would propagate. The reader's weaker concern about A_m = ε^{-1} is a red herring – that is a standard result from Zagier [12], and the worked example is consistent.

So the verdict is conditional: the main theorems are likely correct, but the omitted computation should be supplied or referenced before the paper can be fully trusted. I would send it to peer review and ask the author to expand that part. For someone working on ray class zeta functions or Stark–Shintani, this is worth citing. I'd bring it to a reading group if we had one focused on this area.

Referee report

Major objections (1)

[§5.1, proof of Theorem 5.1.1] The only serious gap in the paper is in the proof of Theorem 5.1.1, in the case x_k = 1, y_{k+1} = 0, x_{k+1} = y_k ∈ (0,1). After stating the auxiliary identity G(ω_k, ω_k + 1 − y_k) + G(ω_{k+1}, (1 − x_{k+1})ω_{k+1}) = G(ω_k, 1 − y_k) + G(ω_{k+1}, (1 − x_{k+1})ω_{k+1} + 1) + B_1(y_k) log ω_{k+1}, the author writes 'We omit the detailed computation' and supplies neither a derivation nor a reference. This identity is nontrivial: G is the logarithmic derivative of Barnes' double zeta function, and the identity relates four different G-values with a Bernoulli correction. Because Theorem 5.1.1 is used directly in the proof of Theorem 5.2.3 and in the derivation of Corollary 5.2.5, a sign or factor error in this computation would propagate to the paper's central statements about ζ'(0,C) and the Stark–Shintani invariant X(C). The author should supply a complete proof of this identity and of the cancellation of B_1(y_k) log(ω_{k+1}ω'_{k+1}) with the Bernoulli terms from Z_{Q_k}(0), or at least a fully detailed derivation.

Minor objections (5)

[§2.1, definition of X] The displayed definition of the set X is garbled: it reads X = {x + y ε_f^{-1} ∈ z + b | x>0, y≥0}, but the subsequent decomposition uses the basis {A_k} and translates by nonnegative integers; the expression involving ε_f^{-1} does not match the structure of Lemma 2.1.3. The author should restate this definition correctly.
[§2.1, Lemma 2.1.3] The proof invokes the identity A_m = ε^{-1} from Zagier [12, §6] without further explanation. As this identity is essential for the periodicity of the cone decomposition, a brief indication or a more precise citation would improve self-containedness.
[§3.3, proof of Proposition 3.3.1(5) and (6)] The manipulations of the double zeta series, including the rearrangement over q ∈ Z, are not fully justified; these steps are standard in Shintani's theory but should be accompanied by a note on convergence and analytic continuation.
[§5.3, example] In the example, the pairs (x_k, y_k) are listed for k=0,...,4, but the product for X1(C) is displayed starting with the pair (7/11, 9/11) rather than (2/11, 1/11); the indexing should be made explicit.
[§1.1, notation] The double use of ⟨·⟩ (fractional part vs. Z-linear span) is acknowledged, but the notation in the main text should be checked for consistency; for instance, in §2.1 the expression ⟨b_k x_k + y_k⟩ uses the fractional-part convention.

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The paper

The paper claims

For every ray class of a real quadratic field, the constant term of the zeta function at s=1 is expressed as an explicit finite sum from continued-fraction data.

The reading

This paper proves a complete Kronecker limit formula for the partial zeta function $\zeta(s,C)$ of a ray class $C$ of a real quadratic field with arbitrary conductor $f$. The constant term of the Laurent expansion at $s=1$ is written as a finite sum over one period of a continued-fraction orbit, with each term built from the digamma function, the dilogarithm, and an explicit integral. The same decomposition yields an elementary formula for $\zeta(0,C)$ and expresses the invariant $X(C)=\exp(-\zeta'(0,C)+\zeta'(0,\overline{C}))$ as a product of double sine functions, together with a simple relation between $X(C)$ and $X(CC_2)$. A reader would care because this brings the arithmetic data of ray class zeta functions—class numbers and special $L$-values—within reach of finite continued-fraction calculations.

What carries the argument

The load-bearing mechanism is the cone decomposition datum $\{(\omega_k,x_k,y_k)\}$ associated with $C$: a purely periodic 'minus' continued fraction $\omega=[\![b_0,\ldots,b_{m-1}]\!]$ of a module generator $\omega$, with $\omega_k$ the periodic shifts and $A_k$ defined by $A_{k+1}=A_k/\omega_{k+1}$. The sectors $\{pA_{k-1}+qA_k\}$ tile the positive cone, and the lattice points in $z+\mathfrak{b}$ are encoded by the pairs $(x_k,y_k)$ through the recurrence $x_{k+1}=\langle b_kx_k+y_k\rangle$, $y_{k+1}=1-x_k$. Proposition 2.1.4 converts $\zeta(s,C)$ into a sum of binary quadratic form zeta sums $Z_{Q_k}(s,x_k,y_k)$. The identity $A_m=\varepsilon^{-1}$ over one period, hence $A_{rm}=\varepsilon_f^{-1}$, is what makes the sum over $k=1,\ldots,rm$ exactly the quotient by the unit group. The later sections recycle the same decomposition with evaluations at $s=0$ and double sine identities.

Extended reading notes

Core claim

The central claim is Theorem 2.2.1: for every ray class $C$ of modulus $f$, with data $(\omega_k,x_k,y_k)$ extracted from the minus continued fraction expansion of a generator of the module, the limit $\lim_{s\to 1}((D^{1/2}N(f))^s\zeta(s,C)-\frac{\log\varepsilon_f}{s-1})$ equals $\sum_{k=1}^{rm} P(\omega_k,\omega'_k,x_k,y_k)$, where $P$ is a specific combination of $F(\omega,x,y)$, the dilogarithm $\operatorname{Li}_2$, and digamma terms. The paper also proves $\zeta(0,C)=\sum_{k=1}^{rm}(B_1(x_k)B_1(y_k)+\frac{b_k}{2}B_2(x_k))$, expresses $X(C)$ as a product of double sine functions $S(\omega_k,z_k)S(\omega'_k,z'_k)$, and shows $X_1(C)=X_1(C^*)^{-1}$, $X_2(C)=X_2(C^*)$ for $C^*=CC_2$.

Load-bearing premise

The proof takes from reference [12, section 6] the fact that the product of the continued fraction coefficients over one period equals the inverse of the fundamental unit $\varepsilon$; if that product failed to equal $\varepsilon^{-1}$, the sum over representatives would not be correctly quotiented by the unit group and the limit formula would not follow.

Editorial extensions

If this is right

The Laurent constant $\rho(C)$ at $s=1$ can be evaluated by a finite continued-fraction calculation once the decomposition data are known, with no transcendental functions beyond the dilogarithm and digamma.
For a ray class character $\chi$ of signature $(+1,-1)$ or $(-1,+1)$, Corollary 5.2.5 writes $L(1,\chi)$ as a finite sum of logarithms of the double-sine factors $X_1(C)$ or $X_2(C)$, weighted by $\chi$.
The value $\zeta(0,C)$ has an elementary Bernoulli-polynomial expression in the same continued-fraction data, which also shows $\zeta(0,C)=\zeta(0,\overline{C})$ for $\overline{C}=CC_1C_2$.
The relations $X_1(C)=X_1(C^*)^{-1}$ and $X_2(C)=X_2(C^*)$ describe how the invariant changes when the signature is altered, feeding the functional-equation route to $L(1,\chi)$.
In the worked example $K=\mathbb{Q}(\sqrt{5})$, $f=(4-\sqrt{5})$, the product splitting forces $X_2(C)=1$, reducing $X(C)$ to $X_1(C)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the period-product identity is as robust as assumed, the formulas form a practical algorithm: one period of continued fraction data yields $\rho(C)$, $\zeta(0,C)$, and $X(C)$ for any modulus, with running time proportional to the period length.
The paper's remark that only infinite places where $\chi$ is positive contribute to $L(1,\chi)$ suggests a testable principle in higher-degree totally real fields: a similarly signed sum of invariants should reproduce $L(1,\chi)$ in, say, a cubic example.
The explicit double-sine product makes the conjectured unit property of $X(C)$ computationally checkable: for small conductors one can evaluate the product numerically and compare with the predicted units of the narrow ray class field.
The example's conclusion $X_2(C)=1$ hints that one of the two double-sine factors may often be trivial, which would simplify the conjectural unit construction.

Original abstract

Let $\zeta(s,C)$ be the partial zeta function attached to a ray class C of a real quadratic field. We study this zeta function at s=1 and s=0, combining some ideas and methods due to Zagier and Shintani. The main results are (1) a generalization of Zagier's formula for the constant term of the Laurent expansion at s=1, (2) some expressions for the value and the first derivative at s=0, related to the theory of continued fractions, and (3) a simple description of the behavior of Shintani's invariant X(C), which is related to $\zeta'(0,C)$, when we change the signature of C.

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Let $\zeta(s,C)$ be the partial zeta function attached to a ray class C of a real quadratic field. We study this zeta function at s=1 and s=0, combining some ideas and methods due to Zagier and Shintani. The main results are (1) a generalization of Zagier's formula for the constant term of the Laurent expansion at s=1, (2) some expressions for the value and the first derivative at s=0, related to the theory of continued fractions, and (3) a simple description of the behavior of Shintani's invariant X(C), which is related to $\zeta'(0,C)$, when we change the signature of C.

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s1	We study this zeta function at s=1 and s=0, combining some ideas and methods due to Zagier and Shintani.	contribution 0.58	0.84	0.03	asserted 0.97
s2	The main results are (1) a generalization of Zagier's formula for the constant term of the Laurent expansion at s=1, (2) some expressions for the value and the first derivative at s=0, related to the theory of continued fractions, and (3) a simple description of the behavior of Shintani's invariant X(C), which is related to $\zeta'(0,C)$, when we change the signature of C.	result 0.91	0.93	0.94	asserted 1.00

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References

Works cited (15)

[12]

D. Zagier, A Kronecker limit formula for real quadratic ﬁelds, Ma th. Ann., 213 (1975), 153–184

View on Pith 1975
[1]

E. W. Barnes, The genesis of the double gamma functions, Proc. London Math. Soc., 31 (1899), 358–381

View on Pith
[2]

E. W. Barnes, The theory of the double gamma function, Philos. T rans. Roy. Soc. (A), 196 (1901), 265–388

View on Pith 1901
[3]

S. Egami, A note on the Kronecker limit formula for real quadratic ﬁelds, Mathematika, 33 (1986), 239–243

View on Pith 1986
[4]

N. Kurokawa, S. Koyama, Multiple sine functions, Forum Math., 15 (2003), 839–876

View on Pith 2003
[5]

C. Meyer, Die Berechnung der Klassenzahl Abelscher K¨ orper ¨ uber quadratischen Zahlk¨ orpern, Akademie-Verlag, Berlin, 1957

View on Pith 1957
[6]

T. Shintani, On evaluation of zeta functions of totally real algebr aic number ﬁelds at non-positive integers, J. Fac. Sci. Univ. Tokyo, 23 (1976), 393–417

View on Pith 1976
[7]

T. Shintani, On a Kronecker limit formula for real quadratic ﬁelds, J. Fac. Sci. Univ. Tokyo, 24 (1977), 167–199

View on Pith 1977
[8]

T. Shintani, On special values of L-functions of algebraic number ﬁelds (Japanese), Sugaku, 29 (1977), 204–216

View on Pith 1977
[9]

T. Shintani, On certain ray class invariants of real quadratic ﬁeld s, J. Math. Soc. Japan, 30 (1978), 139–167

View on Pith 1978
[10]

C. L. Siegel, Advanced analytic number theory, Tata Institute of Fun- damental Research, Bombay, 1980

View on Pith 1980
[11]

H. Yoshida, Absolute CM-Periods, Mathematical Surveys and M ono- graphs, Vol. 106, AMS, 2003

View on Pith 2003
[13]

D. Zagier, Nombres de classes et fractions continues, Soc. Ma th. de France Ast´ erisque, 24-25 (1975), 81–97

View on Pith 1975
[14]

D. Zagier, Valeurs des fonctions zˆ eta des corps quadratique s r´ eels aux entiers n´ egatifs, Soc. Math. de France Ast´ erisque, 41-42 (1977), 135–151

View on Pith 1977
[15]

D. Zagier, Zetafunktionen und quadratische K¨ orper, Spring er, 1981. 24

View on Pith 1981

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [12]

D. Zagier, A Kronecker limit formula for real quadratic ﬁelds, Ma th. Ann., 213 (1975), 153–184

View on Pith 1975

[2] [1]

E. W. Barnes, The genesis of the double gamma functions, Proc. London Math. Soc., 31 (1899), 358–381

View on Pith

[3] [2]

E. W. Barnes, The theory of the double gamma function, Philos. T rans. Roy. Soc. (A), 196 (1901), 265–388

View on Pith 1901

[4] [3]

S. Egami, A note on the Kronecker limit formula for real quadratic ﬁelds, Mathematika, 33 (1986), 239–243

View on Pith 1986

[5] [4]

N. Kurokawa, S. Koyama, Multiple sine functions, Forum Math., 15 (2003), 839–876

View on Pith 2003

[6] [5]

C. Meyer, Die Berechnung der Klassenzahl Abelscher K¨ orper ¨ uber quadratischen Zahlk¨ orpern, Akademie-Verlag, Berlin, 1957

View on Pith 1957

[7] [6]

T. Shintani, On evaluation of zeta functions of totally real algebr aic number ﬁelds at non-positive integers, J. Fac. Sci. Univ. Tokyo, 23 (1976), 393–417

View on Pith 1976

[8] [7]

T. Shintani, On a Kronecker limit formula for real quadratic ﬁelds, J. Fac. Sci. Univ. Tokyo, 24 (1977), 167–199

View on Pith 1977

[9] [8]

T. Shintani, On special values of L-functions of algebraic number ﬁelds (Japanese), Sugaku, 29 (1977), 204–216

View on Pith 1977

[10] [9]

T. Shintani, On certain ray class invariants of real quadratic ﬁeld s, J. Math. Soc. Japan, 30 (1978), 139–167

View on Pith 1978

[11] [10]

C. L. Siegel, Advanced analytic number theory, Tata Institute of Fun- damental Research, Bombay, 1980

View on Pith 1980

[12] [11]

H. Yoshida, Absolute CM-Periods, Mathematical Surveys and M ono- graphs, Vol. 106, AMS, 2003

View on Pith 2003

[13] [13]

D. Zagier, Nombres de classes et fractions continues, Soc. Ma th. de France Ast´ erisque, 24-25 (1975), 81–97

View on Pith 1975

[14] [14]

D. Zagier, Valeurs des fonctions zˆ eta des corps quadratique s r´ eels aux entiers n´ egatifs, Soc. Math. de France Ast´ erisque, 41-42 (1977), 135–151

View on Pith 1977

[15] [15]

D. Zagier, Zetafunktionen und quadratische K¨ orper, Spring er, 1981. 24

View on Pith 1981