## Abstract

We express the coefficients of the Hirzebruch *L*-polynomials in terms of certain alternating multiple zeta values. In particular, we show that every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign. Similar results hold for the polynomials associated to the \(\hat{A}\)-genus.

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## 1 Introduction

The Hirzebruch *L*-polynomials are certain polynomials with rational coefficients,

featured in the Hirzebruch signature theorem, which expresses the signature \(\sigma (M)\) of a smooth compact oriented manifold \(M^{4k}\) as

where \(\mathsf {p}_i\) are taken to be the Pontryagin classes of the tangent bundle of *M*, see [3, Theorem 8.2.2] or [5, Theorem 19.4]. The *k*th polynomial has the form

where the sum is over all partitions \((j_1,\ldots ,j_r)\) of *k*, i.e., sequences of integers \(j_1\ge \cdots \ge j_r \ge 1\) such that \(j_1+\cdots + j_r =k\). The purpose of this note is to establish certain properties of the coefficients \(h_{j_1,\ldots ,j_r}\).

For real numbers \(s_1,\ldots ,s_r>1\), we define the series

where \(n\ge _2 m\) means “\(n\ge m\) with equality only if *n* is even”. Define the symmetrization of this series by

where \(\Sigma _r\) is the symmetric group.

### Theorem 1

The coefficients of the Hirzebruch *L*-polynomials are given by

where \(\alpha _\ell \) counts how many of \(j_1,\ldots ,j_r\) are equal to \(\ell \).

It is well-known that \(h_k\) is positive for all *k*. In [8, Appendix A], it is argued that \(h_{i,j}\) is always negative and that \(h_{i,j,k}\) is always positive (following an argument attributed to Galatius in the case of \(h_{i,j}\)), and it is asked whether it has been proved in general that \((-1)^{r-1}h_{j_1,\ldots ,j_r}\) is positive. We have not been able to locate such a result in the literature, but we can prove it using our formula. It follows from the following result.

### Theorem 2

For all real \(s_1,\ldots ,s_r>1\),

### Corollary 3

The coefficient \(h_{j_1,\ldots ,j_r}\) in the Hirzebruch *L*-polynomial \(\mathsf L_k\) is non-zero for every partition \((j_1,\ldots ,j_r)\) of *k*. It is negative if *r* is even and positive if *r* is odd.

It is remarked in [8] that a similar pattern has been observed in the multiplicative sequence of polynomials associated with the \(\hat{A}\)-genus. The polynomials in question are

These can be treated similarly. Let us write

where the sum is over all partitions \((j_1,\ldots ,j_r)\) of *k*. Consider the series

and its symmetrization

### Theorem 4

The coefficients of the \(\hat{A}\)-polynomials are given by

In particular, the coefficient \(a_{j_1,\ldots ,j_r}\) is negative if *r* is odd and positive if *r* is even.

## 2 Proofs

The first step in our proof is to establish a formula that expresses the coefficient \(h_{j_1,\ldots ,j_r}\) as a linear combination of products \(h_{k_1}\cdots h_{k_\ell }\). This generalizes the formulas for \(h_{i,j}\) and \(h_{i,j,k}\) found in [8]. In the appendix of [2], recursive formulas for computing \(h_{j_1,\ldots ,j_r}\) in terms of products \(h_{k_1}\cdots h_{k_\ell }\) are given. Here we give an explicit closed formula. The result holds for arbitrary multiplicative sequences of polynomials (see [3, §1]).

### Theorem 5

Let \(K_0,K_1,K_2,\ldots \) be a multiplicative sequence of polynomials with

The coefficients satisfy the relation

where \(\alpha _i\) counts how many of \(j_1,\ldots ,j_r\) are equal to *i*, the sum is over all partitions \(\mathcal {P} = \{P_1,\ldots ,P_\ell \}\) of the set \(\{1,2,\ldots ,r\}\),

and

### Proof

A multiplicative sequence of polynomials is determined by its characteristic power series

where \(b_k = \lambda _{1,\ldots ,1}\) is the coefficient of \(\mathsf {p}_1^k\) in \(K_k\). Indeed, if we, as in [3], formally interpret the coefficients \(b_k\) as elementary symmetric functions in \(\beta _1',\ldots ,\beta _m'\) (\(m\ge k\)), so that

then the coefficient \(\lambda _{j_1,\ldots ,j_r}\) is the *monomial symmetric function* in \(\beta _1',\ldots ,\beta _m'\) (see [3, Lemma 1.4.1]).

Note that \(\lambda _k\) equals the power sum \(\sum _i (\beta _i')^k\). The product \(\lambda _{k_1}\cdots \lambda _{k_\ell }\) is then the *power sum symmetric function* evaluated at \(\beta _i'\), and the claim follows from a general formula that expresses the monomial symmetric functions in terms of power sum symmetric functions, see Theorem 8 below. \(\square \)

The characteristic series of the Hirzebruch *L*-polynomials is

where

and \(B_k\) are the Bernoulli numbers,

see [3, §1.5].

As is well-known, the leading coefficient \(h_k\) of \(\mathsf {p}_k\) in \(\mathsf L_k\) is given by

see [3, p.12]. In [8], the formula

involving the Riemann zeta function,

is used to argue that \(h_{i,j}<0\) and \(h_{i,j,k}>0\). From this point on, our argument will depart from that of [8]. A key observation is that we can express \(h_k\) in terms of the alternating zeta function,

instead of the Riemann zeta function. It is well-known, and easily seen, that

Moreover, the following holds for all positive integers *k*,

Combining this with (2), we see that

From (1) we get

where the notation is as in Theorem 5.

The next observation is that the sum in the right hand side bears a striking resemblance with the right hand side of Hoffman’s formula [4, Theorem 2.2] (proved anew in Theorem 7 below), which relates multiple zeta values and products of zeta values—the only difference is that \(\zeta ^*\) appears instead of \(\zeta \). The second step in the proof is then to find a Hoffman-like formula for \(\zeta ^*\). Here is what we were led to write down: Define, for real numbers \(s_1,\ldots ,s_r>1\),

where \(n\ge _2 m\) means “\(n\ge m\) with equality only if *n* is even”. Then symmetrize, and define

Here is our Hoffman-like formula. Together with (5) it implies Theorem 1.

### Theorem 6

The following equality holds for all real \(s_1,\ldots ,s_r>1\),

where the sum is over all partitions \(\mathcal {P}= \{P_1,\ldots ,P_\ell \}\) of \(\{1,2,\ldots ,r\}\) and

### Proof

This will follow by specialization of Theorem 10 below. \(\square \)

Next we turn to the proof of Theorem 2, which says that

for all real \(s_1,\ldots ,s_r>1\).

### Proof of Theorem 2

The proof is in principle not more difficult than the proof that \(\zeta ^*(s)\) is positive; to see this one simply arranges the sum as

and notes that the summands are positive. Since the series is absolutely convergent, we are free to rearrange as we please, as the reader will recall from elementary analysis.

Towards the general case, introduce for \(k\ge 1\) the auxiliary series

Then one can argue using the following two equalities, whose verification we leave to the reader:

Here \(T_{2\ell +2}(s_1,\ldots ,s_{j-1})\) should be interpreted as 1 for \(j=1\). The second equality may be used to show that \(T_{2k}(s_1,\ldots ,s_r)\) is positive by induction on *r*. The first equality then shows that \(T(s_1,\ldots ,s_r)\) is negative. \(\square \)

Finally, we turn to the proof of Theorem 4. The argument turns out to be easier in this case. Recall that the \(\hat{A}\)-genus has characteristic series

Let us write

where the sum is over all partitions \((j_1,\ldots ,j_r)\) of *k*. By using the Cauchy formula (see [3, p.11]) one can calculate the coefficient \(a_k\) of \(\mathsf {p}_k\) in \({\hat{\mathsf {A}}}_k\). The result is

It follows that

for every partition \((k_1,\ldots ,k_\ell )\) of *k*. Theorem 5 then yields

The terms in the sum are clearly positive, so we see already from this expression that \((-1)^ra_{j_1,\ldots ,j_r}>0\) for all partitions \((j_1,\ldots ,j_r)\) of *k*. However, more can be said; the sum in the right hand side now not only resembles but is *equal* to the right hand side of another formula of Hoffman [4, Theorem 2.1]. In our notation this formula says that

This proves Theorem 4.

## 3 Combinatorics of infinite sums

The proofs of Theorems 5 and 6, as well as of Hoffman’s formula, share the same combinatorial underpinnings; this is the topic of the present section.

Recall that a *partition* of a set *S* is a set of non-empty disjoint subsets,

such that \(S = \pi _1\cup \cdots \cup \pi _r\). Write \(\ell (\pi ) = r\) for the *length* of \(\pi \). The set of partitions \(\Pi _S\) is partially ordered by refinement, \(\pi = \{\pi _1\ldots ,\pi _r\} \le \rho = \{\rho _1,\ldots ,\rho _\ell \}\) if and only if there is a partition \(\mathcal {P}= \{P_1,\ldots ,P_\ell \}\) of the set \(\{1,2,\ldots ,r\}\) such that

We will write \(\rho = \mathcal {P}(\pi )\) if (7) holds. Note that for every \(\rho \ge \pi \) there is a unique partition \(\mathcal {P}\) such that \(\rho = \mathcal {P}(\pi )\).

We will consider certain formal power series in indeterminates \(a_n\) for \(a\in S\) and positive integers *n*. For a subset \(T\subseteq S\), write

For a partition \(\pi = \{\pi _1,\ldots ,\pi _r\}\) of *S*, consider the formal power series

It is then immediate that

By applying the Möbius inversion formula (see e.g. [6, Proposition 3.7.2]), we get

The Möbius function of \(\Pi _S\) is given by

where the number

counts how many ‘\(\pi \)-blocks’ \(\rho _i\) consists of, see e.g. [6, Example 3.10.4].

Let us first note that this gives a neat proof of Hoffman’s formula (though we would be surprised if this has not been noticed before). Recall that the multiple zeta function is defined by

for real \(s_1,\ldots ,s_r >1\).

### Theorem 7

(Hoffman [4, Theorem 2.2])

where the sum is over all partitions \(\mathcal {P}= \{P_1,\ldots ,P_\ell \}\) of \(\{1,2,\ldots ,r\}\) and

### Proof

Take \(S= \{1,2,\ldots ,r\}\) and substitute \(a_n\) by \(\frac{1}{n^{s_a}}\) for \(a\in S\) in (8). \(\square \)

Secondly, we will use (8) to express the monomial symmetric functions in terms of power sum symmetric functions. We refer to [7, Chapter 7] for a pleasant introduction to symmetric functions. For an integer partition \(I = (i_1,\ldots ,i_r)\vdash k\), recall that the *power sum symmetric function* \(p_I\) is the formal power series in indeterminates \(x_1,x_2,\ldots \) defined by \(p_I = p_{i_1}\cdots p_{i_r}\), where

The *monomial symmetric function* \(m_I\) is defined as the sum of all pairwise distinct monomials of the form \(x_{\sigma _1}^{i_1}\cdots x_{\sigma _r}^{i_r}\).

### Theorem 8

For every \(k\ge 1\) and every integer partition \(I = (i_1,\ldots ,i_r)\) of *k*,

where the sum is over all partitions \(\mathcal {P}= \{P_1,\ldots ,P_\ell \}\) of the set \(\{1,2,\ldots ,r\}\), the number \(\alpha _j\) counts how many of \(i_1,\ldots ,i_r\) are equal to *j*, and \(J = (j_1,\ldots ,j_\ell )\) is given by

### Proof

Let *S* be any set with *k* elements. Perform the substitution \(a_n = x_n\) for each \(a\in S\) in the equality (8) and note that this takes \(p_\pi \) to \(p_I\) and \(m_\pi \) to \(\alpha _1!\cdots \alpha _k!\, m_I\), where \(I = \big (|\pi _1|,\ldots ,|\pi _r|\big )\) is the integer partition underlying the set partition \(\pi \) (assuming, as we may, \(|\pi _1|\ge \cdots \ge |\pi _r|\)). \(\square \)

Next, we turn to the result that will specialize to our formula in Theorem 6. Consider the following alternating version of \(p_\pi \):

For an *ordered* partition \(\widetilde{\pi } = (\pi _1,\ldots ,\pi _r)\) we define

Then for an unordered partition \(\pi = \{\pi _1, \ldots , \pi _r\}\), define

where the sum is over the *r*! ordered partitions \(\widetilde{\pi }\) whose underlying unordered partition is \(\pi \).

### Lemma 9

For every partition \(\pi \in \Pi _S\),

### Proof

We may without loss of generality assume that \(\pi \) is the minimal element, because the poset \(\left\{ \rho \in \Pi _S \mid \rho \ge \pi \right\} \) is isomorphic to the poset \(\Pi _\pi \) of partitions of the set \(\pi \). Then \(n= \ell (\pi )\) is the number of elements of *S*. The number of partitions of length *k* in \(\Pi _S\) is equal to the Stirling number of the second kind *S*(*n*, *k*), see e.g. [6, Example 3.10.4]. Thus,

By plugging in \(x=-1\) in the well-known identity

where \((x)_k = x(x-1)(x-2) \cdots (x-k+1)\), we see that (9) equals \((-1)^n\). \(\square \)

### Theorem 10

For every partition \(\pi \),

### Proof

By Möbius inversion, the equality is equivalent to

and we proceed to prove (11). It is clear that both sides can be written as linear combinations of series of the form

for various \(\nu = \{\nu _1,\ldots ,\nu _m\} \ge \pi \), where *e* is an assignment of a parity \(e_i \in \{0,1\}\) to each \(\nu _i\). For example, if \(\nu = \{\{a,b\},\{c\}\}\) and *e* assigns 1 to \(\{a,b\}\) and 0 to \(\{c\}\), then

The question is with what coefficients \(m_{\nu ,e}\) will appear in the respective sides of (11). For the left hand side this is not difficult: \(m_{\nu ,e}\) appears in \((-1)^{\ell (\pi )} \overline{p}_\pi \) with coefficient

where \(v_i\) is the number of \(\pi \)-blocks in \(\nu _i\).

The right hand side requires a little more effort—and notation. It is clear that \(m_{\nu ,e}\) appears in \(T_\rho ^\Sigma \) only if \(\nu \ge \rho \) and *e* assigns an even value to \(\nu _i\) whenever \(\nu _i\) consists of more than one \(\rho \)-block. This can be reformulated as saying that \(\nu \ge \rho \ge e(\nu )\), where \(e(\nu )\le \nu \) is the partition that keeps \(\nu _i\) intact if \(e_i\) is odd and splits \(\nu _i\) completely if \(e_i\) is even. Or more precisely, \(e(\nu )\) is the smallest element below \(\nu \) that contains \(\nu _i\) whenever \(e_i\) is odd. Since we symmetrize, there will be repetitions; for \(\nu \ge \rho \ge e(\nu )\), the term involving \(m_{\nu ,e}\) will be repeated \(b_{\rho ,\nu } = b_1! \cdots b_m!\) times in \(T_\rho ^\Sigma \), where \(b_i\) is the number of \(\rho \)-blocks in \(\nu _i\). Thus, the coefficient of \(m_{\nu ,e}\) in \((-1)^{\ell (\rho )} T_\rho ^\Sigma \) is \({\text {sgn}}(\rho ,\nu ,e)b_{\rho ,\nu }\). It follows that the coefficient of \(m_{\nu ,e}\) in \(\sum _{\rho \ge \pi } (-1)^{\ell (\rho )} T_\rho ^\Sigma \) is

where \(e(\nu )\vee \pi \) is the least upper bound of \(e(\nu )\) and \(\pi \).

Put \(\pi _{(i)}=\{\pi _j \in \pi : \pi _j \subseteq \nu _i\}\) and \(\nu _{(i)} = \{\nu _i\}\). We then have an isomorphism of posets \([e(\nu ) \vee \pi ,\nu ] \cong \prod _{i : e_i = 0} [\pi _{(i)},\nu _{(i)}]\). Under this isomorphism \(\rho \in [e(\nu ) \vee \pi ,\nu ] \) is sent to \(\rho _{(i)}=\{\rho _j \in \rho : \rho _j \subseteq \nu _i\} \in [\pi _{(i)},\nu _{(i)}]\). Note that \(b_i\) is the length of \(\rho _{(i)}\). We now find that the sum (13) factors as a product

By Lemma 9, this is equal to

This shows that (13) equals (12), and the theorem is proved. \(\square \)

To prove Theorem 6, take \(S=\{1,2,\ldots ,r\}\) and substitute \(a_n\) by \(\frac{1}{n^{s_a}}\) in (10).

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## Acknowledgements

We thank Don Zagier and Matthias Kreck for valuable comments. The impetus for this work was a question from Oscar Randal-Williams to the first author about certain points in [1]. The first author was supported by the Swedish Research Council through Grant No. 2015-03991.

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Berglund, A., Bergström, J. Hirzebruch *L*-polynomials and multiple zeta values.
*Math. Ann.* **372**, 125–137 (2018). https://doi.org/10.1007/s00208-018-1647-2

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DOI: https://doi.org/10.1007/s00208-018-1647-2