Explorations in the theory of partition zeta functions ∗

We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theory of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the analytic continuations of these “partition zeta functions”, find unusual formulas for the Riemann zeta function, prove identities for multiple zeta values, and see that some of the formulas allow for p-adic interpolation. The second family we study was anticipated by Manin and makes use of modular forms, functions which are intimately related to integer partitions by universal polynomial recurrence relations. We survey recent work on these zeta polynomials which includes the proof of their Riemann Hypothesis. 1 The setting: Visions of Euler In antiquity, storytellers began their narratives by invoking the muse whose divine influence would guide the unfolding imagery. It is fitting, then, that we begin this article by praising the immense curiosity of Euler, whose imagination ranged playfully across almost the entire landscape of modern mathematical thought. Ken Ono Department of Mathematics and Computer Science, Emory University e-mail: [email protected] Larry Rolen Department of Mathematics, The Pennsylvania State University e-mail: [email protected] Robert Schneider Department of Mathematics and Computer Science, Emory University e-mail: [email protected] ∗ The first author is supported by National Science Foundation and the Asa Griggs Candler Fund. 1 ar X iv :1 60 5. 05 53 6v 1 [ m at h. N T ] 1 8 M ay 2 01 6 2 K. Ono, L. Rolen, and R. Schneider Euler made spectacular use of product-sum relations, often arrived at by unexpected avenues, thereby inventing one of the principle archetypes of modern number theory. Among his many profound identities is the product formula for what is now called the Riemann zeta function: ζ (s) := ∞ ∑ n=1 n−s = ∏ p∈P (1− p−s)−1 (1) With this relation, Euler connected the (at the time) cutting-edge theory of infinite series to the ancient set P of prime numbers. Moreover, in solving the famous “Basel problem” posed a century earlier by Pietro Mengoli (1644), Euler showed us how to compute even powers of π using the zeta function, giving explicit formulas of the shape ζ (2N) = π2N× rational. (2) It turns out there are other classes of zeta functions, arising from other Eulerian formulas in the universe of partition theory. Much like the set of positive integers, but perhaps even more richly, the set of integer partitions ripples with striking patterns and beautiful number-theoretic phenomena. In fact, the positive integers N are embedded in the integer partitions P in a number of ways: obviously, positive integers themselves represent the set of partitions into one part; less trivially, the prime decompositions of integers are in bijective correspondence with the set of prime partitions, i.e., the partitions into prime parts (if we map the number 1 to the empty partition / 0), as Alladi and Erdős note [1]. We might also identify the divisors of n with the partitions of n into identical parts, and there are many other interesting ways to associate integers to the set of partitions. Euler found another profound product-sum identity, the generating function for the partition function p(n) ∞ ∏ n=1 (1−qn)−1 = ∞ ∑ n=0 p(n)qn, (3) single-handedly establishing the theory of integer partitions. This formula doesn’t look much like the zeta function identity (1); however, generalizing Euler’s proofs of these theorems leads to a new class of partition-theoretic zeta functions. It turns out that (1) and (3) both arise as specializations of a single “master” product-sum formula. Before we proceed, let us fix some notation. Let P denote the set of all integer partitions. Let λ = (λ1,λ2, . . . ,λr), with λ1 ≥ λ2 ≥ ·· · ≥ λr ≥ 1, denote a generic partition, l(λ ) := r denote its length (the number of parts), and |λ | := λ1 + λ2 + · · ·+λr denote its size (the number being partitioned). We write “λ ` n” to mean λ is a partition of n, and “λi ∈ λ” to indicate λi ∈ N is one of the parts of λ . Then we have the following “master” identity (which is a piece of Theorem 1.1 in [40]). Proposition 1. For an arbitrary function f : N→ C, we have Explorations in the theory of partition zeta functions 3 ∞ ∏ n=1 (1− f (n)qn)−1 = ∑ λ∈P q|λ | ∏ λi∈λ f (λi). The sum on the right is taken over all partitions, and the left-hand product is taken over the parts λi of partition λ . The proof proceeds along similar lines to Euler’s proof of (3), and can be seen immediately if one expands a few terms of the infinite product by hand, without collecting coefficients in the usual manner. This simple identity also has interesting (and sometimes exotic) representations in terms of Eulerian q-series and continued fractions; readers are referred to [40] for details. Remark 1. Note that, collecting like terms, we can re-write such partition sums as standard power series, summed over non-negative integers: ∑ λ∈P cλ q |λ | = ∞ ∑ n=0 qn ∑ λ`n cλ (4) It is immediate from Proposition 1 and (4) that the partition generating function (3) results if we let f ≡ 1 identically. Similarly, if we set f (n) = n−s,q = 1, and sum instead over the subset PP of prime partitions, we arrive at the Euler product formula (1) for the zeta function. This follows from the bijective correspondence between prime partitions and the factorizations of natural numbers noted above. In light of these observations, we might view the Riemann zeta function as the prototype for a new class of combinatorial objects arising from Eulerian methods. 1.1 Partition-theoretic zeta functions Inspired by work of Euler [19], Fine [23], Andrews [2], Bloch and Okounkov [7], Zagier [49], Alladi and Erdős [1], and others, the authors here undertake the study of a class of zeta functions introduced by the third author in [40], resembling the Riemann zeta function ζ (s) but summed over proper subsets of P , as opposed to over natural numbers. In this paper, we review a few of the results from [40], and record a number of further identities relating certain zeta functions arising from the theory of partitions to various objects in number theory such as Riemann zeta values, multiple zeta values, and infinite product formulas. Some of these formulas are related to results in the literature; they are presented here as examples of this new class of partition-theoretic zeta functions. We also give several formulas for the Riemann zeta function, and results on the analytic continuation (or non-existence thereof) of zeta-type series formed in this way. Furthermore, we discuss the p-adic interpolation of these zeta functions in analogy with the classical work of Kubota and Leopoldt on p-adic continuation of the Riemann zeta function [31]. To describe our primary object of study, we must introduce a new statistic related to partitions, to complement the length and size defined above. We define the integer 4 K. Ono, L. Rolen, and R. Schneider of a partition λ , notated as nλ , to be the product of its parts: nλ := λ1λ2...λr (5) This multiplicative statistic may not seem very natural as partitions arise purely additively, with deep additive structures such as Ramanujan congruences dominating the theory. Yet if we let nλ formally replace the usual index n in the Riemann zeta function, and sum instead over appropriate subsets of partitions, we arrive at an analytic-combinatorial object with many nice properties. Definition 1. Over a subset P ′ ⊂P and value s ∈C for which the sum converges, we define a partition zeta function to be the series ζP ′(s) := ∑ λ∈P ′ n−s λ . Very nice relations arise from unique properties of special subsets P ′, e.g. partitions with some distinguishing structure, or into parts sharing some trait. For example, if we let P∗ ⊂P denote partitions into distinct parts, there are interesting closed-form expressions, such as ζP∗(2) = sinhπ π , (6) which follows from Proposition 1 together with a formula of Euler (cf. [40]). By summing instead over partitions into parts ≥ 2 (that is, disallowing 1’s), we arrive at curious identities such as ζP≥2(3) = 3π cosh ( 1 2 π √ 3 ) , (7) which can be found from Proposition 1 together with a formula of Ramanujan [40]. This equation is somewhat surprising, as the Riemann zeta values at odd arguments are famously enigmatic. Other attractive closed sums can be found—and general structures observed, as detailed in [40]—when we restrict our attention to partitions PM whose parts all lie in a subset M ⊂N. It is easy to check from Proposition 1 that we have the Euler product formula ζPM (s) = ∏ k∈M ( 1− k−s )−1 . (8) We see from the right-hand side of (8) that ζPM (s) diverges if 1 ∈M , thus the restriction 1 / ∈M exhibited in the pair of identities above is a necessary one here. In fact, some restriction on the maximum multiplicity of 1 occurring as a part is necessary for any partition zeta function to converge. That is, we must sum over partitions containing 1 with multiplicity at most m≥ 0. Note that the resulting zeta function will equal the one in the case where 1’s are not allowed, multiplied by Explorations in the theory of partition zeta functions 5 m+ 1, as each n−s λ is repeated m+ 1 times in the sum (adjoining 1’s to a partition does not affect its “integer”). We note that if Dirichlet series coefficients an are defined by ζPM (s) =: ∑ n≥1 ann, it is easy to see that an counts the number of ways to write n as a product of integers in M , where each ordering of factors is only counted once. When M = N, then these ways of writing n as a product of smaller numbers are known as multiplicative partitions, and have been studied in a number of places in the literature; for example, the interested reader is referred to [2, 14, 33, 42, 51]. We wish to study partition zeta functions over special subsets of P and arguments s that lead to interesting relations. We begin by highlighting a few nicelooking examples. Let us recall a few identities from [40] as examples of zeta function phenomena induced by suitable partition subsets of the form PM . By summing over partitions into even parts we have a combinatorial formula to compute π: