Partition zeta functions
We exploit transformations relating generalized -series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as , and to connect sums over partitions to the Riemann zeta function, multiple zeta values, and other number-theoretic objects.
addressref=aff1, firstname.lastname@example.org ]\initsRS\fnmRobert \snmSchneider
partitions \kwd-series \kwdzeta functions
1 Introduction, notations and central theorem
One marvels at the degree to which our contemporary understanding of -series, integer partitions, and what is now known as the Riemann zeta function emerged nearly fully-formed from Euler’s pioneering work [1, 2]. Euler discovered the magical-seeming generating function for the partition function
in which the -Pochhammer symbol is defined as for , and if the product converges, where we take and with (the upper half-plane). He also discovered the beautiful product formula relating the zeta function to the set of primes
to solve the so-called Basel problem, finding the exact value of ; he went on to find an exact formula for for every . Euler’s approach to these problems, interweaving infinite products, infinite sums and special functions, permeates number theory.
Very much in the spirit of Euler, here we consider certain series of the form , where the sum is taken over the set of integer partitions , as well as the “empty partition” , and where . We might sum instead over a subset of , and will intend to mean the set of partitions whose parts all lie in .
A few other notations should be fixed and comments made. We call the number of parts of the length of the partition. We call the number being partitioned the size of (also referred to as the weight of the partition). We write to indicate is a partition of (i.e., ), and we allow a slight abuse of notation to let “” indicate is one of the parts of (with multiplicity). Furthermore, for formal transparency, we define the natural number , which we call the integer of , to be the product of its parts, i.e.,
We assume the conventions , and (being an empty product), and take , and throughout, unless otherwise specified. Proofs are postponed until Section 3.
Sums of the form obey many interesting transformations, and often reveal patterns that are otherwise obscure, much as with sums over natural numbers. MacMahon appears to be the first to have considered such summations explicitly, looking at sums over partitions of a given positive integer in . Fine gives a variety of beautiful results and techniques related to sums over partitions in , as do Andrews , Alladi , and other authors. More recent work of Bloch-Okounkov  and Zagier  relates sums over partitions to infinite families of quasimodular forms via the -bracket operator, and Griffin-Ono-Warnaar  use partition sums involving Hall-Littlewood polynomials to produce modular functions. These series have deep connections. It is natural then to wonder, in what other ways might sums over partitions connect to classical number-theoretic objects?
We need to introduce one more notation, in order to state the central theorem. Define by and
where , for an arbitrary function . When the infinite product converges, let . We think of as a generalization of the -Pochhammer symbol. Note that if we set equal to a constant , then does specialize to the -Pochhammer symbol, as and .
The product on the right-hand side of identity (4) above is taken over the parts of . Note that the summation in (7) converges for outside the unit circle (it may converge inside the circle as well). Note also that, by L‘Hospital’s rule, any power series with constant term zero can be written as the limit
It is obvious that if is completely multiplicative, then , where is the so-called ”integer” of defined above. We record one more, obvious consequence of Theorem 1.1, as we assume it throughout this paper. As before, let , and take to be the set of partitions into elements of . Then clearly by setting if in Theorem 1.1, we see
The remaining summations in the theorem (aside from (7), which may not converge) are taken over .
The partition generating function (1) is true.
Assuming , if we take , if is prime and otherwise, then Theorem 1.1 yields another classical fact, plus a formula giving the zeta function as a sum over primes.
The Euler product formula (2) for the zeta function is true. We also have the identity
2 Partition-theoretic zeta functions
A multitude of nice specializations of Theorem 1.1 may be obtained. We would like to focus on an interesting class of partition sums arising from Euler’s sine function (3) combined with Theorem 1.1. Taking (as we have done in Corollary 1.3), we begin by noting an easy partition-theoretic formula that may be used to compute the value of .
Let denote the set of partitions into multiples of . Recall from above that the “integer” of a partition is the product of its parts.
Summing over partitions into even parts, we have the formula
We notice that the form of the sum of the right-hand side resembles . Based on this similarity, we wonder if there exists a nice partition-theoretic analog of possessing something of a familiar zeta function structure—perhaps Corollary 2.1 gives an example of such a function? However, in this case it is not so: the above identity arises from different types of phenomena from those associated with . We have an infinite family of formulas of the following shapes.
Summing over partitions into multiples of any whole number , we have
and increasingly complicated formulas can be computed for , .
Examples like these are appealing, but their right-hand sides are not entirely reminiscent of the Riemann zeta function, aside from the presence of . Certainly they are not as tidy as expressions of the form . Based on the previous corollaries, a reasonable first guess at a partition-theoretic analog of might be to define
Of course, neither side of this identity converges, but we do obtain convergent expressions if we omit the first term and perhaps subsequent terms of the product to yield , where denotes the set of partitions into parts greater than or equal to . For instance, we have the following formula.
Summing over partitions into parts greater than or equal to , we have
While it is an interesting expression, stemming from an identity of Ramanujan , once again this formula does not seem to evoke the sort of structure we anticipate from a zeta function—of course, the value of is not even known. We need to find the “right” subset of to sum over, if we hope to find a nice partition-theoretic zeta function. As it turns out, there are subsets of that naturally produce analogs of for certain values of .
We define a partition-theoretic generalization of the Riemann zeta function by the following sum over all partitions of fixed length at argument , :
This is a fairly natural formation, being similar in shape (and notation) to the weight multiple zeta function , which is instead summed over length- partitions into distinct parts; Hoffman gives interesting formulas relating (in different notation) to combinations of multiple zeta functions , which exhibit rich structure.
We have immediately that and . Using Theorem 1.1 and proceeding (see Section 3) much as Euler did to find the value of , we are able to find explicit values for at every positive integer . Somewhat surprisingly, we find that in these cases is a rational multiple of .
For , we have the identity
For example, we have the following values:
Corollary 2.4 reveals that is indeed of the form for all positive , like the zeta values given by Euler (we note that is the highest zeta value Euler published) . We have more: we can find explicitly for all . These values are finite combinations of well-known zeta values, and are also of the form .
For we have the identity
and increasingly complicated formulas can be computed for .
The summation on the far right above may be shortened by noting the symmetry of the summands around the term.
It would be desirable to understand the value of at other arguments ; the proof we give below (see Section 3) does not shed much light on this question, being based very closely on Euler’s formula (1.3), which forces be a power of . Also, if we solve Corollary 2.3 for , we conclude that , which is the value of under analytic continuation. Can be extended via analytic continuation for values of ? In a larger sense we wonder: do nice zeta function analogs exist if we sum over other interesting subsets of ?
We do have a few general properties shared by convergent series summed over large subclasses of . First we need to refine some of our previous notations.
Take any subset of partitions . Then for , on analogy to classical zeta function theory, when these expressions converge we define
As important special cases, we have and . It is also easy to see that and if we assume absolute convergence. Moreover, given absolute convergence, we may write as classical Dirichlet series related to multiplicative partitions: we have and (see  for more about multiplicative partitions).
As previously, take and take to denote partitions into elements of (thus ). Note that is divergent if and, when is finite (thus there is no restriction on the value of ), if for any and even integer . Similarly, when is finite, is divergent if for any and odd integer . Clearly if , then from the product representations we also have and .
Many interesting subsets of partitions have the form , in particular those to which Theorem 1.1 most readily applies. Note that such subsets are partition ideals of order 1, in the sense of Andrews . With the above notations, we have the following useful “doubling” formulas.
If converges over , then
Furthermore, for we have the identity
If we take , then (13) reduces to the classical identity , where is Liouville’s function. Another specialization of Corollary 2.6 leads to new information about : we may extend the domain of to if we take , , . We find inherits analytic continuation from the sum on the right-hand side below.
For , we have
This resembles a well-known series shuffle product formula for multiple zeta values .
Another interesting subset of is the set of partitions into distinct parts; also of interest is the set of partitions into distinct elements of (thus ). However, partitions into distinct parts are not immediately compatible with the identities in Theorem 1.1. Happily, we have a dual theorem that leads us to zeta functions summed over for any .
Let us recall the infinite product from Theorem 1.1.
Note that there is not a nice “inverted” sum of the form (7) here.
Just as with Theorem 1.1, we may write arbitrary power series as limiting cases, and we have the obvious identity
with the remaining summations in Theorem 2.8 being taken over elements of . For completeness, we record another obvious but useful consequence of Theorems 1.1 and 2.8. The following statement might be viewed as a generalized eta quotient formula, with coefficients given explicitly by finite combinatorial sums.
For defined on , consider the double product
where the sign is fixed for fixed , but may vary as varies. Then the coefficients are given by the -tuple sum
in which we have set and with the sign as associated to each above.
The or signs in the formula for indicate partitions arising from the numerator or denominator, respectively, of the double product. One may replace with to effect further sign changes.
Analogous corollaries to those following Theorem 1.1 are available, but we wish right away to apply this theorem to the problem at hand, the investigation of partition zeta functions. We have as well as . It is immediate then from (15) that for we also have the following relations, where the sum on the left-hand side of each equation is taken over the partitions into distinct elements of :
Note that and are finite sums (and entire functions of ) if is a finite set, unlike and . Note also that identically if , with zeros when is finite at the values for any and even. Unlike , we can see from (19) that is well-defined on (thus both and are well-defined over all subsets of ); when is finite, has zeros at for and odd. Morever, we have and . Here is an example of a zeta sum of this form.
Summing over partitions into distinct parts, we have that
Zeta sums over partitions into distinct parts admit an important special case: as we remarked beneath definition (11), the multiple zeta function can be written
For we have the identities
and increasingly complicated formulas of the shape can be computed for multiple zeta values of the form .
If converges over , then
Furthermore, for we have
We note that by recursive arguments, from (13) and (21) together with (8), we have these curious product formulas connecting sums over partitions into distinct parts to their counterparts involving unrestricted partitions:
Now, if we take then (21) becomes the classical identity , where is the Möbius function. We might view the simple quantity as a partition-theoretic generalization of ; it specializes to the Möbius function (when considering partitions into distinct prime parts), and also to Liouville’s function (considering unrestricted prime partitions), as we saw above. K. Alladi has observed this correspondence as well (personal communication, December 22, 2015).
It is fascinating—and rather mysterious—that partitions (which are defined additively, with no connection to multiplication) into parts that are prime numbers (defined multiplicatively) should have significant number-theoretic connections.
The literature abounds with product formulas which, when fed through the machinery of the identities noted here, produce nice partition zeta sum variants; the interested reader is referred to  as a starting point for further study.
3 Proofs of theorems and corollaries
Proof of Theorem 1.1.
and multiply out all the terms (without collecting coefficients in the usual way). The result is the partition sum in (4).
recalling the notation (as well as ) from the theorem, which is (5). Similarly, we can show
Thus we have
which leads to (6).
term-by-term into the sum (5) and simplify to find the desired expression.
We notice that the expression on the left-hand side is also present on the far right in the denominator. We replace this term in the denominator with the entire right-hand side of the equation; reiterating this process indefinitely gives (8). ∎
The seriesenjoy other nice, golden ratio-like relationships. For instance, because
it is easy to see that
which resembles the formula involving the golden ratio and its reciprocal.
Proof of Corollary 1.3.
As noted above, we assume . Let , if is prime and otherwise; then by (4)
Consider the prime decomposition of a positive integer , . We will associate this decomposition to the unique partition into prime parts , where is repeated times (thus is equal to ). Every positive integer is associated to exactly one partition into prime parts (with associated to ), and conversely: there is a bijective correspondence between and (Alladi and Erdős give an interesting study  along these lines). Therefore we see by absolute convergence that
Proof of Corollary 2.1.
This is actually a special case of the subsequent Corollary 2.2, setting in the first equation (see below). ∎