In his study of Nekrasov–Okounkov type formulas on “partition theoretic” expressions for families of infinite products, Han discovered seemingly unrelated q-series that are supported on precisely the same terms as these infinite products. In collaboration with Ono, Han proved one instance of this occurrence that exhibited a relation between the numbers a(n) that are given in terms of hook lengths of partitions, with the numbers b(n) that equal the number of 3-core partitions of n. Recently Han revisited the q-series with coefficients a(n) and b(n), and numerically found a third q-series whose coefficients appear to be supported on the same terms. Here we prove Han’s conjecture about this third series by proving a general theorem about this phenomenon.
In their study of supersymmetric gauge theory, Nekrasov and Okounkov discovered a striking infinite product identity . This surprising theorem relates the sum over products of partition hook lengths to the powers of Euler products and has been generalized in many ways to give expressions for many infinite product q-series. The original identity is given by
where the sum is over integer partitions, is the integer partitioned by λ, and denotes the multiset of classical hook lengths associated to a partition λ.
The Nekrasov–Okounkov formula specializes in the case and to two classical q-series identities. The first is a special case of Euler’s Pentagonal Number Theorem, and the second gives Jacobi’s famous identity for the product .
In , Han extended the Nekrasov–Okounkov identity to consider the number of t-core partitions of n. While working on this generalization, Han investigated the nonvanishing of infinite product coefficients. For example, he considered the infinite product,
and conjectured in  that the coefficient of is not equal to 0 for , positive integers such that and . Letting and , Han reformulated the famous conjecture of Lehmer that the coefficients of
In , Han formulated a conjecture comparing the nonvanishing of terms of with another infinite Euler product given by
Recall is the series given by
Based on numerical evidence, Han conjectured that the non-zero coefficients of and are supported on the same terms; assuming the notation above, if and only if .
This conjecture is proved in a joint paper with Ono . In addition to proving the conjecture, Han and Ono proved precisely for those non-negative integers n for which is odd for some prime .
Recently Han discovered another series that appears to be supported on the same terms as and . This series is given by,
Based on numerical evidence, Han conjectured if and only if if and only if .
Here we prove the following general theorem that produces infinitely many modular forms, including those in Equations (1.2) and (1.3) that are supported precisely on the same terms as Equation (1.1).
It is convenient to normalize Equation (1.1) as shown below.
Suppose thatis an even weight newform with trivial nebentypus that has complex multiplication byand a level of the form, where. Then the coefficientsif and only if. More precisely,for those non-negative integers n for whichis odd for some prime .
Here we letandis the usual Fourier expansion at infinity.
Consider the normalized function ofandgiven by,
Theorem 1 implies the original work of Han and Ono. As explained in ,is a weight 4 newform with complex multiplication bywith level 9. Theorem 1 also implies Han’s new conjecture concerning coefficients of (1.3) becauseis the weight 2 complex multiplication form for the elliptic curve with complex multiplication bygiven bywith level.
It turns out that more is true about the relationship between the two series in Equations (1.2) and (1.3). Ifis prime, then we have thatdivides. We will prove this statement in Section 3.1.
To prove Theorem 1, we make use of the known description of Equation (1.1), the generating function for the 3-core partition function, and then generalize the work in  to extend to this situation.
We begin by recalling the exact formula for the coefficients of the modular form , (2.1), defined below. Recall that the Dedekind’s eta function, denoted , is defined by the infinite product
The coefficients are given by
(Lemma 2.5 of ).Assuming the notation above, we have that
The following lemma describes the nonvanishing conditions for the Equation (2.1) as described in .
Assume the notation above. Thenif and only if n is a non-negative integer for whichis odd for some prime .
To prove the original conjecture, Han and Ono recalled the exact formula for the coefficients described in . The modular form , (2.2), is given by
where and , the upper half of the complex plane. This normalized series , such that , is an example of a special type of modular form. This modular form is in , the space of weight 4 cusp forms on . Note that is a newform with complex multiplication. Using this theory, Han and Ono proved the following theorem.
(Theorem 2.1 of ).Assume the notation above. Then the following are true:
Iforis prime, then .
Ifis prime, then
where x and y are integers for whichand .
The theorem above shows that satisfies the same nonvanishing conditions demonstrated by as noted in Lemma 3, proving the original conjecture.
3. Proof of Theorem 1.1
We now briefly recall the theory of newforms with complex multiplication (see Chapter 12 of  or Section 1.2 of ). Let be the fundamental discriminant of an imaginary quadratic field . Let be the ring of integers of K. Let Λ be a nontrivial ideal in and denote the group of fractional ideals prime to Λ. Then ϕ defines a homomorphism
such that for each with , we have
Let be the Dirichlet character defined as
for every integer n coprime to Λ. Consider the function defined by
where the sum is over the integral ideals that are prime to Λ and is the norm of the ideal . This function is a cusp form in . When p does not divide the level, notice that if p is inert in K, then .
The cusp form is a “newform” in the sense of Atkin and Lehner . Therefore, is a normalized cusp form that is an eigenform of all the Hecke operators and all the Atkin–Lehner involutions for primes and . The following theorem describes the vanishing Hecke eigenvalues when there is a prime p such that divides the level.
(Theorem 2.27 (3) of ).Supposeis a newform. If p is a prime for which, then .
This information gives the following nonvanishing conditions on newforms with complex multiplication.
Suppose thatis an even weight newform with trivial nebentypus and complex multiplication bywith level of the formwhere. Thenif and only iforis prime.
Proof of Lemma 6.
The level of is and therefore 3 is the only prime that divides the level. Since , we know always divides the level, therefore by Theorem 5 in , . When for prime, p is inert and therefore . ☐
The following are true about .
If m and n are coprime positive integers, then
For every positive integer t, we have that .
Ifis prime and t is a positive integer, thenif t is odd andif t is even.
If, then .
Proof of Corollary 7.
Claim is well known to hold for all normalized Hecke eigenforms.
Claim follows as .
To prove Claim , observe that every newform is a Hecke eigenform. Moreover, since , the Hecke eigenvalue of is .Therefore, for every integer n and prime , we have that
The left hand side of the equation is the statement that is the Hecke eigenvalue. The right hand side of the equation is the action of the Hecke operator . Let and be prime. Since for , this equation becomes
Claim follows from induction as and .
To prove Claim , let p be a prime such that . Suppose that . This implies that α is totally imaginary, but then
which is false. Claim then follows by induction. ☐
Proof of Theorem 1.
The theorem follows by combining Lemma 3, Corollary 7 and Lemma 6. ☐
3.1. The Series in Equation (1.3)
We normalize the function using the following series,
The series is a modular form given by
In , Martin and Ono gave a complete description of all weight 2 newforms that are products and quotients of the Dedekind eta-function. The descriptions in  include formulas for the coefficients. Since these coefficients are Hecke multiplicative, it suffices to give the formula for only p prime. Specifically, for , we have the following theorem.
(Theorem 2 in ).Assuming the notation above, the following are true.
If, then .
If, thenwhereandand .
Recall 4 from  gave the following conditions on the coefficients of : If is prime, then
where x and y are integers for which and .
Here we show that for primes and n being even:
Let and . Then
Since and , we have and as mentioned in a remark, .
Conflicts of Interest
The authors declare no conflict of interest.
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