Half Lerch Sum in q-Series Theory

• Half Lerch sum is a q-hypergeometric series defined as a one-sided specialization of the Appell–Lerch sum, connecting partition statistics and arithmetic properties.
• It admits equivalent expansions that reveal combinatorial interpretations, where coefficients count representations such as perfect squares and related forms.
• Extensions include restricted-divisor formulas and mock-modular frameworks that link half Lerch sums to Jacobi forms and Ramanujan’s mock theta functions.

Searching arXiv for recent and relevant papers on the half Lerch sum and closely related Appell–Lerch/Lerch-sum literature. The half Lerch sum is a $q$-hypergeometric series closely related to Appell–Lerch sums, partition statistics, and arithmetic questions on coefficient distributions. In a prevalent usage, it denotes

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$

a one-sided specialization of the classical Appell–Lerch sum studied by Xinhua Xiong and later by Tian and Zhou (Xiong, 2016, Tian et al., 7 Sep 2025). Related literature also uses the term for the $v=\tfrac12$ specialization of a two-sided Lerch sum and, in analytic-continuation problems, for the $k=\tfrac12$ case of a partial Lerch transcendent (Bagis, 2018, Sousa, 2021). This suggests that the phrase denotes a family of “half” constructions rather than a single universally standardized object.

1. Definition within the Appell–Lerch framework

Tian and Zhou place the half Lerch sum inside the classical level-$1$ Appell–Lerch sum

$$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$

and state that setting $z=0$ and then taking “one-half” of the two-sided sum produces

$$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$

In this normalization, $\phi(q)$ is the same series that Xiong denotes by $h(q)$ (Tian et al., 7 Sep 2025).

Xiong describes the same object as an interesting $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$0-hypergeometric series coming from a Lerch sum and from rank and crank moments for partitions and overpartitions. He also emphasizes its close kinship to the classical Lerch sum

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$1

with $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$2, which motivates the name “half Lerch sum” (Xiong, 2016).

A basic structural fact is that geometric-series expansion yields

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$3

from which Tian and Zhou deduce in particular that $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$4 for all $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$5 (Tian et al., 7 Sep 2025).

2. Equivalent expansions and combinatorial interpretation

Xiong gives an equivalent expansion

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$6

This immediately turns the coefficient $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$7 into a representation-counting function: $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$8 counts, with small multiplicities, the number of representations of $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$9 as a perfect square, a double square $v=\tfrac12$0, or a value of the form $v=\tfrac12$1 with $v=\tfrac12$2 (Xiong, 2016).

The coefficient formula is made explicit by the identity

$v=\tfrac12$3

In Xiong’s treatment this elementary combinatorial-arithmetic decomposition is the basic device for proving all small-value results (Xiong, 2016).

The same series appears naturally in partition-theoretic generating functions. Xiong records that Andrews–Chan–Kim–Osburn showed

$v=\tfrac12$4

while Larsen–Rust–Swisher obtained the overpartition analogue

$v=\tfrac12$5

In both cases, the pure half-Lerch factor is the central $v=\tfrac12$6-series factor (Xiong, 2016).

Xiong also notes that $v=\tfrac12$7 admits a Hecke-type double-sum reformulation, but with linear dependence on the auxiliary summation variable in the exponent; he therefore distinguishes it from the usual Hecke sums occurring for true Maass forms (Xiong, 2016).

3. Small values, parity, and early conjectures

Xiong formulated Andrews-style conjectures for the value distribution of $v=\tfrac12$8. The central conjecture was that for each positive integer $v=\tfrac12$9, the equation $k=\tfrac12$0 has infinitely many solutions $k=\tfrac12$1, and in particular

$k=\tfrac12$2

Further conjectures proposed positive lower density for the sets $k=\tfrac12$3, $k=\tfrac12$4, and $k=\tfrac12$5, with the strongest qualitative claim being that almost all nonzero values of $k=\tfrac12$6 are $k=\tfrac12$7 (Xiong, 2016).

The paper also proves a substantial list of small-value theorems by elementary factorization arguments derived from the coefficient identity above. The following families are stated explicitly.

Value of $k=\tfrac12$8	Infinite families stated in the literature	Source
$k=\tfrac12$9	$1$0 with $1$1 an odd prime and $1$2 odd; certain square-free products $1$3 of odd primes with $1$4	(Xiong, 2016)
$1$5	$1$6; $1$7 with $1$8 odd prime	(Xiong, 2016)
$1$9	$$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$0 with $$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$1; $$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$2 or $$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$3 with $$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$4 an odd prime	(Xiong, 2016)
$$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$5	$$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$6 with odd primes $$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$7	(Xiong, 2016)

Xiong additionally proves the parity criterion

$$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$8

A stated consequence is that $$A(z;\tau)=w^{1/2}\sum_{n\in\mathbb Z}\frac{(-1)^n q^{\binom{n+1}{2}}}{1-wq^n}, \qquad w=e^{2\pi i z},\quad q=e^{2\pi i\tau},$$9 is even for almost all $z=0$0 (Xiong, 2016).

For the counting functions of zeros, ones, and twos, Xiong derives square-root scale upper and lower bounds. For example, the number of $z=0$1 with $z=0$2 is bounded between $z=0$3 and $z=0$4, while the counts for $z=0$5 and $z=0$6 are also controlled between lower bounds of order $z=0$7 and upper bounds of order $z=0$8 (Xiong, 2016).

4. Restricted divisor formula and higher-level generalization

Tian and Zhou extend the level-$z=0$9 half Lerch sum to a family of one-sided level-$$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$0 Appell-type series. For integers $$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$1 and odd $$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$2, they define coefficients $$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$3 by

$$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$4

so that $$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$5 recovers the original half Lerch sum (Tian et al., 7 Sep 2025).

Their main theorem gives an exact restricted-divisor description. They prove

$$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$6

where

$$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$7

This converts the coefficients into counts of divisors lying in a short interval centered at the $$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$8-scale. In particular, it yields $$\phi(q):=\sum_{n\ge 0} h(n)\,q^n =\sum_{n\ge 1}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n}.$$9 for all $\phi(q)$0, and for $\phi(q)$1 it gives a complete divisor-sum description of the coefficients $\phi(q)$2 (Tian et al., 7 Sep 2025).

The restricted-divisor formula resolves Xiong’s principal conjecture. Tian and Zhou show that for each fixed $\phi(q)$3, there are infinitely many $\phi(q)$4 with $\phi(q)$5; hence

$\phi(q)$6

Specializing again to $\phi(q)$7, this proves that the original half Lerch sum takes arbitrarily large values infinitely often (Tian et al., 7 Sep 2025).

They also establish an asymptotic density statement for zeros: $\phi(q)$8 where

$\phi(q)$9

Thus almost all coefficients vanish. This later result is significant for interpreting the earlier conjectural picture: Xiong’s 2016 summary proposed positive lower density for several small values, whereas Tian and Zhou’s 2025 theorem shows density one of zeros (Xiong, 2016, Tian et al., 7 Sep 2025).

The proof strategy combines series manipulations, explicit prime-factor constructions such as $h(q)$0, $h(q)$1 for large primes $h(q)$2, and a two-parameter construction $h(q)$3 for carefully chosen primes $h(q)$4. For the zero-density theorem, they invoke Ford’s theorem on integers having a divisor in a short interval (Tian et al., 7 Sep 2025).

5. Mock-modular and Jacobi-form aspects

A second major usage of the term arises in the analytic theory of Appell–Lerch sums. Bagis studies the normalized Lerch sum

$h(q)$5

and then specializes to the “half” case by taking $h(q)$6, equivalently $h(q)$7. In this normalization the half-Lerch sum becomes

$h(q)$8

up to the stated theta prefactor (Bagis, 2018).

Bagis proves that Lerch sums of this type admit integral representations involving Jacobi theta-functions and a secant kernel. For the half specialization, the paper records both secant-kernel formulas and the integral

$h(q)$9

This places the half-Lerch sum in the analytic framework of Appell–Lerch integrals and makes explicit the theta-kernel structure behind the series (Bagis, 2018).

Bagis also states the mock-modular interpretation: $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$00 is the holomorphic part of a weight-$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$01 Maass–Jacobi form whose shadow is proportional to a unary theta series. Writing

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$02

with $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$03 the standard real-analytic correction, the completed object satisfies the classical Jacobi-type modular transformation

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$04

after the nonholomorphic correction is included (Bagis, 2018).

This framework also connects the half-Lerch sum to Ramanujan’s mock theta functions. Bagis states that Ramanujan’s third-order mock theta function $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$05 is recovered, up to a simple $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$06-power and normalization, by an appropriate specialization of $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$07. He further gives explicit Fourier-coefficient evaluations for related half-sums and describes the completion

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$08

as a genuine Jacobi form (Bagis, 2018).

6. Terminological extensions and adjacent usages

The phrase “half Lerch” is also used in a distinct analytic-continuation setting. Guillera and Sondow study the partial Lerch sum

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$09

and specialize to $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$10. In that context, the “half-Lerch” case is

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$11

for which they derive an explicit representation involving edge terms, $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$12, and two convergent integrals. They also show that for $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$13, the partial sums converge to the full Lerch transcendent with an exponentially small error term in $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$14 (Sousa, 2021).

A separate, historically unrelated usage appears in the theory of Fermat quotients. Dobson discusses Lerch’s sums

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$15

for an odd prime $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$16, and then derives “short sums” or “half-sums” from the complementarity relation

$$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$17

These half-sum identities concern congruences for Fermat quotients and block decompositions of harmonic sums; they are not Appell–Lerch $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$18-series (Dobson, 2011).

This terminological spread is a common source of confusion. In the $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$19-series and mock-modular literature, “half Lerch sum” usually refers to the one-sided series $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$20 or to the $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$21 specialization $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$22. In analytic continuation, it may refer instead to the half-order case $$h(q)=\sum_{n=1}^{\infty} h(n)\,q^n =\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{\binom{n+1}{2}}}{1-q^n},$$23 of a partial Lerch transcendent, while in arithmetic congruence theory “half-sum” describes a shortening of Lerch’s harmonic sums rather than a Lerch-type generating function (Bagis, 2018, Sousa, 2021, Dobson, 2011).