---
title: Half Lerch Sum in q-Series Theory
url: https://www.emergentmind.com/topics/half-lerch-sum
type: topic
---

# Half Lerch Sum in q-Series Theory

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 [1605.09508; 2509.06032]. 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 [1808.07970; 2101.06835]. 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 \(h(q)\) [2509.06032].

Xiong describes the same object as an interesting \(q\)-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
\[
L(u,v;\tau)=\sum_{n\in\mathbb Z}\frac{(-1)^n\,e^{\pi i n(n+1)\tau+2\pi i n u}}{1-e^{2\pi i n\tau+2\pi i v}},
\]
with \(q=e^{2\pi i\tau}\), which motivates the name “half Lerch sum” [1605.09508].

A basic structural fact is that geometric-series expansion yields
\[
\phi(q)=\sum_{j\ge 1}\;\sum_{0\le r<j} q^{\,j(j+r)}(1+q^j),
\]
from which Tian and Zhou deduce in particular that \(h(n)\ge 0\) for all \(n\) [2509.06032].

## 2. Equivalent expansions and combinatorial interpretation

Xiong gives an equivalent expansion
\[
h(q)=\sum_{n=1}^{\infty}\sum_{j=1}^{n-1}2\,q^{\,n(n+j)}
+\sum_{n=1}^{\infty}\bigl(q^{n^2}+q^{2n^2}\bigr).
\]
This immediately turns the coefficient \(h(n)\) into a representation-counting function: \(h(n)\) counts, with small multiplicities, the number of representations of \(n\) as a perfect square, a double square \(2m^2\), or a value of the form \(x(x+y)\) with \(1\le y\le x-1\) [1605.09508].

The coefficient formula is made explicit by the identity
\[
h(n)
=
2\#\{(x,y): n=x(x+y),\;1\le y\le x-1\}
+\#\{x: x^2=n\}
+\#\{x:2x^2=n\}.
\]
In Xiong’s treatment this elementary combinatorial-arithmetic decomposition is the basic device for proving all small-value results [1605.09508].

The same series appears naturally in partition-theoretic generating functions. Xiong records that Andrews–Chan–Kim–Osburn showed
\[
\sum_{n=0}^{\infty} M_1^+(n)\,q^n
=
\sum_{n=1}^{\infty}
\frac{(-1)^{n+1}q^{\frac{n(n+1)}2}}{(1-q^n)(q)_\infty},
\]
while Larsen–Rust–Swisher obtained the overpartition analogue
\[
\sum_{n=0}^{\infty}\overline M_1^+(n)\,q^n
=
\frac{(-q)_\infty}{(q)_\infty}
\sum_{n=1}^{\infty}
\frac{(-1)^{n+1}q^{\frac{n(n+1)}2}}{1-q^n}.
\]
In both cases, the pure half-Lerch factor is the central \(q\)-series factor [1605.09508].

Xiong also notes that \(h(q)\) 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 [1605.09508].

## 3. Small values, parity, and early conjectures

Xiong formulated Andrews-style conjectures for the value distribution of \(h(n)\). The central conjecture was that for each positive integer \(k\), the equation \(h(n)=k\) has infinitely many solutions \(n\), and in particular
\[
\limsup_{n\to\infty} h(n)=+\infty.
\]
Further conjectures proposed positive lower density for the sets \(\{n:h(n)=0\}\), \(\{n:h(n)=1\}\), and \(\{n:h(n)=2\}\), with the strongest qualitative claim being that almost all nonzero values of \(h(n)\) are \(2\) [1605.09508].

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 \(h(n)\) | Infinite families stated in the literature | Source |
|---|---|---|
| \(0\) | \(n=p^d\) with \(p\) an odd prime and \(d\) odd; certain square-free products \(p_1p_2\cdots p_k\) of odd primes with \(p_k>2p_1p_2\cdots p_{k-1}\) | [1605.09508] |
| \(1\) | \(n=2^k\); \(n=p^{2k}\) with \(p\) odd prime | [1605.09508] |
| \(2\) | \(n=p_1p_2\) with \(p_1<p_2<2p_1\); \(n=p(p+1)\) or \(n=(p-1)p\) with \(p\) an odd prime | [1605.09508] |
| \(3\) | \(n=p_1p_2\) with odd primes \(p_1<p_2<\sqrt2\,p_1\) | [1605.09508] |

Xiong additionally proves the parity criterion
\[
h(n)\equiv 1 \pmod 2
\quad\Longleftrightarrow\quad
n \text{ is a perfect square or twice a perfect square}.
\]
A stated consequence is that \(h(n)\) is even for almost all \(n\) [1605.09508].

For the counting functions of zeros, ones, and twos, Xiong derives square-root scale upper and lower bounds. For example, the number of \(n\le X\) with \(h(n)=0\) is bounded between \((2-\sqrt2)\sqrt X-1\) and \((4-2\sqrt2)\sqrt X\), while the counts for \(h(n)=1\) and \(h(n)=2\) are also controlled between lower bounds of order \(\sqrt X\log X\) and upper bounds of order \(\sqrt X\) [1605.09508].

## 4. Restricted divisor formula and higher-level generalization

Tian and Zhou extend the level-\(1\) half Lerch sum to a family of one-sided level-\(k\) Appell-type series. For integers \(m\ge 0\) and odd \(k\ge 1\), they define coefficients \(h_{m,k}(n)\) by
\[
\sum_{n\ge 0} h_{m,k}(n)\,q^n
=
\sum_{n\ge 1}\frac{(-1)^{n-1}q^{\,k\binom n2 + mn}}{1-q^n},
\]
so that \((m,k)=(1,1)\) recovers the original half Lerch sum [2509.06032].

Their main theorem gives an exact restricted-divisor description. They prove
\[
h_{m,k}(n)
=
\#\bigl\{d\mid n:\; u_{m,k}(n)<2kd\le 2u_{m,k}(n)\bigr\},
\]
where
\[
u_{m,k}(n)
=
\sqrt{\,2kn+\left(m-\tfrac k2\right)^2\,}
-\left(m-\tfrac k2\right).
\]
This converts the coefficients into counts of divisors lying in a short interval centered at the \(\sqrt n\)-scale. In particular, it yields \(h_{m,k}(n)\ge 0\) for all \(n\), and for \((m,k)=(1,1)\) it gives a complete divisor-sum description of the coefficients \(h(n)\) [2509.06032].

The restricted-divisor formula resolves Xiong’s principal conjecture. Tian and Zhou show that for each fixed \(\ell\ge 0\), there are infinitely many \(n\) with \(h_{m,k}(n)=\ell\); hence
\[
\limsup_{n\to\infty} h_{m,k}(n)=+\infty.
\]
Specializing again to \((m,k)=(1,1)\), this proves that the original half Lerch sum takes arbitrarily large values infinitely often [2509.06032].

They also establish an asymptotic density statement for zeros:
\[
\#\{n\le x: h_{m,k}(n)=0\}
=
x+O\!\bigl(x(\log x)^{-\delta}(\log\log x)^{-3/2}\bigr),
\]
where
\[
\delta=1-\frac{1+\log\log 2}{\log 2}\approx 0.08607.
\]
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 [1605.09508; 2509.06032].

The proof strategy combines series manipulations, explicit prime-factor constructions such as \(h_{m,k}(kp)=0\), \(h_{m,k}(kp^2)=1\) for large primes \(p\), and a two-parameter construction \(h_{m,k}(k\,p^t q^t)=t+1\) for carefully chosen primes \(p<q\). For the zero-density theorem, they invoke Ford’s theorem on integers having a divisor in a short interval [2509.06032].

## 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
\[
p(u,v;\tau)
:=
\frac{a^{1/2}}{\vartheta(v;\tau)}
\sum_{n\in\mathbb Z}
\frac{(-b)^n q^{\,n(n+1)/2}}{1-aq^n},
\qquad
q=e^{2\pi i\tau},\quad a=e^{2\pi iu},\quad b=e^{2\pi iv},
\]
and then specializes to the “half” case by taking \(v=\tfrac12\), equivalently \(b=-1\). In this normalization the half-Lerch sum becomes
\[
H(z,\tau)
:=
\sum_{n\in\mathbb Z}
\frac{(-1)^n q^{\,n(n+1)/2}}{1-e^{2\pi i z}q^n}
=
p\!\left(z,\tfrac12;\tau\right)
\]
up to the stated theta prefactor [1808.07970].

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
\[
H(z,\tau)
=
\int_{-\infty}^{\infty}
\frac{e^{-\pi \tau t^2+2\pi i t(z-\tau/2)}}{\cosh(\pi t)}\,dt.
\]
This places the half-Lerch sum in the analytic framework of Appell–Lerch integrals and makes explicit the theta-kernel structure behind the series [1808.07970].

Bagis also states the mock-modular interpretation: \(H(z,\tau)\) is the holomorphic part of a weight-\(\tfrac12\) Maass–Jacobi form whose shadow is proportional to a unary theta series. Writing
\[
H(z,\tau)=H^+(z,\tau)+\tfrac12 R(z,\tau),
\]
with \(R(z,\tau)\) the standard real-analytic correction, the completed object satisfies the classical Jacobi-type modular transformation
\[
H\!\left(\frac{z}{\tau},-\frac1\tau\right)
=
(-i\tau)^{1/2}e^{\pi i z^2/\tau}H(z,\tau)
\]
after the nonholomorphic correction is included [1808.07970].

This framework also connects the half-Lerch sum to Ramanujan’s mock theta functions. Bagis states that Ramanujan’s third-order mock theta function \(f(q)\) is recovered, up to a simple \(q\)-power and normalization, by an appropriate specialization of \(z\). He further gives explicit Fourier-coefficient evaluations for related half-sums and describes the completion
\[
\widehat H(z,\tau)
=
H(z,\tau)
+\frac{i}{2}\int_{-\overline z}^{\,i\infty}
\frac{\vartheta_4(w,\tau)}{(-i(\tau+w))^{1/2}}\,dw
\]
as a genuine Jacobi form [1808.07970].

## 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
\[
S_N(k,b;z)=\sum_{n=0}^{N-1}\frac{e^{nz}}{(n+b)^k},
\]
and specialize to \(k=\tfrac12\). In that context, the “half-Lerch” case is
\[
S_N\!\left(\tfrac12,b;z\right)
=
\sum_{n=0}^{N-1}\frac{e^{nz}}{(n+b)^{1/2}},
\]
for which they derive an explicit representation involving edge terms, \(\operatorname{erfc}\), and two convergent integrals. They also show that for \(\Re(z)<0\), the partial sums converge to the full Lerch transcendent with an exponentially small error term in \(N\) [2101.06835].

A separate, historically unrelated usage appears in the theory of Fermat quotients. Dobson discusses Lerch’s sums
\[
s(k,N):=\sum_{j=\lfloor kp/N\rfloor+1}^{\lfloor (k+1)p/N\rfloor}\frac1j
\]
for an odd prime \(p\), and then derives “short sums” or “half-sums” from the complementarity relation
\[
s(N-1-k,N)=-s(k,N).
\]
These half-sum identities concern congruences for Fermat quotients and block decompositions of harmonic sums; they are not Appell–Lerch \(q\)-series [1103.3907].

This terminological spread is a common source of confusion. In the \(q\)-series and mock-modular literature, “half Lerch sum” usually refers to the one-sided series \(h(q)\) or to the \(v=\tfrac12\) specialization \(H(z,\tau)\). In analytic continuation, it may refer instead to the half-order case \(k=\tfrac12\) 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 [1808.07970; 2101.06835; 1103.3907].

Source: https://www.emergentmind.com/topics/half-lerch-sum