The special values of $L$-functions at $s=1$ of theta products of weight 3

TL;DR

This paper computes the special values at s=1 of L-functions associated with theta products of weight 3, expressing them through generalized hypergeometric functions, advancing understanding of their arithmetic properties.

Contribution

It provides explicit formulas for L-values of theta products of weight 3 in terms of hypergeometric functions, a novel connection in the field.

Findings

Explicit formulas for L-values at s=1 for specific theta products

Expression of these values in terms of generalized hypergeometric functions

Enhanced understanding of the arithmetic nature of these L-values

Abstract

In this paper, we compute the special values of $L$-functions at $s=1$ of some theta products of weight $3$, and express them in terms of special values of generalized hypergeometric functions.

Equations

$$L(f,s):=\sum _{n=1}^{\infty }\frac{a_n}{n^s},\mathrm{Re}(s)>k+1.$$

$${}_{p+1}{F}_p\left[\begin{array}{c}{a}_1,a_2,\dots ,a_{p+1}\\ b_1,b_2,\dots ,b_p\end{array}|z\right]:=\sum _{n=0}^{\infty }\frac{(a_1{)}_n\cdots (a_{p+1}{)}_n}{(b_1{)}_n\cdots (b_p{)}_n}\frac{z^n}{(1{)}_n},$$

$$L(f,m)=\frac{(-1{)}^{m-1}}{\Gamma (m)}{\int }_0^{1}f(q)(\log q{)}^{m-1}\frac{dq}{q}.$$

$$L(f,1)={\int }_0^{1}f(q)\frac{dq}{q}.$$

$${\int }_0^1(\text{mbox}polynomialsin{\alpha }^l(1-\alpha {)}^m){}_2{F}_1({\alpha }^n)\frac{d\alpha }{\alpha (1-\alpha )}.$$

$$2f(q)={\theta }_2(q^4){\theta }_3^4(q^4){\theta }_4(q^4)={\theta }_2(q^4){\theta }_4^5(q^4)+{\theta }_2^5(q^4){\theta }_4(q^4).$$

$$L(f,1)=\frac{{\Gamma }^2\left(\frac{1}{4}\right)}{16\sqrt{\pi }}\left({}_3{F}_2\left[\begin{array}{c}\frac{1}{4},\frac{1}{2},\frac{1}{2}\\ \frac{3}{2},1\end{array}|1\right]+{}_3{F}_2\left[\begin{array}{c}\frac{5}{4},\frac{1}{2},\frac{1}{2}\\ \frac{3}{2},1\end{array}|1\right]\right).$$

$$(b-a){}_3{F}_2\left[\begin{array}{c}a,b,c\\ e,f\end{array}|z\right]+a{}_3{F}_2\left[\begin{array}{c}a+1,b,c\\ e,f\end{array}|z\right]=b{}_3{F}_2\left[\begin{array}{c}a,b+1,c\\ e,f\end{array}|z\right],$$

$$L(f,1)=\frac{{\Gamma }^4\left(\frac{1}{4}\right)}{8\sqrt{2}{\pi }^2}.$$

$${\theta }_2(q)$$

$${\theta }_3(q)$$

$${\theta }_4(q)$$

$${\theta }_3^4(q)={\theta }_2^4(q)+{\theta }_4^4(q).$$

$${\theta }_3^2(q)={}_2{F}_1\left[\begin{array}{c}\frac{1}{2},\frac{1}{2}\\ 1\end{array}|\alpha \right],\frac{dq}{q}=\frac{d\alpha }{\alpha (1-\alpha ){{}_2{F}_1}^2\left[\begin{array}{c}\frac{1}{2},\frac{1}{2}\\ 1\end{array}|\alpha \right]}.$$

$$\Gamma \left[\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}\right]:=\frac{\Gamma (a_1)\cdots \Gamma (a_p)}{\Gamma (b_1)\cdots \Gamma (b_q)}.$$

$$L(f,1)=\frac{1}{16}{\int }_0^1{\theta }_2^4(q){\theta }_4^2(q)\frac{dq}{q}=\frac{1}{16}{\int }_0^1{\left(\frac{{\theta }_2(q)}{{\theta }_3(q)}\right)}^4{\left(\frac{{\theta }_4(q)}{{\theta }_3(q)}\right)}^2{\theta }_3^6(q)\frac{dq}{q}.$$

$$\frac{1}{16}{\int }_0^1\alpha (1-\alpha {)}^{1/2}{}_2{F}_1\left[\begin{array}{c}\frac{1}{2},\frac{1}{2}\\ 1\end{array}|\alpha \right]\frac{d\alpha }{\alpha (1-\alpha )}.$$

$${}_{p+1}{F}_p\left[\begin{array}{c}{a}_1,a_2,\dots ,a_{p+1}\\ b_1,b_2,\dots ,b_p\end{array}|z\right]$$

$$=\Gamma \left[\begin{array}{c}{b}_1\\ a_1,b_1-a_1\end{array}\right]{\int }_0^1{t}^{a_1}(1-t{)}^{b_1-a_1}{}_p{F}_{p-1}\left[\begin{array}{c}{a}_2,\dots ,a_{p+1}\\ b_2,\dots ,b_p\end{array}|zt\right]\frac{dt}{t(1-t)},$$

$$L(f,1)=\frac{1}{16}\Gamma \left[\begin{array}{c}1,\frac{1}{2}\\ \frac{3}{2}\end{array}\right]{}_3{F}_2\left[\begin{array}{c}1,\frac{1}{2},\frac{1}{2}\\ \frac{3}{2},1\end{array}|1\right].$$

$${}_2{F}_1\left[\begin{array}{c}a,b\\ c\end{array}|1\right]=\Gamma \left[\begin{array}{c}c,c-a-b\\ c-a,c-b\end{array}\right],$$

$$L(f,1)=\frac{{\Gamma }^2\left(\frac{1}{2}\right)}{16}=\frac{\pi }{16}.$$

$$L(f,1)=\frac{1}{8}{\int }_0^1{\theta }_2(q){\theta }_3^{5/2}(q){\theta }_4^{5/2}(q)\frac{dq}{q}.$$

$$L(f,1)=\frac{1}{8}\Gamma \left[\begin{array}{c}\frac{1}{4},\frac{5}{8}\\ \frac{7}{8}\end{array}\right]{}_3{F}_2\left[\begin{array}{c}\frac{1}{4},\frac{1}{2},\frac{1}{2}\\ \frac{7}{8},1\end{array}|1\right].$$

$${}_3{F}_2\left[\begin{array}{c}a,b,c\\ \frac{1+a+b}{2},2c\end{array}|1\right]=\Gamma \left[\begin{array}{c}\frac{1}{2},\frac{1+2c}{2},\frac{1+a+b}{2},\frac{1-a-b+2c}{2}\\ \frac{1+a}{2},\frac{1+b}{2},\frac{1-a+2c}{2},\frac{1-b+2c}{2}\end{array}\right],$$

$$L(f,1)=\Gamma \left[\begin{array}{c}\frac{1}{4},\frac{4}{8},\frac{1}{2}\\ \frac{7}{8},\frac{3}{4},\frac{3}{4}\end{array}\right].$$

$$\Gamma (n)\Gamma (1-n)=\frac{\pi }{\sin n\pi },$$

$$\Gamma (nz)=\frac{n^{nz-1/2}}{(2\pi {)}^{(n-1)/2}}\prod _{k=0}^{n-1}\Gamma \left(z+\frac{k}{n}\right).$$

$$a(q)$$

$$b(q)$$

$$c(q)$$

$$a^3(q)=b^3(q)+c^3(q),$$

$$a(q)={}_2{F}_1\left[\begin{array}{c}\frac{1}{3},\frac{2}{3}\\ 1\end{array}|\alpha \right],\frac{dq}{q}=\frac{d\alpha }{\alpha (1-\alpha ){{}_2{F}_1}^2\left[\begin{array}{c}\frac{1}{3},\frac{2}{3}\\ 1\end{array}|\alpha \right]}.$$

Full text

1 Introduction and Main Results

For a modular form $f$ of weight $k$ with the $q$-expansion $f(q)=\sum_{n=1}^{\infty}a_{n}q^{n}$, we define its $L$-function $L(f,s)$ by

[TABLE]

It is well-known (cf. [12]) that $L(f,s)$ has meromorphic continuation to $\mathbb{C}$, and satisfies a functional equation under $s\leftrightarrow k-s$. In this paper, we consider $L$-functions of modular forms which are products of the Jacobi theta series $\theta_{2}(q)$, $\theta_{3}(q)$, $\theta_{4}(q)$ (these are modular forms of weight $1/2$) or the Borwein theta series $a(q)$, $b(q)$, $c(q)$ (these are modular forms of weight $1$). For the definitions of these theta series, see section 2 and 3 respectively.

In 2010s, for some modular forms, it was found that special values of $L$-functions can be expressed in terms of special values of generalized hypergeometric functions

[TABLE]

where $(a)_{n}:=\Gamma(a+n)/\Gamma(a)$ denotes the Pochhammer symbol. For example,

Otsubo [8] expressed $L(f,2)$ for some theta products $f(q)$ of weight $2$ in terms of ${{}_{3}F_{2}}(1)$ by using his regulator formula and Bloch’s theorem. 2. 2.

Rogers [9], Rogers-Zudilin [11], Zudilin [14] and the author [7] expressed $L(f,2)$ for some theta products $f(q)$ of weight $2$ in terms of ${{}_{3}F_{2}}(1)$ by an analytic method. Further, Zudilin [14] expressed $L(f,3)$ for the theta product corresponding to the elliptic curve of conductor $32$ in terms of ${{}_{4}F_{3}}(1)$. 3. 3.

Rogers-Wan-Zucker [10] expressed $L(f,2)$ (resp. $L(f,3)$, $L(f,4)$) for some quotients $f(q)$ of the Dedekind eta function $\eta(q):=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})$ of weight $3$ (resp. $4$, $5$) in terms of special values of generalized hypergeometric functions or the gamma function $\Gamma(s)$ by an analytic method.

Note that all of these results are in the case of cusp forms.

In this paper, we compute $L(f,1)$ for theta products (not necessarily cusp forms) of weight $3$, and express them in terms of special values of generalized hypergeometric functions.

Our main results are the following.

Theorem 1**.**

We have the following table.

[TABLE]

[TABLE]

Theorem 2**.**

We have the following table.

[TABLE]

Here we normalized $f(q)=\sum_{n=1}^{\infty}a_{n}q^{n}$ so that $a_{1}=1$.

We remark that, for theta products $f(q)$ of weight $3$ which are considered in [10, Theorem 5], the values $L(f,2)$ (hence $L(f,1)$ by the functional equation) are expressed in terms of special values of the gamma function, not generalized hypergeometric functions. It is new that the values of $L$-functions at $s=1$ of theta products of weight $3$ are expressed in terms of special values of generalized hypergeometric functions.

Our strategy to compute $L(f,1)$ is the same as that used in [7], [9], [10], [11] and [14]. For a modular form $f(q)=\sum_{n=1}^{\infty}a_{n}q^{n}$ and $m\in\mathbb{Z}_{\geqq 1}$, the value $L(f,m)$ is obtained by the Mellin transformation of $f(q)$

[TABLE]

The case $m=1$ is special since the logarithm in the integral above vanishes:

[TABLE]

Since the Jacobi theta series and the Borwein theta series have connections with hypergeometric functions (see (2) and (3) below), we can reduce (1) to an integral of the form

[TABLE]

Then, we obtain a hypergeometric evaluation of $L(f,1)$ after some computation.

Finally, we remark that one of the results of Rogers-Wan-Zucker [10] can be recovered from our results. Let $f(q)=\frac{1}{2}\theta_{2}(q^{4})\theta_{3}^{4}(q^{4})\theta_{4}(q^{4})$. By the Jacobi identity (see section 2), we have

[TABLE]

Therefore, by Theorem 1 (iv), (xiv), we obtain

[TABLE]

If we use the following contiguous relation

[TABLE]

then we have

[TABLE]

This is the second last formula in [10, Theorem 5].

2 Proof of Theorem 1

We define the Jacobi theta series $\theta_{2}(q)$, $\theta_{3}(q)$ and $\theta_{4}(q)$ by

[TABLE]

These are many relations between $\theta_{2}(q)$, $\theta_{3}(q)$ and $\theta_{4}(q)$. One of the most important relations is the Jacobi identity [4, p.35, (2.1.10)]

[TABLE]

Further, the Jacobi theta series have a connection with hypergeometric functions. Let $\alpha:=\theta_{2}^{4}(q)/\theta_{3}^{4}(q)$. Note that we have $1-\alpha=\theta_{4}^{4}(q)/\theta_{3}^{4}(q)$ by the Jacobi identity. Then we have

[TABLE]

The former is [2, p.101, Entry 6], and the latter follows from the former and [1, p.87, Entry 30].

Proof of Theorem 1.

Theorem 1 follows from (2). First, we show the formula (xiii). For simplicity, we denote

[TABLE]

By (1), we have

[TABLE]

If we use (2), the integral above is equal to

[TABLE]

Since generalized hypergeometric functions have the integral representation [13, p.108, (4.1.2)]

[TABLE]

we have

[TABLE]

Note that this ${{}_{3}F_{2}}$ reduces to a ${{}_{2}F_{1}}$. By Gauss’ summation formula [13, p.28, (1.7.6)]

[TABLE]

we obtain

[TABLE]

Next we show the formula (v). Since we have $\theta_{3}(q)\theta_{4}(q)=\theta_{4}^{2}(q^{2})$ [4, p.34, (2.1.7 ii)], we obtain

[TABLE]

If we use (2) and the integral representation of generalized hypergeometric functions, we obtain

[TABLE]

By the Watson summation formula [13, p.54, (2.3.3.13)]

[TABLE]

we have

[TABLE]

Therefore we obtain the formula if we simplify this $\Gamma$-factor using the reflection formula and multiplication formula for $\Gamma(s)$

[TABLE]

Similar computations lead to the remaining formulas. Note that we use the Watson summation formula for the formulas (xi), (xii) and (xv).

∎

3 Proof of Theorem 2

We define the Borwein theta series $a(q)$, $b(q)$ and $c(q)$ [5], [6] by

[TABLE]

where $\omega$ denotes a primitive 3rd root of unity.

There are many relations analogous to those of the Jacobi theta series. For example, J.M. Borwein and P.B. Borwein proved [5]

[TABLE]

which is a cubic analogue of the Jacobi identity. Further, like the Jacobi theta series, the Borwein theta series have a connection with hypergeometric functions. Let $\alpha:=c^{3}(q)/a^{3}(q)$. Note that we have $1-\alpha=b^{3}(q)/a^{3}(q)$ by the cubic identity above. Then we have

[TABLE]

The former is [3, p.97, (2.26)], and the latter follows from the former and [1, p.87, Entry 30]).

Proof of Theorem 2.

Like the cases of Jacobi products, Theorem 2 follows from (3). For example, the formula (i) follows from the following computations.

By (1), we have

[TABLE]

If we use (3), the integral above is equal to

[TABLE]

By the integral representation of ${{}_{3}F_{2}}(z)$, we obtain

[TABLE]

Note that this ${{}_{3}F_{2}}$ reduces to a ${{}_{2}F_{1}}$, hence, by Gauss’ theorem, we have

[TABLE]

If we use the reflection formula to simplify the $\Gamma$-factor, we obtain the formula.

Similar computations lead to the formulas (ii) and (iii).

Next we prove the formula (iv). Since we have $a(q^{3})=(a(q)+2b(q))/3$, $c(q)=(a(q)-b(q))/3$ [6, Lemma 2.1 (iii) and (2.1)], we obtain

[TABLE]

Therefore we have

[TABLE]

If we substitute $\alpha\mapsto 1-\alpha^{3}$ and $\alpha\mapsto\frac{1-u}{1+2u}$, the integral above is equal to

[TABLE]

We have the cubic transformation [3, p.96, Theorem 2.3]

[TABLE]

hence we obtain

[TABLE]

If we integrate term-by-term, we have

[TABLE]

By the multiplication formula, we obtain

[TABLE]

hence we have

[TABLE]

We can simplify the $\Gamma$-factor by using the multiplication formula again, then we obtain the formula.

We can prove the formula (v) by similar computations.

∎

Acknowledgment

This paper is based on the author’s doctor’s thesis at Chiba University. I would like to thank Noriyuki Otsubo for valuable comments.