Riemann-Xi integral for the second error-function transformation, exact evaluations of its confluent-hypergeometric integrals, and the Delta/Nabla parity reversal (open questions of Dixit-Roy-Zaharescu)

Dixit, Roy and Zaharescu (arXiv:1605.08904; Acta Arith. 177 (2017) 1-37) proved two complementary modular-type transformations for an integral analogue of the partial theta function, $\int_{0}^{\infty}\frac{e^{-\pi\alpha^{2}x^{2}}\sin(\sqrt{\pi}\alpha xz)}{e^{2\pi x}-1}\,dx$, involving the error function $\mathrm{erf}(z)$ and the imaginary error function $\mathrm{erfi}(z)$, whose difference recovers Ramanujan's transformation for an integral analogue of theta functions. Their concluding section poses a coherent bundle of open questions as a single problem: (A) find, or prove the nonexistence of, a single integral involving the Riemann $\Xi$-function that equals both sides of the second error-function transformation, analogous to the known $\Xi$-integral for the first; (B) two exact evaluation problems for $\hat{J}_{k}(\alpha)=\int_{0}^{\infty}\frac{xe^{-\alpha x^{2}}}{e^{2\pi x}-1}{}_{1}F_{1}\!\left(-k;\tfrac{3}{2};2\alpha x^{2}\right)\,dx$ — Question 1 (even $k$, at the self-dual point $\alpha=\pi$) and Question 2 (odd $k$, $\alpha\neq\pi$); and (C) an intrinsic explanation for the $\Delta/\nabla$ parity reversal in the character analogues. Paris (2018) gives a partial answer to (B), expressing $\hat{J}_{k}$ as a Gauss ${}_{2}F_{1}$ term plus a gamma ratio plus a remainder integral with an exponentially small bound in $k$ and noting that closed forms for the remainders are doubtful; (A), (C) and the exact evaluations in (B) remain open.

Background

A Mordell integral is a definite integral combining a Gaussian with the Bose kernel $1/(e^{2\pi x}-1)$, of the type $h(z;\tau)=\int_{\mathbb{R}}\frac{e^{\pi i\tau x^{2}-2\pi zx}}{e^{2\pi x}-1}\,dx$ for $\mathrm{Im}\,\tau>0$; these integrals pervade Ramanujan's Lost Notebook and play a central role in Zwegers' thesis on mock theta functions (Ramanujan's q-series that acquire modular transformations only after adjoining non-holomorphic integral corrections of Mordell type). On page 198 of the Lost Notebook Ramanujan studied $F_{\omega}(z)=\int_{0}^{\infty}\frac{\sin(\pi xz)}{\tanh(\pi x)}e^{-\pi\omega x^{2}}\,dx$, which satisfies the transformation $F_{\omega}(z)=\frac{-i}{\sqrt{\omega}}e^{-\pi z^{2}/(4\omega)}F_{1/\omega}\!\left(\frac{iz}{\omega}\right)$, proved (with the other claimed properties) by Berndt and Xu, who accordingly called $F_{\omega}$ an integral analogue of a theta function: theta satisfies Jacobi's modular transformation, and $F_{\omega}$ satisfies an integral analogue of it. In the same spirit, Dixit-Roy-Zaharescu (DRZ) call the half-line integral $\int_{0}^{\infty}\frac{e^{-\pi\alpha^{2}x^{2}}\sin(\sqrt{\pi}\alpha xz)}{e^{2\pi x}-1}\,dx$ an integral analogue of a partial theta function, since the partial theta function sums the theta series over only half of the integer lattice while this integral runs over only half of the line. Here $\mathrm{erf}(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\,dt$ is the error function and $\mathrm{erfi}(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\,dt$ its imaginary analogue; ${}_{1}F_{1}(a;c;z)$ is Kummer's confluent hypergeometric function; $\xi(s)=\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is the Riemann $\xi$-function ($\zeta$ the Riemann zeta function, $\Gamma$ the gamma function) and $\Xi(t)=\xi(1/2+it)$.

DRZ proved two complementary transformations (their Theorems 1.1 and 1.2), both of the modular form $G(z,\alpha)=G(iz,\beta)$ for $\alpha,\beta>0$, $\alpha\beta=1$. The first equates $\sqrt{\alpha}e^{z^{2}/8}\left(\mathrm{erf}(z/2)-4\int_{0}^{\infty}\frac{e^{-\pi\alpha^{2}x^{2}}\sin(\sqrt{\pi}\alpha xz)}{e^{2\pi x}-1}\,dx\right)$ with its $\beta$-partner and, crucially, with a third member: a single integral $\frac{z}{8\pi^{2}}\int_{0}^{\infty}\Gamma\!\left(\frac{-1+it}{4}\right)\Gamma\!\left(\frac{-1-it}{4}\right)\Xi\!\left(\frac{t}{2}\right)\Delta\!\left(\alpha,z,\frac{1+it}{2}\right)\,dt$, where $\Delta(x,z,s)=\omega(x,z,s)+\omega(x,z,1-s)$ with $\omega(x,z,s)=x^{1/2-s}e^{-z^{2}/8}{}_{1}F_{1}\!\left(1-\tfrac{s}{2};\tfrac{3}{2};\tfrac{z^{2}}{4}\right)$. The second transformation (their Theorem 1.2) is the complementary identity with the integrals taken over $(-\infty,0]$ instead of $[0,\infty)$ and a plus sign; subtracting the two transformations recovers Ramanujan's transformation for the integral analogue of theta functions. The analogous theta-function identity admits a $\Xi$-integral representation whose $z=0$ case is the classical formula Hardy used in 1914 to prove that infinitely many zeros of $\zeta(s)$ lie on the critical line $\mathrm{Re}\,s=1/2$; DRZ therefore argue that a $\Xi$-integral equal to both sides of the second transformation would be a first step toward an integral analogue of Hardy's formula and hence toward new information on the non-trivial zeros of $\zeta(s)$. Writing the transformations in an alternate form, DRZ also obtained, for non-negative integers $k$, the exact evaluation $\int_{0}^{\infty}\frac{xe^{-\pi x^{2}}}{e^{2\pi x}-1}{}_{1}F_{1}\!\left(-2k-1;\tfrac{3}{2};2\pi x^{2}\right)\,dx=-\tfrac{1}{4\pi}{}_{2}F_{1}\!\left(-2k-1,1;\tfrac{3}{2};2\right)$ (their Corollary 1.8, with ${}_{2}F_{1}$ the Gauss hypergeometric function), valid precisely at the self-dual point $\alpha=\pi$, together with an approximation formula valid for all $\alpha>0$ but exact only in limiting regimes — a trade-off that motivates their Questions 1 and 2 below. Finally, for a real primitive Dirichlet character $\chi$ modulo $q$ (a completely multiplicative function on the integers inducing a multiplicative map on residues coprime to $q$; $\chi$ is called even if $\chi(-1)=1$ and odd if $\chi(-1)=-1$, and its Gauss sum is $G(\chi)=\sum_{m=1}^{q}\chi(m)e^{2\pi im/q}$), with the character analogue $\Xi(t,\chi)$ of $\Xi(t)$ built from the Dirichlet $L$-function (the series $L(s,\chi)=\sum_{n=1}^{\infty}\chi(n)n^{-s}$), DRZ proved character analogues of their first transformation in which the $\Xi(t,\chi)$-integral involves the kernel $\Delta$ when $\chi$ is even and the kernel $\nabla(x,z,s)=\rho(x,z,s)+\rho(x,z,1-s)$, $\rho(x,z,s)=x^{1/2-s}e^{-z^{2}/8}{}_{1}F_{1}\!\left(\tfrac{1-s}{2};\tfrac{1}{2};\tfrac{z^{2}}{4}\right)$, when $\chi$ is odd — exactly the opposite assignment to the one in the same authors' earlier Riesz-type-criteria paper, a reversal they arrive at only by computation.

Problem Statement

Resolve the open questions posed in the concluding section (Section 7) of Dixit-Roy-Zaharescu 'Error functions, Mordell integrals and an integral analogue of partial theta function' (arXiv:1605.08904; Acta Arith. 177 (2017) 1-37). The three questions jointly complete the analytic theory of the second error-function transformation and the hypergeometric integrals it generates. The three parts are independent sub-problems: each is a self-contained open question, and a complete answer to any one of them — including a proof of nonexistence for (A) — resolves that part without requiring the others. (The same concluding section also contains a separate remark, outside the scope of the present problem, on generalizing a class-number identity of Mordell for $\Lambda(w)=\sum_{n\geq1}F(n)e^{i\pi nw}$, where $F(D)$ counts classes of positive definite binary quadratic forms of determinant $-D$.)

(A) Find an explicit single integral involving the Riemann $\Xi$-function that is equal to both sides of the second error-function transformation $\sqrt{\alpha}e^{z^{2}/8}\left(\mathrm{erf}(z/2)+4\int_{-\infty}^{0}\frac{e^{-\pi\alpha^{2}x^{2}}\sin(\sqrt{\pi}\alpha xz)}{e^{2\pi x}-1}\,dx\right)=\sqrt{\beta}e^{-z^{2}/8}\left(\mathrm{erfi}(z/2)+4\int_{-\infty}^{0}\frac{e^{-\pi\beta^{2}x^{2}}\sinh(\sqrt{\pi}\beta xz)}{e^{2\pi x}-1}\,dx\right)$ ($\alpha\beta=1$), analogous to the $\Xi$-integral known for the first transformation; or prove that no such integral exists. (The authors document why reversing the proof of the first transformation fails: the error-function terms do not cancel against $\frac{2}{\pi}\int_{0}^{\infty}\frac{e^{-\pi\alpha^{2}x^{2}+2\pi x}\sin(\sqrt{\pi}\alpha xz)}{x}\,dx$, and the Mellin transform of $e^{-ax^{2}-cx}\sin(bx)$ involves parabolic cylinder functions rather than a single ${}_{1}F_{1}$, so genuinely new representations are required.)

(B) For $\hat{J}_{k}(\alpha)=\int_{0}^{\infty}\frac{xe^{-\alpha x^{2}}}{e^{2\pi x}-1}{}_{1}F_{1}\!\left(-k;\tfrac{3}{2};2\alpha x^{2}\right)\,dx$, evaluate exactly:

$$\int_{0}^{\infty}\frac{xe^{-\pi x^{2}}}{e^{2\pi x}-1}{}_{1}F_{1}\!\left(-2k;\tfrac{3}{2};2\pi x^{2}\right)\,dx$$

for $k\in\mathbb{Z}^{+}\cup\{0\}$ (Question 1 of the source — the even-index case at the self-dual point $\alpha=\pi$, where the known modular relation degenerates and the odd-index integral is exactly evaluated by Corollary 1.8); and

$$\int_{0}^{\infty}\frac{xe^{-\alpha x^{2}}}{e^{2\pi x}-1}{}_{1}F_{1}\!\left(-2k-1;\tfrac{3}{2};2\alpha x^{2}\right)\,dx$$

for $\alpha\neq\pi$ a positive real and $k\in\mathbb{Z}^{+}\cup\{0\}$ (Question 2 of the source — the odd-index case away from $\alpha=\pi$), for which an exact evaluation is asked; the bounded-remainder approximation of Paris (2018) does not count as a resolution.

(C) Explain intrinsically, beyond the computational manipulation employed, why in the character analogue of the first transformation the $\Xi(t,\chi)$-integral involves the $\Delta$ kernel when $\chi$ is even and the $\nabla$ kernel when $\chi$ is odd, which is exactly the opposite of the assignment in Theorems 1.3-1.5 of the same authors' Riesz-type-criteria paper (J. Number Theory 160 (2016) 385-408). An accepted answer identifies a structural property of the character analogue that forces this kernel assignment, rather than re-deriving the identity by evaluating the character sums.

References

1. Atul Dixit, Arindam Roy, Alexandru Zaharescu, 'Error functions, Mordell integrals and an integral analogue of a partial theta function', arXiv:1605.08904 (2016); published as Acta Arithmetica 177(1) (2017) 1-37, DOI 10.4064/aa8207-5-2016, https://doi.org/10.4064/aa8207-5-2016
2. R. B. Paris, 'A note on an integral of Dixit, Roy and Zaharescu', arXiv:1804.07527 (2018), https://arxiv.org/abs/1804.07527 (partial answer to Questions 1-2: Gauss-hypergeometric-plus-gamma representation with remainder integral and exponentially small bound in k)
3. Atul Dixit, Arindam Roy, Alexandru Zaharescu, 'Riesz-type criteria and theta transformation analogues', Journal of Number Theory 160 (2016) 385-408, DOI 10.1016/j.jnt.2015.08.005, https://doi.org/10.1016/j.jnt.2015.08.005 (the earlier character-analogue paper whose Delta/Nabla assignment is reversed)
4. B. C. Berndt, P. Xu, 'An integral analogue of theta functions and Gauss sums in Ramanujan's Lost Notebook', Mathematical Proceedings of the Cambridge Philosophical Society 147 (2009) 257-265, DOI 10.1017/s0305004109002552, https://doi.org/10.1017/s0305004109002552
5. G. H. Hardy, 'Sur les zeros de la fonction zeta(s) de Riemann', Comptes Rendus de l'Academie des Sciences, Paris 158 (1914) 1012-1014, zbMATH 2617076, https://zbmath.org/2617076 (the note whose z=0 theta-identity equality Hardy used to prove infinitely many zeta zeros on the critical line)
6. Atul Dixit, 'Modular type transformations and integrals involving the Riemann Xi-function', The Mathematics Student 87(3-4) (2018) 47-59, zbMATH 7844157, https://zbmath.org/7844157 (survey)
7. Atul Dixit, Rahul Kumar, 'Superimposing theta structure on a generalized modular relation', arXiv:2005.08316 (2020); published in Research in the Mathematical Sciences 8(3) (2021), article 41, DOI 10.1007/s40687-021-00277-0, https://doi.org/10.1007/s40687-021-00277-0 (builds on DRZ Theorem 1.1, the first transformation)
8. Atul Dixit, Rajat Gupta, 'Koshliakov zeta functions I: Modular relations', arXiv:2108.00810 (2021); published in Advances in Mathematics 393 (2021) 108093, DOI 10.1016/j.aim.2021.108093, https://doi.org/10.1016/j.aim.2021.108093 (modular relations involving integrals containing Xi)
9. Atul Dixit, Rahul Kumar, 'A modular relation involving a generalized digamma function and asymptotics of some integrals containing Xi(t)', arXiv:2211.08763 (2022); published in The Hardy-Ramanujan Journal 45 (2022), DOI 10.46298/hrj.2023.10913, https://doi.org/10.46298/hrj.2023.10913 (generalizes DRZ Theorem 1.10)
10. Atul Dixit, Sumukha Sathyanarayana, N. Guru Sharan, 'Modular relations involving generalized digamma functions', arXiv:2306.10991 (2023); published in Journal of Mathematical Analysis and Applications 539(1) (2024) 128479, DOI 10.1016/j.jmaa.2024.128479, https://doi.org/10.1016/j.jmaa.2024.128479 (modular relations involving a generalized digamma function)
11. Cormac O'Sullivan, 'A generalization of the Riemann-Siegel formula', arXiv:1811.01130 (2018); published in Mathematische Zeitschrift 303(1) (2022), article 20, DOI 10.1007/s00209-022-03164-8, https://doi.org/10.1007/s00209-022-03164-8 (Mordell-integral background)
12. Dandan Chen, Rong Chen, Sander Zwegers, 'A generalization of the Mordell integral', arXiv:2511.00493 (2025); published in Proceedings of the American Mathematical Society 154(1) (2026) 1-10, DOI 10.1090/proc/16686, https://doi.org/10.1090/proc/16686 (a generalization of the Mordell integral)
13. Hari Mohan Srivastava, Mohd Idris Qureshi, Mohd Shahid Baboo, Bhawna Gupta, 'Evaluations of Some General Classes of Mordell Integrals by Applying the Mellin-Barnes-Type Contour Integration', Axioms 14(8) (2025) 633, DOI 10.3390/axioms14080633, https://doi.org/10.3390/axioms14080633 (evaluations of a class of Mordell integrals)