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Superimposing theta structure on a generalized modular relation

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Abstract

A generalized modular relation of the form \(F(z, w, \alpha )=F(z, iw,\beta )\), where \(\alpha \beta =1\) and \(i=\sqrt{-1}\), is obtained in the course of evaluating an integral involving the Riemann \(\Xi \)-function. This modular relation involves a surprising new generalization of the Hurwitz zeta function \(\zeta (s, a)\), which we denote by \(\zeta _w(s, a)\). We show that \(\zeta _w(s, a)\) satisfies a beautiful theory generalizing that of \(\zeta (s, a)\). In particular, it is shown that for \(0<a<1\) and \(w\in \mathbb {C}\), \(\zeta _w(s, a)\) can be analytically continued to Re\((s)>-1\) except for a simple pole at \(s=1\). The theories of functions reciprocal in a kernel involving a combination of Bessel functions and of a new generalized modified Bessel function \({}_1K_{z,w}(x)\), which are also essential to obtain the generalized modular relation, are developed.

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Notes

  1. See [18, Theorem 1.2] for the corrected versions of equations (19) and (21) of [54].

  2. Note that there is a typo in this formula in that \(\pi \) should not be present.

  3. The interchange of the order of \(\lim _{w\rightarrow 0}\) and integration can be easily justified.

  4. Note that \(\zeta _w(s, a)\) is defined for Re\((s)>1\) by (1.1.14), and for \(-1<\text{ Re }(s)\le 1, s\ne 1\), by (1.1.27). The latter is inferred from the paragraph following (1.1.27).

  5. The first equality in (2.1.4) is actually valid for \(z\in \mathbb {C}\backslash \{-1,1\}\). The condition \(-1<\)Re\((z)<1\) is required only for both equalities to hold. Same is the case with (2.1.2).

  6. This result is, in fact, true for Re\((a)>1\).

  7. We do not give the details of the argument here as a similar one is established in full detail in Lemma 5.1.1.

  8. This formula, which is given for \(b>0\) in [52], can be seen to be true for Re\((b)>0\) by analytic continuation.

  9. This result actually holds for \(\text{ Re }(a)>0\) and \(\text{ Re }(s)>-1, s\ne 1\), by analytic continuation.

  10. According to Remark 7 below, this result is actually valid for \(-1<\mathrm {Re}(s)<2\).

  11. Koshliakov [40] stated them only for \(-\frac{1}{2}<z<\frac{1}{2}\), however, they are easily seen to be true for \(-\frac{1}{2}<\text{ Re }(z)<\frac{1}{2}\).

  12. Throughout the analysis z and w will be fixed complex numbers in some domains of their respective complex planes.

  13. The integrals \(\int _0^\infty x^{-s} \int _0^\infty \psi (t,z, w)\left( \sin (\pi z)J_{2z}(4\sqrt{xt})-\cos (\pi z)L_{2z}(4\sqrt{xt}))\right) {\text {d}}t\,{\text {d}}x\) and \(\int _0^\infty \psi (t,z, w)\) \(\times \int _0^\infty x^{-s}\left( \sin (\pi z)J_{2z}(4\sqrt{xt})-\cos (\pi z)L_{2z}(4\sqrt{xt}))\right) {\text {d}}x\,{\text {d}}t\) are absolutely convergent for \(\frac{3}{4}<\)Re\((s)<1-|\)Re(z)|. So the result follows from Fubini’s theorem.

  14. There is a typo on page 1117 of [25], namely, all instances of |Re(z)|/2 should be replaced by |Re(z)|.

  15. The conditions in the corresponding theorem given in [24], namely Theorem 1.9, are too restrictive. It is actually valid for \(z, w\in \mathbb {C}\), and \(|\arg (x)|<\pi \).

  16. There is typo in the argument of \({}_0F_2\). It should be \(-\frac{a^2y}{4}\) instead of \(-\frac{a^2y}{2}\). Also, \(y^{z-1/2}\) is missing from the integrand.

  17. In [25, Lemma 4.2], the condition given was \(-1<\mathrm {Re}(z)<1\) which is not correct.

  18. Even though this result is true only for \(-\frac{3}{4}<\text{ Re }(z)<\frac{3}{4}\), later while actually using the reciprocal functions in this result, we will replace z by z/2, in which case the result then actually holds for \(-\frac{3}{2}<\text{ Re }(z)<\frac{3}{2}\).

  19. This result also holds for Re\((s)<1\) if \(0<a<1\). See [5, p. 257, Theorem 12.6].

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Acknowledgements

The authors sincerely thank the referee for his/her important suggestions which improved the readability of the paper. They also thank Nico M. Temme for his help, Neer Bhardwaj for providing them a copy of [58], and Richard B. Paris, Don Zagier, M. Lawrence Glasser, Robert Maier, Alexandru Zaharescu and Alexandre Kisselev for interesting discussions. The first author’s research is partially supported by the SERB-DST Grant ECR/2015/000070 and partially by the SERB MATRICS Grant MTR/2018/000251. He sincerely thanks SERB for the support.

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  1. Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat, 382355, India

    Atul Dixit & Rahul Kumar

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Dixit, A., Kumar, R. Superimposing theta structure on a generalized modular relation. Res Math Sci 8, 41 (2021). https://doi.org/10.1007/s40687-021-00277-0

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