Author:
Guodong Hua School of Mathematics and Statistics, Weinan Normal University, Weinan 714099, China
Qindong Mathematical Research Institute, Weinan Normal University, Weinan 714099, China
School of Mathematics, Shandong University, Shandong, Jinan 250100, China

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In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

  • [1]

    Banerjee, S. and Pandey, M. K. Signs of Fourier coefficients of cusp form at sum of two squares. Proc. Indian Acad. Sci. Math. Sci. 130, 1 (2020), Paper No. 2, 9 pp.

    • Search Google Scholar
    • Export Citation
  • [2]

    Bourgain, J. Decoupling, exponential sums and the Riemann zeta function. J. Amer. Math. Soc. 30 (2017), 205224.

  • [3]

    Clozel, L. and Thorne, J. A. Level-raising and symmetric power functoriality. I. Compos. Math. 150 (2014), 729748.

  • [4]

    Clozel, L. and Thorne, J. A. Level-raising and symmetric power functoriality. II. Ann. of Math. 181 (2015), 303359.

  • [5]

    Clozel, L. and Thorne, J. A. Level-raising and symmetric power functoriality. III. Duke Math. J. 166 (2017), 325402.

  • [6]

    Deligne, P. La Conjecture de Weil. I. Inst. Hautes Études Sci. Pull. Math. 43 (1974), 273307.

  • [7]

    Dieulefait, L. Automorphy of symm5(GL(2)) and base change. J. Math. Pures Appl. (9) 104, (4) (2015), 619656.

  • [8]

    Gelbart, S. and Jacqet, H. A relation between automorphic representations of 𝐺L(2) and 𝐺L(3). Ann. Sci. École Norm. Sup. 11, 4 (1978), 471542.

    • Search Google Scholar
    • Export Citation
  • [9]

    Gun, S., Kohnen, W., and Rath, P. Simultaneous sign change of Fourier-coefficients of two cusp forms. Arch. Math. 105 (2015), 413424.

    • Search Google Scholar
    • Export Citation
  • [10]

    Hua, G. D. The simultaneous sign changes of Hecke eigenvalues over an integral binary quadratic form. Acta Math. Hungar. 167, 2 (2022), 476491.

    • Search Google Scholar
    • Export Citation
  • [11]

    A. Ivić, Exponential pairs and the zeta function of Riemann. Stud. Sci. Math. Hungar. 15 (1980), 157181.

  • [12]

    Iwaniec, H. Topics in Classical Automorphic Forms. Grad. Stud. Math., Vol. 17, Amer. Math. Soc., (Providence, 1997).

  • [13]

    Iwaniec, H. and Kowalski, E. Analytic Number Theory. Amer. Math. Soc. Colloquium Publ., Vol. 53, Amer. Math. Soc., (Providence, 2004).

  • [14]

    Jacqet, H. and Shalika, J. A. On the Euler products and the classification of automorphic representations I. Amer. J. Math. 103, 3 (1981), 499558.

    • Search Google Scholar
    • Export Citation
  • [15]

    Jacqet, H. and Shalika, J. A. On the Euler products and the classification of automorphic forms II. Amer. J. Math. 103 (1981), 777815.

    • Search Google Scholar
    • Export Citation
  • [16]

    Kim, H. and Shahidi, F. Functorial products for 𝐺L2 × 𝐺L3 and functorial symmetric cube for GL2, with an appendix by C. J. Bushnell and G. Heniart. Ann. of Math. 155 (2002), 837893.

    • Search Google Scholar
    • Export Citation
  • [17]

    Kim, H. and Shahidi, F. Cuspidality of symmetric power with applications. Duke Math. J. 112 (2002), 177107.

  • [18]

    Kim, H. Functoriality for the exterior square of 𝐺L4 and symmetric fourth of 𝐺L2, Appendix 1 by D. Ramakrishan, Appendix 2 by H. Kim and P. Sarnak. J. Amer. Math. Soc. 16 (2003), 139183.

    • Search Google Scholar
    • Export Citation
  • [19]

    Knopp, M., Kohnen, W., and Pribitkin, W. On the signs of Fourier coefficients of cusp forms. Ramanujan J. 7 (2003), 269277.

  • [20]

    Kohnen, W. and Sengupta, J. Signs of Fourier coefficients of two cusp forms of different weights. Proc. Amer. Math. Soc. 137 (2009), 35633567.

    • Search Google Scholar
    • Export Citation
  • [21]

    Kohnen, W. and Martin, Y. Sign changes of Fourier coefficients of cusp forms supported on prime power indices. Int. J. Number Theory 10, 8 (2014), 19211927.

    • Search Google Scholar
    • Export Citation
  • [22]

    Kumari, M. and Murty, M. R. Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms. Int. J. Number Theory 14, 8 (2018), 22912301.

    • Search Google Scholar
    • Export Citation
  • [23]

    Liu, J. Y. and Ye, Y. B. Perron’s formula and the prime number theorem for automorphic L-functions. Pure Appl. Math. Q. 3, No. 2, 2007, pp. 481497.

    • Search Google Scholar
    • Export Citation
  • [24]

    Lau, Y.-K. and , G. S. Sums of Fourier coefficients of cusp forms. Q. J. Math. 62, 3 (2011), 687716.

  • [25]

    Lao, H. X. and Luo, S. Sign changes and non-vanishing of Fourier coefficients of holomorphic cusp forms. Rocky Mountain J. Math. 51, 5 (2021), 17011714.

    • Search Google Scholar
    • Export Citation
  • [26]

    Liu, H. F. On the asymptotic distribution of Fourier coefficients of cusp forms. Bull. Braz. Math. Soc. (N.S.) 54, 21 (2023), 17 pp.

  • [27]

    Murty, M. R. Oscillations of the Fourier coefficients of modular forms. Math. Ann. 262 (1983), 431446.

  • [28]

    Meher, J., Shankhadhar, K. D., and Viswanadham, G. K. A short note on sign changes. Proc. Indian Acad. Sci. (Math. Sci.) 123, 3 (2013), 315320.

    • Search Google Scholar
    • Export Citation
  • [29]

    Newton, J. and Thorne, J. A. Symmetric power functoriality for holomorphic modular forms. Publ. Math. Inst. Hautes Études Sci. 134 (2021), 1116.

    • Search Google Scholar
    • Export Citation
  • [30]

    Newton, J. and Thorne, J. A. Symmetric power functoriality for holomorphic modular forms. II. Publ. Math. Inst. Hautes Études Sci. 134 (2021), 117152.

    • Search Google Scholar
    • Export Citation
  • [31]

    Perelli, A. General L-functions. Ann. Mat. Pura Appl. 130 (1982), 287306.

  • [32]

    Ramachandra, K. and Sankaranarayanan, A. Notes on the Riemann zeta-function. J. Indian Math. Soc. (N.S.) 57, 1-4 (1991), 6777.

  • [33]

    Rudnick, Z. and Sarnak, P. Zeros of principal L-functions and random matrix theory. Duke Math. J. 81 (1996), 269322.

  • [34]

    Siegel, C. L. Berechnung von Zetafunctionen an ganzzanhligen Stellen. Nachr. Akad. Wiss. Göttingen Math. Phys. K1. II 2 (1969), 87102.

    • Search Google Scholar
    • Export Citation
  • [35]

    Shahidi, F. On certain L-functions. Amer. J. Math. 103 (1981), 297355.

  • [36]

    Shahidi, F. Third symmetric power L-functions for 𝐺L(2). Compos. Math. 70 (1989), 245273.

  • [37]

    Vaishya, L. Signs of Fourier coefficients of cusp forms at integers represented by an integral binary quadratic form. Proc. Indian Acad. Sci. Math. Sci. 131, 2 (2021), Paper No. 41, 14 pp.

    • Search Google Scholar
    • Export Citation
  • [38]

    Venkatasubbareddy, K. and Sankaranarayanan, A. On the tetra, penta, hexa, hepta and octa product L-functions. Eur. J. Math. 9, 1 (2023), Paper No. 17, 24 pp.

    • Search Google Scholar
    • Export Citation
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Mathematica Pannonica
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