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Summation formulae of multiplicative functions over arithmetic progressions and applications
Published online by Cambridge University Press: 12 April 2024
Abstract
In this paper, we investigate the asymptotic distribution of a class of multiplicative functions over arithmetic progressions without the Ramanujan conjecture. We also apply these results to some interesting arithmetic functions in automorphic context, such as coefficients of automorphic L-functions, coefficients of their Rankin–Selberg.
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Footnotes
This work was supported by the National Key Research and Development Program of China (Grant No. 2021YFA1000700), the National Natural Science Foundation of China (Grant Nos. 12271297 and 12031008), and the Natural Science Foundation of Shandong Province (Grant No. ZR2022YQ03).
References
Barnet-Lamb, T., Geraghty, D., Harris, M., and Taylor, R.,
A family of Calabi–Yau varieties and potential automorphy II
. Publ. Res. Inst. Math. Sci. 47(2011), 29–98.10.2977/prims/31CrossRefGoogle Scholar
Barthel, L. and Ramakrishnan, D.,
A nonvanishing result for twists of
$L$
-functions of
$\mathrm{GL}(n)$
. Duke Math. J. 74(1994), 681–700.10.1215/S0012-7094-94-07425-5CrossRefGoogle Scholar
$L$
-functions of
$\mathrm{GL}(n)$
. Duke Math. J. 74(1994), 681–700.10.1215/S0012-7094-94-07425-5CrossRefGoogle ScholarChandrasekharan, K. and Narasimhan, R.,
Functional equations with multiple gamma factors and the average order of arithmetical functions
. Ann. of Math. (2) 76(1962), 93–136.10.2307/1970267CrossRefGoogle Scholar
Deligne, P.,
La conjecture de Weil. I
. Inst. Hautes Études Sci. Publ. Math. 43(1974), 273–307.10.1007/BF02684373CrossRefGoogle Scholar
Duke, W. and Iwaniec, H.,
Estimates for coefficients of
$L$
-functions. I. In: Automorphic forms and analytic number theory (Montreal, PQ, 1989), University of Montreal, Montreal, QC, 1990, pp. 43–47.Google Scholar
$L$
-functions. I. In: Automorphic forms and analytic number theory (Montreal, PQ, 1989), University of Montreal, Montreal, QC, 1990, pp. 43–47.Google ScholarDuke, W. and Iwaniec, H.,
Estimates for coefficients of
$L$
-functions. II. In: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), University of Salerno, Salerno, 1992, pp. 71–82.Google Scholar
$L$
-functions. II. In: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), University of Salerno, Salerno, 1992, pp. 71–82.Google ScholarDuke, W. and Iwaniec, H.,
Estimates for coefficients of
$L$
-functions. III. In: David, S. (ed.), Séminaire de Théorie des Nombres, Paris, 1989–90, Progress in Mathematics, 102, Birkhäuser, Boston, MA, 1992, pp. 113–120.Google Scholar
$L$
-functions. III. In: David, S. (ed.), Séminaire de Théorie des Nombres, Paris, 1989–90, Progress in Mathematics, 102, Birkhäuser, Boston, MA, 1992, pp. 113–120.Google ScholarDuke, W. and Iwaniec, H.,
Estimates for coefficients of
$L$
-functions. IV
. Amer. J. Math. 116(1994), 207–217.10.2307/2374986CrossRefGoogle Scholar
$L$
-functions. IV
. Amer. J. Math. 116(1994), 207–217.10.2307/2374986CrossRefGoogle ScholarDuke, W. and Iwaniec, H.,
Convolution
$L$
-series
. Compositio Math. 91(1994), 145–158.Google Scholar
$L$
-series
. Compositio Math. 91(1994), 145–158.Google ScholarHeath-Brown, D. R.,
The divisor function
${d}_3\hspace{-0.6pt}(n)$
in arithmetic progressions
. Acta Arith. 47(1986), 29–56.10.4064/aa-47-1-29-56CrossRefGoogle Scholar
${d}_3\hspace{-0.6pt}(n)$
in arithmetic progressions
. Acta Arith. 47(1986), 29–56.10.4064/aa-47-1-29-56CrossRefGoogle ScholarIwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004.Google Scholar
Jacquet, H., Principal
$L$
-functions of the linear group automorphic forms, representations and L-functions
. In: Borel, A. and Casselman, W. (eds.), Proceedings Symposia Pure Mathematics, vol. XXXIII, part 2, American Mathematical Society, Providence, RI, 1979, pp. 63–86.10.1090/pspum/033.2/546609CrossRefGoogle Scholar
$L$
-functions of the linear group automorphic forms, representations and L-functions
. In: Borel, A. and Casselman, W. (eds.), Proceedings Symposia Pure Mathematics, vol. XXXIII, part 2, American Mathematical Society, Providence, RI, 1979, pp. 63–86.10.1090/pspum/033.2/546609CrossRefGoogle ScholarJiang, Y., Lü, G., and Wang, Z.,
Exponential sums with multiplicative coefficients without the Ramanujan conjecture
. Math. Ann. 379(2021), 589–632.10.1007/s00208-020-02108-zCrossRefGoogle Scholar
Kuo, W.,
Summatory functions of elements in Selberg’s class. II
. C. R. Math. Acad. Sci. Soc. R. Can. 25(2003), 55–62.Google Scholar
Luo, W.,
Nonvanishing of
$L$
-functions for
$\mathrm{GL}(n,{\mathbb{A}}_{\mathbb{Q}})$
. Duke Math. J. 128(2005), 199–207.Google Scholar
$L$
-functions for
$\mathrm{GL}(n,{\mathbb{A}}_{\mathbb{Q}})$
. Duke Math. J. 128(2005), 199–207.Google ScholarLuo, W., Rudnick, Z., and Sarnak, P.,
On Selberg’s eigenvalue conjecture
. Geom. Funct. Anal. 5(1995), 387–401.10.1007/BF01895672CrossRefGoogle Scholar
Miller, S. D. and Schmid, W.,
A general Voronoi summation formula for
$\mathrm{GL}(n,\mathbb{Z})$
. In: Geometry and analysis, No. 2, Advanced Lectures in Mathematics, 18, International Press, Somerville, 2011, pp. 173–224.Google Scholar
$\mathrm{GL}(n,\mathbb{Z})$
. In: Geometry and analysis, No. 2, Advanced Lectures in Mathematics, 18, International Press, Somerville, 2011, pp. 173–224.Google ScholarMüller, W. and Speh, B.,
Absolute convergence of the spectral side of the Arthur trace formula for
$\mathrm{GL}_n$
, with appendix by E. M. Lapid
. Geom. Funct. Anal. 14(2004), 58–93.10.1007/s00039-004-0452-0CrossRefGoogle Scholar
$\mathrm{GL}_n$
, with appendix by E. M. Lapid
. Geom. Funct. Anal. 14(2004), 58–93.10.1007/s00039-004-0452-0CrossRefGoogle ScholarNewton, J. and Thorne, J. A.,
Symmetric power functoriality for holomorphic modular forms
. Publ. Math. Inst. Hautes Études Sci. 134(2021), 1–116.10.1007/s10240-021-00127-3CrossRefGoogle Scholar
Shiu, P.,
A Brun–Titchmarsh theorem for multiplicative functions
. J. Reine Angew. Math. 313(1980), 161–170.Google Scholar
Smith, R. A.,
The average order of a class of arithmetic functions over arithmetic progressions with applications to quadratic forms
. J. Reine Angew. Math. 317(1980), 74–87.Google Scholar
Smith, R. A.,
The generalized divisor problem over arithmetic progressions
. Math. Ann. 260(1982), 255–268.10.1007/BF01457239CrossRefGoogle Scholar
Smith, R. A.,
Fourier coefficients of modular forms over arithmetic progressions. I, II, with remarks by M. R. Murty
. C. R. Math. Acad. Sci. Soc. R. Can. 15(1993), 85–90, 91–98.Google Scholar
Soundararajan, K. and Thorner, J.,
Weak subconvexity without a Ramanujan hypothesis, with an appendix by Farrell Brumley
. Duke Math. J. 168(2019), 1231–1268.10.1215/00127094-2018-0065CrossRefGoogle Scholar