Continuous Lower Bounds for Moments of the Mixed Product of Twisted L-functions

Ying Zhang

doi:10.54097/6f5gh954

Authors

Ying Zhang

DOI:

https://doi.org/10.54097/6f5gh954

Keywords:

Twisted L-functions, Cuspidal Holomorphic Hecke Eigenform, Moments Lower Bounds

Abstract

Moments of central values in family of L-functions are important subjects in analytic number theory. The twisted L -functions is an important class of automorphic L-functions. Recently, using the lower bound principle of Heap and Soundararajan, Chen et al. have obtained the sharp lower bounds for all positive real k-th (K>1) moments of the mixed product of two twisted L-functions at the central value, when there exists a divisor q₀ such that q₀ |q and q^η q₀q^1/2-ηHere q is the modulus of Dirichlet character and η is a small real number. Based on Chen et al. and Blomer et al. the work of on the first moment of the mixed product of two twisted L-function at the central value, when the modulus q of the Dirichlet character is prime, in this paper, we study lower bounds for all positive real k-th (K>1) moments of the mixed product of two distinct twisted L-function at the central value for the prime modulus case and obtain the sharp lower bounds.

Downloads

Download data is not yet available.

References

[1] V. Blomer. “Shifted convolution sums and subconvexity bounds for automorphic L-functions,” Int. Math. Res. Not. IMRN 73:3905–3926, 2004.

[2] V. Blomer, ´ E. Fouvry, E. Kowalski, Ph. Michel and D. Mili´cevi´c. “On moments of twisted L-functions,” Amer. J. Math. 139(3):707–768, 2017.

[3] V. Blomer and D. Mili´cevi´c. “The second moment of twisted modular L-functions,” Geom. Funct. Anal. 25(2):453–516, 2015.

[4] J. B. Conrey, D. W. Farmer, J. P. Keating, et al. “Integral moments of L-functions,” Proc. London Math. Soc. (3), 91(1):33–104, 2005.

[5] G. Chen and X. He. “Lower bound for higher moments of the mixed product of twisted L-functions,” J. Number Theory 222:233–248, 2021.

[6] P. Deligne. “La conjecture de Weil. I, Inst,” Hautes ´ Etudes Sci. Publ. Math. (43):273–307, 1974.

[7] P. Gao, X. He and X. Wu. “Bounds for moments of modular L-functions to a fixed modulus,” Acta Arith. 205(2):161–189, 2022.

[8] W. Heap and K. Soundararajan. “Lower bounds for moments of zeta and L-functions revisited,” Mathematika 68(1):1–14, 2022.

[9] H. Iwaniec and E. Kowalski. “Analytic number theory, volume 53 of American Mathematical Society Collo quium Publications,” American Mathematical Society, Providence, RI, 2004.

[10] J. P. Keating and N. C. “Snaith. Random matrix theory and ζ(1/2+it),” Comm. Math. Phys. 214(1):57–89, 2000.

[11] J. P. Keating and N. C. Snaith. “Random matrix theory and L-functions at s = 1/2,” Comm. Math. Phys. 214(1):91–110, 2000.

[12] J. B. Conrey, D. W. Farmer, J. P. Keating, et al. “Integral moments of L-functions,” Proc. London Math. Soc. (3), 91(1):33–104, 2005.

[13] E. Kowalsiki, Ph. Michel and W. Sawin. “Bilinear forms with Kloosterman sums and applications,” Ann. of Math. 186:413–500, 2017.

[14] M. Radziwill and K, Soundararajan. “Moments and distribution of central L-values of quadratic twists of elliptic curves,” Invent. Math. 202:1029-1068, 2015.

[15] Z. Rudnick and K. Soundararajan. “Lower bounds for moments of L-functions,” Proc. Natl. Acad. Sci. USA 102(19):6837–6838, 2005.

[16] Z. Rudnick and K. Soundararajan. “Lower bounds for moments of L-functions: symplectic and orthogonal examples,” In Multiple Dirichlet series, automorphic forms, and analytic number theory, volume 75 of Proc. Sympos. Pure Math, pages 293–303, Amer. Math. Soc., Providence, RI, 2006.

[17] K. Soundararajan. “Moments of the Riemann zeta function,” Ann. of Math. (2) 170(2):981–993, 2009.

[18] G. Chen, W. Li, and T. Wang. “Continuous lower bounds for moments of the mixed product of twisted L-functions,” Number Theory. 2025.