Abstract
For a positive integer
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12271135
Award Identifier / Grant number: 11871187
Funding source: Fundamental Research Funds for the Central Universities
Award Identifier / Grant number: PA2021KCPY0040
Funding statement: This work is supported in part by the National Natural Science Foundation of China (Grant nos. 12271135, 11871187) and the Fundamental Research Funds for the Central Universities of China.
Acknowledgements
The author would like to express our heartfelt thanks to the anonymous referees for their careful reading and helpful suggestion.
References
[1] S. Bettin, H. M. Bui, X. Li and M. Radziwiłł, A quadratic divisor problem and moments of the Riemann zeta-function, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 12, 3953–3980. 10.4171/JEMS/999Search in Google Scholar
[2] S. Bettin, V. Chandee and M. Radziwiłł, The mean square of the product of the Riemann zeta-function with Dirichlet polynomials, J. Reine Angew. Math. 729 (2017), 51–79. 10.1515/crelle-2014-0133Search in Google Scholar
[3] V. Blomer, É. Fouvry, E. Kowalski, P. Michel and D. Milićević, On moments of twisted L-functions, Amer. J. Math. 139 (2017), no. 3, 707–768. 10.1353/ajm.2017.0019Search in Google Scholar
[4] V. Blomer, É. Fouvry, E. Kowalski, P. Michel and D. Milićević, Some applications of smooth bilinear forms with Kloosterman sums (in Russian), Tr. Mat. Inst. Steklova 296 (2017), 24–35; translation in Proc. Steklov Inst. Math. 296 (2017), no. 1, 18–29. Search in Google Scholar
[5] V. Blomer, P. Humphries, R. Khan and M. B. Milinovich, Motohashi’s fourth moment identity for non-archimedean test functions and applications, Compos. Math. 156 (2020), no. 5, 1004–1038. 10.1112/S0010437X20007101Search in Google Scholar
[6] H. M. Bui and D. R. Heath-Brown, A note on the fourth moment of Dirichlet L-functions, Acta Arith. 141 (2010), no. 4, 335–344. 10.4064/aa141-4-3Search in Google Scholar
[7] V. Chandee and X. Li, The eighth moment of Dirichlet L-functions, Adv. Math. 259 (2014), 339–375. 10.1016/j.aim.2014.03.020Search in Google Scholar
[8] J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith, Integral moments of L-functions, Proc. Lond. Math. Soc. (3) 91 (2005), no. 1, 33–104. 10.1112/S0024611504015175Search in Google Scholar
[9] J. B. Conrey, H. Iwaniec and K. Soundararajan, The sixth power moment of Dirichlet L-functions, Geom. Funct. Anal. 22 (2012), no. 5, 1257–1288. 10.1007/s00039-012-0191-6Search in Google Scholar
[10] W. Duke, J. B. Friedlander and H. Iwaniec, A quadratic divisor problem, Invent. Math. 115 (1994), no. 2, 209–217. 10.1007/BF01231758Search in Google Scholar
[11] É. Fouvry, E. Kowalski and P. Michel, Algebraic trace functions over the primes, Duke Math. J. 163 (2014), no. 9, 1683–1736. 10.1215/00127094-2690587Search in Google Scholar
[12] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965. Search in Google Scholar
[13] G. H. Hardy and J. E. Littlewood, Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119–196. 10.1007/BF02422942Search in Google Scholar
[14] A. J. Harper, Sharp conditional bounds for moments of the Riemann zeta function, preprint (2013), https://arxiv.org/abs/1305.4618. Search in Google Scholar
[15] D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. Lond. Math. Soc. (3) 38 (1979), no. 3, 385–422. 10.1112/plms/s3-38.3.385Search in Google Scholar
[16] D. R. Heath-Brown, The fourth power mean of Dirichlet’s L-functions, Analysis 1 (1981), no. 1, 25–32. 10.1524/anly.1981.1.1.25Search in Google Scholar
[17] C. P. Hughes and M. P. Young, The twisted fourth moment of the Riemann zeta function, J. Reine Angew. Math. 641 (2010), 203–236. 10.1515/crelle.2010.034Search in Google Scholar
[18] A. Ivić and Y. Motohashi, On the fourth power moment of the Riemann zeta-function, J. Number Theory 51 (1995), no. 1, 16–45. 10.1006/jnth.1995.1033Search in Google Scholar
[19]
J. P. Keating and N. C. Snaith,
Random matrix theory and
[20] B. Kerr, I. E. Shparlinski, X. Wu and P. Xi, Bounds on bilinear forms with Kloosterman sums, J. Lond. Math. Soc. (2023), 10.1112/jlms.12753. 10.1112/jlms.12753Search in Google Scholar
[21]
H. H. Kim,
Functoriality for the exterior square of
[22] E. Kowalski, P. Michel and W. Sawin, Bilinear forms with Kloosterman sums and applications, Ann. of Math. (2) 186 (2017), no. 2, 413–500. 10.4007/annals.2017.186.2.2Search in Google Scholar
[23] H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer, Berlin, 1971. 10.1007/BFb0060851Search in Google Scholar
[24] Y. Motohashi, Spectral Theory of the Riemann Zeta-Function, Cambridge Tracts in Math. 127, Cambridge University, Cambridge, 1997. 10.1017/CBO9780511983399Search in Google Scholar
[25] N. Ng, The sixth moment of the Riemann zeta function and ternary additive divisor sums, Discrete Anal. 2021 (2021), Paper No. 6. Search in Google Scholar
[26] V. V. Rane, A note on the mean value of L-series, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 3, 273–286. 10.1007/BF02838080Search in Google Scholar
[27] I. E. Shparlinski and T. Zhang, Cancellations amongst Kloosterman sums, Acta Arith. 176 (2016), no. 3, 201–210. 10.4064/aa8365-6-2016Search in Google Scholar
[28] K. Soundararajan, The fourth moment of Dirichlet L-functions, Analytic Number Theory, Clay Math. Proc. 7, American Mathematical Society, Providence (2007), 239–246. Search in Google Scholar
[29] K. Soundararajan, Moments of the Riemann zeta function, Ann. of Math. (2) 170 (2009), no. 2, 981–993. 10.4007/annals.2009.170.981Search in Google Scholar
[30] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University, New York, 1986. Search in Google Scholar
[31] B. Topacogullari, The fourth moment of individual Dirichlet L-functions on the critical line, Math. Z. 298 (2021), no. 1–2, 577–624. 10.1007/s00209-020-02610-9Search in Google Scholar
[32] W. Wang, Fourth power mean value of Dirichlet’s L-functions, International Symposium in Memory of Hua Loo Keng, Vol. I (Beijing 1988), Springer, Berlin (1991), 293–321. 10.1007/978-3-662-07981-2_18Search in Google Scholar
[33] X. Wu, The fourth moment of Dirichlet L-functions at the central value, Math. Ann. (2022), 10.1007/s00208-022-02483-9. 10.1007/s00208-022-02483-9Search in Google Scholar
[34] M. P. Young, A short proof of Levinson’s theorem, Arch. Math. (Basel) 95 (2010), no. 6, 539–548. 10.1007/s00013-010-0199-9Search in Google Scholar
[35] M. P. Young, The fourth moment of Dirichlet L-functions, Ann. of Math. (2) 173 (2011), no. 1, 1–50. 10.4007/annals.2011.173.1.1Search in Google Scholar
[36] N. I. Zavorotnyĭ, On the fourth moment of the Riemann zeta function, Automorphic Functions and Number Theory, Part I, II (in Russian), Akad. Nauk SSSR, Vladivostok (1989), 69–125. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- On the arithmetic of polynomial semidomains
- Extra-special Leibniz superalgebras
- Restricted iso-minimum condition
- On the spectral large sieve inequality for symmetric-squares
- Simple 𝔰𝔩(V)-modules which are free over an abelian subalgebra
- Epsilon-strongly graded rings: Azumaya algebras and partial crossed products
- Non-weight modules over N = 1 Lie superalgebras of Block type
- Simpler foundations for the hyperbolic plane
- Characterizations of the mixed radial-angular central Campanato space via the commutators of Hardy type
- The fourth moment of Dirichlet L-functions along the critical line
- Diagonal restriction of Eisenstein series and Kudla–Millson theta lift
- Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term
- Jantzen filtration of Weyl modules for general linear supergroups
- Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems
Articles in the same Issue
- Frontmatter
- On the arithmetic of polynomial semidomains
- Extra-special Leibniz superalgebras
- Restricted iso-minimum condition
- On the spectral large sieve inequality for symmetric-squares
- Simple 𝔰𝔩(V)-modules which are free over an abelian subalgebra
- Epsilon-strongly graded rings: Azumaya algebras and partial crossed products
- Non-weight modules over N = 1 Lie superalgebras of Block type
- Simpler foundations for the hyperbolic plane
- Characterizations of the mixed radial-angular central Campanato space via the commutators of Hardy type
- The fourth moment of Dirichlet L-functions along the critical line
- Diagonal restriction of Eisenstein series and Kudla–Millson theta lift
- Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term
- Jantzen filtration of Weyl modules for general linear supergroups
- Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems