Details

Sandro Bettin ^[a] and John Brian Conrey ^[b,c]

Averages of long Dirichlet polynomials

Pages: 1-27
Received: 26 February 2020
Accepted in revised form: 22 June 2020
Mathematics Subject Classification (2010): Primary 11N37, Secondary 11M06, 11M50.
Keywords: Moments conjecture, divisor function in short intervals, divisor function in arithmetic progressions, Riemann zeta-function.
Authors address:
[a]: DIMA - Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy
[b]: American Institute of Mathematics, 600 East Brokaw Road, San Jose, CA 95112, USA
[c]: School of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom

S. Bettin is member of the INdAM group GNAMPA and his work is partially supported by PRIN 2017 ''Geometric, algebraic and analytic methods in arithmetic''. J. B. Conrey is supported in part by a grant from the NSF.

Abstract: We consider the asymptotic behavior of the mean square of truncations of the Dirichlet series of \(\zeta(s)^k\). We discuss the connections of this problem with that of the variance of the divisor function in short intervals and in arithmetic progressions, reviewing the recent results on this topic. Finally, we show how these results can all be proved assuming a suitable version of the moments conjecture.

References

[1]: E. Basor, F. Ge and M. O. Rubinstein, Some multidimensional integrals in number theory and connections with the Painlevé V equation, J. Math. Phys. 59 (2018), 091404, 14 pp. MR3825378
[2]: S. Bettin, The second moment of the Riemann zeta function with unbounded shifts, Int. J. Number Theory 6 (2010), 1933-1944. MR2755480
[3]: 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), 3953-3980. MR4176783
[4]: V. Blomer, The average value of divisor sums in arithmetic progressions, Q. J. Math. 59 (2008), 275-286. MR2444061
[5]: J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith, Autocorrelation of random matrix polynomials, Comm. Math. Phys. 237 (2003), 365-395. MR1993332
[6]: J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith, Integral moments of \(L\)-functions, Proc. London Math. Soc. 91 (2005), 33-104. MR2149530
[7]: J. B. Conrey and A. Ghosh, A conjecture for the sixth power moment of the Riemann zeta-function, Internat. Math. Res. Notices 1998 (1998), 775-780. MR1639551
[8]: J. B. Conrey and S. M. Gonek, High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), 577-604. MR1828303
[9]: J. B. Conrey, H. Iwaniec and K. Soundararajan, The sixth power moment of Dirichlet \(L\)-functions, Geom. Funct. Anal. 22 (2012), 1257-1288. MR2989433
[10]: J. B. Conrey and J. P. Keating, Moments of zeta and correlations of divisor-sums: I, Philos. Trans. Roy. Soc. A 373 (2015), no. 2040, 20140313, 11 pp. MR3338122
[11]: J. B. Conrey and J. P. Keating, Moments of zeta and correlations of divisor-sums: II, Advances in the theory of numbers, 75-85, Fields Inst. Commun., 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015. MR3409324
[12]: J. B. Conrey and J. P. Keating, Moments of zeta and correlations of divisor-sums: III, Indag. Math. (N.S.) 26 (2015), 736-747. MR3425874
[13]: J. B. Conrey and J. P. Keating, Moments of zeta and correlations of divisor-sums: IV, Res. Number Theory 2 (2016), Art. 24, 24 pp. MR3574307
[14]: J. B. Conrey and J. P. Keating, Moments of zeta and correlations of divisor-sums: V, Proc. London Math. Soc. 118 (2019), 729-752. MR3938710
[15]: J. B. Conrey and B. Rodgers, Averages of quadratic twists of long Dirichlet polynomials, preprint.
[16]: J. B. Conrey, M. O. Rubinstein and N. C. Snaith, Moments of the derivative of characteristic polynomials with an application to the Riemann zeta function, Comm. Math. Phys. 267 (2006), 611-629. MR2249784
[17]: J. B. Conrey and N. C. Snaith, Applications of the \(L\)-functions ratios conjectures, Proc. London Math. Soc. 94 (2007), 594-646. MR2325314
[18]: G. Coppola and S. Salerno, On the symmetry of the divisor function in almost all short intervals, Acta Arith. 113 (2004), 189-201. MR2049565
[19]: R. de la Bretèche and D. Fiorilli, Major arcs and moments of arithmetical sequences, Amer. J. Math. 142 (2020), 45-77. MR4060871
[20]: F. Ge and G. Liu, A combinatorial identity and the finite dual of infinite dihedral group algebra, arXiv:2005.01410, preprint, 2020.
[21]: D. Goldfeld and J. Hoffstein, Eisenstein series of \(\frac12\)-integral weight and the mean value of real Dirichlet \(L\)-series, Invent. Math. 80 (1985), 185-208. MR0788407
[22]: D. A. Goldston, S. M. Gonek and H. L. Montgomery, Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals, J. Reine Angew. Math. 537 (2001), 105-126. MR1856259
[23]: O. Gorodetsky and B. Rodgers, The variance of the number of sums of two squares in \(\mathbb F_q[T]\) in short intervals, arXiv:1810.06002, preprint, 2018.
[24]: C. Hall, J. P. Keating and E. Roditty-Gershon, Variance of arithmetic sums and L-functions in \(\mathbb F_q[t]\), Algebra Number Theory 13 (2019), 19-92. MR3917915
[25]: 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), 119-196. MR1555148
[26]: A. J. Harper and K. Soundararajan, Lower bounds for the variance of sequences in arithmetic progressions: primes and divisor functions, Q. J. Math. 68 (2017), 97-123. MR3658285
[27]: C. P. Hughes and M. P. Young, The twisted fourth moment of the Riemann zeta function, J. Reine Angew. Math. 641 (2010), 203-236. MR2643931
[28]: A. E. Ingham, Mean-values theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. 27 (1927), 273-300. MR1575391
[29]: A. Ivić, On the mean square of the divisor function in short intervals, J. Théor. Nombres Bordeaux 21 (2009), 251-261. MR2541424
[30]: A. Ivić, On the divisor function and the Riemann zeta-function in short intervals, Ramanujan J. 19 (2009), 207-224. MR2511672
[31]: M. Jutila, On the divisor problem for short intervals, Ann. Univ. Turku. Ser. A I 186 (1984), 23-30. MR0748516
[32]: J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in \(\mathbb F_q[t]\) and matrix integrals, Math. Z. 288 (2018), 167-198. MR3774409
[33]: J. P. Keating and N. C. Snaith, Random matrix theory and \(\zeta(1/2 + it)\), Comm. Math. Phys. 214 (2000), 57-89. MR1794265
[34]: E. Kowalski and G. Ricotta, Fourier coefficients of GL(N) automorphic forms in arithmetic progressions, Geom. Funct. Anal. 24 (2014), 1229-1297. MR3248485
[35]: Y.-K. Lau and L. Zhao, On a variance of Hecke eigenvalues in arithmetic progressions, J. Number Theory 132 (2012), 869-887. MR2890517
[36]: S. Lester, On the variance of sums of divisor functions in short intervals, Proc. Amer. Math. Soc. 144 (2016), 5015-5027. MR3556248
[37]: K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. 183 (2016), 1015-1056. MR3488742
[38]: K. Matomäki and M. Radziwiłł, A note on the Liouville function in short intervals, arXiv:1502.02374, preprint, 2015.
[39]: M. B. Milinovich and C. L. Turnage-Butterbaugh, Moments of products of automorphic L-functions, J. Number Theory 139 (2014), 175-204. MR3173191
[40]: H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc. 8 (1974), 73-82. MR0337775
[41]: Y. Motohashi, On the distribution of the divisor function in arithmetic progressions, Acta Arith. 22 (1973), 175-199. MR0340196
[42]: B. Rodgers and K. Soundararajan, The variance of divisor sums in arithmetic progressions, Forum Math. 30 (2018), 269-293. MR3769996
[43]: B. Saffari and R. C. Vaughan, On the fractional parts of \(x/n\) and related sequences, II, Ann. Inst. Fourier (Grenoble) 27 (1977), v, 1-30. MR0480388
[44]: S. Shimomura, Shifted fourth moment of the Riemann zeta-function, Acta Math. Hungar. 137 (2012), 104-129. MR2966982
[45]: E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., Oxford University Press, New York, 1986. MR0882550