##
**Moments of the Riemann zeta function.**
*(English)*
Zbl 1251.11058

The \(2k\)-th moment of the Riemann zeta-function is defined as,
\[
M_k(T) := \int_{0}^{T} |\zeta(\tfrac 12 + it)|^{2k}\, dt
\]
Historically the study of moments of \(L\)-functions was connected with the desire to establish the Lindelöf Hypothesis. More recently moments have emerged as a useful tool in the study of subconvexity, non-vanishing at the central point, etc.

A folklore conjecture states that for any \(k > 0\) there exists a constant \(C_k > 0\) such that as \(T \rightarrow \infty\) \[ M_k(T) \sim C_k T (\log T)^{k^2} \] Only recently has a precise value of \(C_k\) been conjectured for all \(k > 0\) by J. P. Keating and N. C. Snaith [“Random matrix theory and \(\zeta(1/2 + it)\)”, Commun. Math. Phys 214, No. 1, 57–89 (2000; Zbl 1051.11048)]. An alternative approach to this conjecture has been devised by A. Diaconu, D. Goldfeld and J. Hoffstein [“Multiple Dirichle series and moments of zeta and \(L\)-functions”, Compos. Math 139, No. 3, 297–360 (2003; Zbl 1053.11071)]. Previously, J.B. Conrey and S.M. Gonek [“High moments of the Riemann zeta-function”, Duke Math. J. 107, No. 3, 577–604 (2001; Zbl 1006.11048)] and J. B. Conrey and A. Ghosh [“A conjecture for the sixth power moment of the Riemann zeta-function”, Int. Math. Res. Not. 1998, No. 15, 775–780 (1998; Zbl 0920.11060)] have obtained conjectures for \(k = 4\) and \(k = 3\) respectively.

We have asymptotic formulas for \(M_k(T)\) only when \(k = 1\) or \(k = 2\) following the work of Hardy-Littlewood and Ingham. Lower bounds of the correct order of magnitude have been established for all \(k\) rational by D. R. Heath-Brown [“Fractional moments of the Riemann zeta-function”, J. Lond. Math. Soc., II., Ser. 24, 65–78 (1981; Zbl 0431.10024)] building on results of K. Ramachandra [“Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series. II.”, Hardy-Ramanujan J. 3, 1–24 (1980; Zbl 0426.10046)]. Recently these lower bounds have been extended to all \(k > 1\) by K. Soundararajan and the reviewer [“Continuous lower bounds for moments of zeta and L-functions”, Mathematika 59, No. 1, 119–128 (2013; Zbl 1273.11128)]. The upper bound problem is more difficult and requires the assumption of the Riemann Hypothesis. Heath-Brown obtains upper bounds for \(M_k(T)\) of the correct order of magnitude when \(0 < k < 2\) D. R. Heath-Brown [“Fractional moments of the Riemann zeta-function”, J. Lond. Math. Soc., II., Ser. 24, 65–78 (1981; Zbl 0431.10024)]. This has recently been extended by the reviewer to \(k < 2.18\) [M. Radziwill, “The 4.36th moment of the Riemann zeta-function”, Int. Math. Res. Not., No. 18, 4345–4259 (2012; Zbl 1290.11120)].

In the paper under review, assuming the Riemann Hypothesis, Soundararajan establishes a beautiful and far-reaching bound for \(M_k(T)\), valid for all \(k > 0\) and missing the correct order of magnitude only by a small power of \(\log T\). Precisely, it is shown that for all \(k > 0\), \[ M_k(T) \ll T (\log T)^{k^2 + \varepsilon}. \] A. Ivić [“On mean value results for the Riemann zeta-function in short intervals”, Hardy-Ramanujan J. 32, 4–23 (2009; Zbl 1221.11177)] has remarked that \(\varepsilon\) can be taken to be \((\log\log\log T)^{-1}\). The methods of this paper have already found many applications for example (in no particular order) in the study of moments of \(\zeta'(\rho)\) [M. B. Milinovich, “Upper bounds for moments of \(\zeta'(\rho)\)”, Bull. Lond. Math. Soc 42, No. 1, 28–44 (2010; Zbl 1223.11102)], the study of the second moment of quadratic twists of modular \(L\)-functions [K. Soundararajan and M. Young, “The second moment of quadratic twists of modular \(L\)-functions”, J. Eur. Math. Soc 12, No. 5, 1097–1116 (2010; Zbl 1213.11165)], the study of the size of \(L\)-functions at \(s = 1\) Xiannan Li, [“Upper bounds on \(L\)-functions at the edge of the critical strip”, Int. Math. Res. Not., No. 4, 727–755 (2010; Zbl 1219.11136)], the study of shifted moments V. Chandee [“On the correlation of shifted values of the Riemann zeta-function”, Q. J. Math. 62, 545–572 (2011; Zbl 1290.11117)], etc. The method is very general and works for \(L\)-functions in families. Certainly many more applications are yet to come.

The idea behind the proof is as follows: Following Selberg \(\log |\zeta(\tfrac 12 + it)|\) can be decomposed into the sum of a Dirichlet polynomial over primes and a sum over the zeros of \(\zeta(s)\). Soundararajan notices that the zeros make a negative contribution, and therefore we can obtain an upper bound for \(\log |\zeta(\tfrac 12 + it)|\) solely in terms of the Dirichlet polynomial over primes. The frequency with which this Dirichlet polynomial is large can be estimated quite accurately in a very long range. This allows the author to show that the measure of those \(T \leq t \leq 2T\) for which \(\log |\zeta(\tfrac 12 + it)| > V\) is \[ \ll T \exp \bigg ( - (1 + o(1)) \frac{V^2}{\log\log T} \bigg ) \] when \(V < \log\log T \log\log\log T\). For larger \(V\) this bound degenerates into \( \ll T \exp( - c V \log V) \) which is still sufficient for applications to moments. An integration by parts then leads to the desired estimate \(M_k(T) \ll T ( \log T)^{k^2 + \varepsilon}\). We refer the reader to the paper for more precise bounds for the frequency of large values of \(\log |\zeta(\tfrac 12 + it)|\).

The author also derives point-wise bounds for \(\log |\zeta(\tfrac 12 + it)|\) refining previous work of Littlewood. It is shown that, on the Riemann Hypothesis, \[ \log |\zeta(\tfrac 12 + it)| < \exp \bigg ( \big ( \frac{3}{8} + o(1) \big) \frac{\log t}{\log\log t} \bigg ). \] The constant \(3/8\) has been recently improved to \(\log 2 / 2\) by V. Chandee and K. Soundararajan [“Bounding \(|\zeta(\frac 12 + it)|\) on the Riemann Hypothesis”, Bull. Lond. Math. Soc. 43, No. 2, 243–250 (2011; Zbl 1238.11078)].

A folklore conjecture states that for any \(k > 0\) there exists a constant \(C_k > 0\) such that as \(T \rightarrow \infty\) \[ M_k(T) \sim C_k T (\log T)^{k^2} \] Only recently has a precise value of \(C_k\) been conjectured for all \(k > 0\) by J. P. Keating and N. C. Snaith [“Random matrix theory and \(\zeta(1/2 + it)\)”, Commun. Math. Phys 214, No. 1, 57–89 (2000; Zbl 1051.11048)]. An alternative approach to this conjecture has been devised by A. Diaconu, D. Goldfeld and J. Hoffstein [“Multiple Dirichle series and moments of zeta and \(L\)-functions”, Compos. Math 139, No. 3, 297–360 (2003; Zbl 1053.11071)]. Previously, J.B. Conrey and S.M. Gonek [“High moments of the Riemann zeta-function”, Duke Math. J. 107, No. 3, 577–604 (2001; Zbl 1006.11048)] and J. B. Conrey and A. Ghosh [“A conjecture for the sixth power moment of the Riemann zeta-function”, Int. Math. Res. Not. 1998, No. 15, 775–780 (1998; Zbl 0920.11060)] have obtained conjectures for \(k = 4\) and \(k = 3\) respectively.

We have asymptotic formulas for \(M_k(T)\) only when \(k = 1\) or \(k = 2\) following the work of Hardy-Littlewood and Ingham. Lower bounds of the correct order of magnitude have been established for all \(k\) rational by D. R. Heath-Brown [“Fractional moments of the Riemann zeta-function”, J. Lond. Math. Soc., II., Ser. 24, 65–78 (1981; Zbl 0431.10024)] building on results of K. Ramachandra [“Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series. II.”, Hardy-Ramanujan J. 3, 1–24 (1980; Zbl 0426.10046)]. Recently these lower bounds have been extended to all \(k > 1\) by K. Soundararajan and the reviewer [“Continuous lower bounds for moments of zeta and L-functions”, Mathematika 59, No. 1, 119–128 (2013; Zbl 1273.11128)]. The upper bound problem is more difficult and requires the assumption of the Riemann Hypothesis. Heath-Brown obtains upper bounds for \(M_k(T)\) of the correct order of magnitude when \(0 < k < 2\) D. R. Heath-Brown [“Fractional moments of the Riemann zeta-function”, J. Lond. Math. Soc., II., Ser. 24, 65–78 (1981; Zbl 0431.10024)]. This has recently been extended by the reviewer to \(k < 2.18\) [M. Radziwill, “The 4.36th moment of the Riemann zeta-function”, Int. Math. Res. Not., No. 18, 4345–4259 (2012; Zbl 1290.11120)].

In the paper under review, assuming the Riemann Hypothesis, Soundararajan establishes a beautiful and far-reaching bound for \(M_k(T)\), valid for all \(k > 0\) and missing the correct order of magnitude only by a small power of \(\log T\). Precisely, it is shown that for all \(k > 0\), \[ M_k(T) \ll T (\log T)^{k^2 + \varepsilon}. \] A. Ivić [“On mean value results for the Riemann zeta-function in short intervals”, Hardy-Ramanujan J. 32, 4–23 (2009; Zbl 1221.11177)] has remarked that \(\varepsilon\) can be taken to be \((\log\log\log T)^{-1}\). The methods of this paper have already found many applications for example (in no particular order) in the study of moments of \(\zeta'(\rho)\) [M. B. Milinovich, “Upper bounds for moments of \(\zeta'(\rho)\)”, Bull. Lond. Math. Soc 42, No. 1, 28–44 (2010; Zbl 1223.11102)], the study of the second moment of quadratic twists of modular \(L\)-functions [K. Soundararajan and M. Young, “The second moment of quadratic twists of modular \(L\)-functions”, J. Eur. Math. Soc 12, No. 5, 1097–1116 (2010; Zbl 1213.11165)], the study of the size of \(L\)-functions at \(s = 1\) Xiannan Li, [“Upper bounds on \(L\)-functions at the edge of the critical strip”, Int. Math. Res. Not., No. 4, 727–755 (2010; Zbl 1219.11136)], the study of shifted moments V. Chandee [“On the correlation of shifted values of the Riemann zeta-function”, Q. J. Math. 62, 545–572 (2011; Zbl 1290.11117)], etc. The method is very general and works for \(L\)-functions in families. Certainly many more applications are yet to come.

The idea behind the proof is as follows: Following Selberg \(\log |\zeta(\tfrac 12 + it)|\) can be decomposed into the sum of a Dirichlet polynomial over primes and a sum over the zeros of \(\zeta(s)\). Soundararajan notices that the zeros make a negative contribution, and therefore we can obtain an upper bound for \(\log |\zeta(\tfrac 12 + it)|\) solely in terms of the Dirichlet polynomial over primes. The frequency with which this Dirichlet polynomial is large can be estimated quite accurately in a very long range. This allows the author to show that the measure of those \(T \leq t \leq 2T\) for which \(\log |\zeta(\tfrac 12 + it)| > V\) is \[ \ll T \exp \bigg ( - (1 + o(1)) \frac{V^2}{\log\log T} \bigg ) \] when \(V < \log\log T \log\log\log T\). For larger \(V\) this bound degenerates into \( \ll T \exp( - c V \log V) \) which is still sufficient for applications to moments. An integration by parts then leads to the desired estimate \(M_k(T) \ll T ( \log T)^{k^2 + \varepsilon}\). We refer the reader to the paper for more precise bounds for the frequency of large values of \(\log |\zeta(\tfrac 12 + it)|\).

The author also derives point-wise bounds for \(\log |\zeta(\tfrac 12 + it)|\) refining previous work of Littlewood. It is shown that, on the Riemann Hypothesis, \[ \log |\zeta(\tfrac 12 + it)| < \exp \bigg ( \big ( \frac{3}{8} + o(1) \big) \frac{\log t}{\log\log t} \bigg ). \] The constant \(3/8\) has been recently improved to \(\log 2 / 2\) by V. Chandee and K. Soundararajan [“Bounding \(|\zeta(\frac 12 + it)|\) on the Riemann Hypothesis”, Bull. Lond. Math. Soc. 43, No. 2, 243–250 (2011; Zbl 1238.11078)].

Reviewer: Maksym Radziwill (Stanford)

### Citations:

Zbl 1051.11048; Zbl 1053.11071; Zbl 1006.11048; Zbl 0920.11060; Zbl 0431.10024; Zbl 0426.10046; Zbl 1221.11177; Zbl 1223.11102; Zbl 1213.11165; Zbl 1219.11136; Zbl 1238.11078; Zbl 1273.11128; Zbl 1290.11120; Zbl 1290.11117
PDFBibTeX
XMLCite

\textit{K. Soundararajan}, Ann. Math. (2) 170, No. 2, 981--993 (2009; Zbl 1251.11058)

### References:

[1] | J. B. Conrey and A. Ghosh, ”On mean values of the zeta-function,” Mathematika, vol. 31, iss. 1, pp. 159-161, 1984. · Zbl 0528.10026 · doi:10.1112/S0025579300010767 |

[2] | 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., vol. 91, iss. 1, pp. 33-104, 2005. · Zbl 1075.11058 · doi:10.1112/S0024611504015175 |

[3] | H. Davenport, Multiplicative Number Theory, 3rd ed., New York: Springer-Verlag, 2000. · Zbl 1002.11001 |

[4] | A. Diaconu, D. Goldfeld, and J. Hoffstein, ”Multiple Dirichlet series and moments of zeta and \(L\)-functions,” Compositio Math., vol. 139, iss. 3, pp. 297-360, 2003. · Zbl 1053.11071 · doi:10.1023/B:COMP.0000018137.38458.68 |

[5] | D. A. Goldston and S. M. Gonek, ”A note on \(S(t)\) and the zeros of the Riemann zeta-function,” Bull. Lond. Math. Soc., vol. 39, iss. 3, pp. 482-486, 2007. · Zbl 1127.11058 · doi:10.1112/blms/bdm032 |

[6] | D. R. Heath-Brown, ”Fractional moments of the Riemann zeta function,” J. London Math. Soc., vol. 24, iss. 1, pp. 65-78, 1981. · Zbl 0431.10024 · doi:10.1112/jlms/s2-24.1.65 |

[7] | D. R. Heath-Brown, ”Fractional moments of the Riemann zeta-function, II,” Quart. J. Math. Oxford Ser., vol. 44, iss. 174, pp. 185-197, 1993. · Zbl 0798.11031 · doi:10.1093/qmath/44.2.185 |

[8] | M. Jutila, ”On the value distribution of the zeta function on the critical line,” Bull. London Math. Soc., vol. 15, iss. 5, pp. 513-518, 1983. · Zbl 0493.10043 · doi:10.1112/blms/15.5.513 |

[9] | N. M. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, Providence, RI: Amer. Math. Soc., 1999. · Zbl 0958.11004 |

[10] | J. P. Keating and N. C. Snaith, ”Random matrix theory and \(\zeta(1/2+it)\),” Comm. Math. Phys., vol. 214, iss. 1, pp. 57-89, 2000. · Zbl 1051.11048 · doi:10.1007/s002200000261 |

[11] | J. P. Keating and N. C. Snaith, ”Random matrix theory and \(L\)-functions at \(s=1/2\),” Comm. Math. Phys., vol. 214, iss. 1, pp. 91-110, 2000. · Zbl 1051.11047 · doi:10.1007/s002200000262 |

[12] | K. Ramachandra, ”Some remarks on the mean value of the Riemann zeta function and other Dirichlet series. I,” Hardy-Ramanujan J., vol. 1, p. 15, 1978. |

[13] | K. Ramachandra, ”Some remarks on the mean value of the Riemann zeta function and other Dirichlet series. II,” Hardy-Ramanujan J., vol. 3, pp. 1-24, 1980. · Zbl 0426.10046 |

[14] | K. Ramachandra, ”Some remarks on the mean value of the Riemann zeta function and other Dirichlet series. III,” Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 5, iss. 1, pp. 145-158, 1980. · Zbl 0448.10031 |

[15] | K. Ramachandra and A. Sankaranarayanan, ”On some theorems of Littlewood and Selberg. I,” J. Number Theory, vol. 44, iss. 3, pp. 281-291, 1993. · Zbl 0778.11047 · doi:10.1006/jnth.1993.1054 |

[16] | Z. Rudnick and K. Soundararajan, ”Lower bounds for moments of \(L\)-functions,” Proc. Natl. Acad. Sci. USA, vol. 102, iss. 19, pp. 6837-6838, 2005. · Zbl 1159.11317 · doi:10.1073/pnas.0501723102 |

[17] | 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, Friedberg, S., Bump, D., Goldfeld, D., and Hoffstein, J., Eds., Providence, RI: Amer. Math. Soc., 2006, pp. 293-303. · Zbl 1120.11039 |

[18] | A. Selberg, ”Contributions to the theory of the Riemann zeta-function,” Arch. Math. Naturvid., vol. 48, iss. 5, pp. 89-155, 1946. · Zbl 0061.08402 |

[19] | A. Selberg, ”Old and new conjectures and results about a class of Dirichlet series,” in Proc. of the Amalfi Conference on Analytic Number Theory, Bombieri, E., Perelli, A., Salerno, S., and Zannier, U., Eds., Salerno: Univ. Salerno, 1992, pp. 367-385. · Zbl 0787.11037 |

[20] | K. Soundararajan, ”Mean-values of the Riemann zeta-function,” Mathematika, vol. 42, iss. 1, pp. 158-174, 1995. · Zbl 0830.11032 · doi:10.1112/S0025579300011438 |

[21] | K. Soundararajan, ”Extreme values of zeta and \(L\)-functions,” Math. Ann., vol. 342, iss. 2, pp. 467-486, 2008. · Zbl 1186.11049 · doi:10.1007/s00208-008-0243-2 |

[22] | E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., New York: Oxford University Press, 1986. · Zbl 0601.10026 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.