Moments of the Riemann zeta function.

*(English)*Zbl 1251.11058The \(2k\)-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)

##### References:

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