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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)].

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M50 Relations with random matrices
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##### References:
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