Sharp upper bounds for fractional moments of the Riemann zeta function

TL;DR

This paper derives precise upper bounds for the fractional moments of the Riemann zeta function on the critical line for all real exponents between 0 and 2, advancing previous results in the field.

Contribution

It provides sharp upper bounds for the $2k$th moment of the Riemann zeta function for all real $k$ in [0,2], improving earlier bounds by Ramachandra, Heath-Brown, and Bettin-Chandee-Radziwi extlangle.

Findings

Established sharp upper bounds for fractional moments of zeta on the critical line.

Extended bounds to all real $k$ in [0,2], filling a gap in the literature.

Improved upon previous bounds by notable researchers in the field.

Abstract

We establish sharp upper bounds for the $2k$th moment of the Riemann zeta function on the critical line, for all real $0 \leqslant k \leqslant 2$. This improves on earlier work of Ramachandra, Heath-Brown and Bettin-Chandee-Radziwi\l\l

Equations

$$I_k(T)={\int }_T^{2T}|\zeta (\frac{1}{2}+it){|}^{2k}dt,$$

$$I_k(T)\ll T(\log T{)}^{k^2}.$$

$$T_{j}:=\exp\Big{(}\frac{\log T}{(\log_{j}T)^{2}}\Big{)}.$$

$${\mathcal{P}}_j(s):=\sum _{T_{j-1}⩽p<T_j}\frac{1}{p^s},\text{ and }{P}_j={\mathcal{P}}_j(1)=\sum _{T_{j-1}⩽p<T_j}\frac{1}{p}.$$

$$P_{j}\sim\log\frac{\log T_{j}}{\log T_{j-1}}=2\log\Big{(}\frac{\log_{j-1}T}{\log_{j}T}\Big{)}=2\log_{j}T-2\log_{j+1}T,$$

$${\mathcal{N}}_j(s;\alpha ):=\sum _{\begin{array}{c}p|n⟹T_{j-1}⩽p<T_j\\ \Omega (n)⩽500P_j\end{array}}\frac{{\alpha }^{\Omega (n)}g(n)}{n^s}$$

$$|\zeta (s){|}^{2k}$$

$$\displaystyle+\sum_{2\leqslant v\leqslant\ell}\Big{(}k|\zeta(s)|^{4}\prod_{2\leqslant j<v}|\mathcal{N}_{j}(s,k-2)|^{2}+(2-k)\prod_{2\leqslant j<v}|\mathcal{N}_{j}(s;k)|^{2}\Big{)}\Big{|}\frac{\mathcal{P}_{v}(s)}{50P_{v}}\Big{|}^{2\lceil 50P_{v}\rceil}.$$

$${\int }_T^{2T}\prod _{2⩽j⩽\ell }|{\mathcal{N}}_j(\frac{1}{2}+it;k){|}^{2}dt\ll T(\log T{)}^{k^2},$$

$$\int_{T}^{2T}\prod_{2\leqslant j<v}|\mathcal{N}_{j}(\tfrac{1}{2}+it;k)|^{2}|\mathcal{P}_{v}(\tfrac{1}{2}+it)|^{2r}dt\ll T(\log T_{v-1})^{k^{2}}\big{(}r!P_{v}^{r}\big{)}.$$

$${\int }_T^{2T}|\zeta (\frac{1}{2}+it){|}^4\prod _{2⩽j⩽\ell }|{\mathcal{N}}_j(\frac{1}{2}+it;k-2){|}^{2}dt\ll T(\log T{)}^{k^2},$$

$${\int }_T^{2T}|\zeta (\frac{1}{2}+it){|}^4$$

$$\displaystyle\ll T(\log T)^{4}(\log T_{v-1})^{k^{2}-4}\Big{(}18^{r}r!P_{v}^{r}\exp(P_{v})\Big{)}.$$

$${\int }_T^{2T}|\zeta (\frac{1}{2}+it){|}^{2k}dt$$

$$\displaystyle\times\Big{(}\frac{\lceil 50P_{v}\rceil!P_{v}^{\lceil 50P_{v}\rceil}}{(50P_{v})^{2\lceil 50P_{v}\rceil}}+\Big{(}\frac{\log T}{\log T_{v-1}}\Big{)}^{4}\frac{18^{\lceil 50P_{v}\rceil}\lceil 50P_{v}\rceil!P_{v}^{\lceil 50P_{v}\rceil}\exp(P_{v})}{(50P_{v})^{2\lceil 50P_{v}\rceil}}\Big{)}.$$

$$\displaystyle\ll T(\log T)^{k^{2}}\Big{(}1+\sum_{2\leqslant v\leqslant\ell}\Big{(}\frac{\log T}{\log T_{v-1}}\Big{)}^{4}\exp(-50P_{v})\Big{)}$$

$$\displaystyle\ll T(\log T)^{k^{2}}\Big{(}1+\sum_{2\leqslant v\leqslant\ell}(\log_{v-1}T)^{8}\Big{(}\frac{\log_{v}T}{\log_{v-1}T}\Big{)}^{100}\Big{)}\ll T(\log T)^{k^{2}}.$$

$$\exp (2\alpha \text{Re }{\mathcal{P}}_j(s))⩽{\left(1-e^{-P_j}\right)}^{-1}|{\mathcal{N}}_j(s;\alpha ){|}^2.$$

$$|\exp(\alpha{\mathcal{P}}_{j}(s))|\leqslant\Big{|}\sum_{m\leqslant 500P_{j}}\frac{\alpha^{m}{\mathcal{P}}_{j}(s)^{m}}{m!}\Big{|}+2\cdot\frac{(100P_{j})^{500P_{j}}}{\lceil 500P_{j}\rceil!}.$$

$$|\exp(\alpha\mathcal{P}_{j}(s))|^{2}\leqslant(1-e^{-P_{j}})^{-1}\Big{|}\sum_{m\leqslant 500P_{j}}\frac{\alpha^{m}\mathcal{P}_{j}(s)^{m}}{m!}\Big{|}^{2}.$$

$$\frac{{\mathcal{P}}_j(s{)}^m}{m!}=\frac{1}{m!}\sum _{T_{j-1}⩽p_1,\dots ,p_m<T_j}\frac{1}{(p_1\dots p_m{)}^s}=\sum _{\begin{array}{c}p|n⟹T_{j-1}⩽p<T_j\\ \Omega (n)=m\end{array}}\frac{g(n)}{n^s},$$

$$|\zeta(s)|^{2k}\leqslant\frac{k}{2}|\zeta(s)|^{4}\prod_{2\leqslant j\leqslant\ell}e^{(-4+2k)\text{Re }\mathcal{P}_{j}(s)}+\Big{(}1-\frac{k}{2}\Big{)}\prod_{2\leqslant j\leqslant\ell}e^{2k\text{Re }\mathcal{P}_{j}(s)}.$$

$$\leqslant\prod_{2\leqslant j\leqslant\ell}(1-e^{-P_{j}})^{-1}\Big{(}\frac{k}{2}|\zeta(s)|^{4}\prod_{2\leqslant j\leqslant\ell}|\mathcal{N}_{j}(s;k-2)|^{2}+\Big{(}1-\frac{k}{2}\Big{)}\prod_{2\leqslant j\leqslant\ell}|\mathcal{N}_{j}(s;k)|^{2}\Big{)}.$$

$$|\zeta (s){|}^{2k}$$

$${\int }_T^{2T}$$

$$\displaystyle\ll T\prod_{2\leqslant j<v}\Big{(}\sum_{\begin{subarray}{c}p|n_{j}\implies T_{j-1}\leqslant p<T_{j}\\ \Omega(n_{j})\leqslant 500P_{j}\end{subarray}}\frac{k^{2\Omega(n_{j})}}{n_{j}}\Big{)}\sum_{\begin{subarray}{c}p|n\implies T_{v-1}\leqslant p<T_{v}\\ \Omega(n)=r\end{subarray}}\frac{(r!g(n))^{2}}{n}.$$

$$\sum_{\begin{subarray}{c}p|n_{j}\implies T_{j-1}\leqslant p<T_{j}\end{subarray}}\frac{k^{2\Omega(n_{j})}}{n_{j}}\leqslant\prod_{T_{j-1}\leqslant p<T_{j}}\Big{(}1+\frac{k^{2}}{p}+\frac{k^{4}}{p^{2}}+\ldots\Big{)}\ll\Big{(}\frac{\log T_{j}}{\log T_{j-1}}\Big{)}^{k^{2}},$$

$$\sum _{\begin{array}{c}p|n⟹T_{v-1}⩽p<T_v\\ \Omega (n)=r\end{array}}\frac{(r!g(n){)}^2}{n}⩽r!\sum _{\begin{array}{c}p|n⟹T_{v-1}⩽p<T_v\\ \Omega (n)=r\end{array}}\frac{r!g(n)}{n}=r!P_v^r.$$

$$\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it)|^{4}\cdot\Big{|}\sum_{n\leqslant T^{\theta}}\frac{a(n)}{n^{1/2+it}}\Big{|}^{2}\Phi\Big{(}\frac{t}{T}\Big{)}dt,$$

$$A_{z_1,z_2,z_3,z_4}=\frac{\zeta (1+z_1+z_3)\zeta (1+z_1+z_4)\zeta (1+z_2+z_3)\zeta (1+z_2+z_4)}{\zeta (2+z_1+z_2+z_3+z_4)},$$

$$B_{z_{1},z_{2},z_{3},z_{4}}(n)=\prod_{p^{n_{p}}\|n}\Big{(}\sum_{j\geqslant 0}\frac{\sigma_{z_{1},z_{2}}(p^{n_{p}+j})\sigma_{z_{3},z_{4}}(p^{j})}{p^{j}}\Big{)}\Big{(}\sum_{j\geqslant 0}\frac{\sigma_{z_{1},z_{2}}(p^{j})\sigma_{z_{3},z_{4}}(p^{j})}{p^{j}}\Big{)}^{-1}$$

$$F(z_{1},z_{2},z_{3},z_{4})=A_{z_{1},z_{2},z_{3},z_{4}}\sum_{m,n}\frac{a(n)\overline{a(m)}}{[m,n]}B_{z_{1},z_{2},z_{3},z_{4}}\Big{(}\frac{n}{(m,n)}\Big{)}B_{z_{3},z_{4},z_{1},z_{2}}\Big{(}\frac{m}{(m,n)}\Big{)}.$$

$${\int }_{\mathbb{R}}$$

Full text

The first author is supported by European Research Council grant no.  670239. The second author acknowledges the support of a Sloan fellowship. The third author is partially supported by a grant from the National Science Foundation, and by a Simons Investigator grant from the Simons Foundation

1. Introduction

This paper is concerned with the moments of the Riemann zeta function on the critical line: namely, with the quantity

[TABLE]

where $k>0$ is real and $T$ is large. The problem of understanding the behavior of these moments is central in the theory of the Riemann zeta-function. The classical work of Hardy and Littlewood [6], and Ingham [8] established asymptotic formulae for $I_{k}(T)$ in the cases $k=1$ and $2$, and these still remain the only situations where an asymptotic is known. Lacking an asymptotic, much work has been focussed on the problems of obtaining sharp upper and lower bounds for these moments. Lower bounds of the form $I_{k}(T)\gg_{k}T(\log T)^{k^{2}}$ are established for all $k\geqslant 1$ in Radziwiłł and Soundararajan [9] unconditionally, and for all $k\geqslant 0$ conditionally on the Riemann Hypothesis in papers of Heath-Brown and Ramachandra, see [11, 12, 5]. Upper bounds of the form $I_{k}(T)\ll_{k}T(\log T)^{k^{2}}$ are known when $k=1/n$ for natural numbers $n$ (due to Heath-Brown [5]) and when $k=1+1/n$ for natural numbers $n$ (by work of Bettin, Chandee, and Radziwiłł [2]). Conditionally on the Riemann Hypothesis, the work of Harper [4], refining earlier work of Soundararajan [13], establishes that $I_{k}(T)\ll_{k}T(\log T)^{k^{2}}$ for all $k\geqslant 0$. This paper adds to our knowledge on moments by establishing a sharp upper bound for $I_{k}(T)$ for all real $0\leqslant k\leqslant 2$.

Theorem 1**.**

Let $0\leqslant k\leqslant 2$. Then, for $T\geqslant 10$,

[TABLE]

The proof of the theorem is based on the method introduced in Radziwiłł and Soundararajan [10] which enunciates that if in a family of $L$-values, asymptotics for a particular moment can be established with a little room to spare, then sharp upper bounds may be obtained for all smaller moments. Theorem 1 is an illustration of this principle, and combines the ideas of [10] together with knowledge of the fourth moment of $\zeta(s)$ twisted by short Dirichlet polynomials (see the work of Hughes and Young [7], and Betin, Bui, Li, and Radziwiłł [1]).

2. Plan of the Proof of Theorem 1

Throughout, $\log_{j}$ will denote the $j$-fold iterated logarithm. Let $T$ be large, and let $\ell$ denote the largest integer such that $\log_{\ell}T\geqslant 10^{4}$. Define a sequence $T_{j}$ by setting $T_{1}=e^{2}$, and for $2\leqslant j\leqslant\ell$ by

[TABLE]

Note that $e^{2}=T_{1}<T_{2}<\ldots<T_{\ell}\leqslant T^{10^{-8}}$.

For each $2\leqslant j\leqslant\ell$, set

[TABLE]

Note that for large $T$,

[TABLE]

so that $P_{\ell}\geqslant 10^{4}$, $P_{\ell-1}\geqslant\exp(10^{4})$, and so on. Further, define

[TABLE]

where $g(n)$ denotes the multiplicative function given on prime powers by $g(p^{r})=1/r!$.

The motivation for these definitions is the following. Typically one might expect that $\zeta(\frac{1}{2}+it)^{\alpha}$ is similar to $\prod_{j\leqslant\ell}\exp(\alpha{\mathcal{P}}_{j}(\frac{1}{2}+it))$. Now most of the time, $|{\mathcal{P}}_{j}(\tfrac{1}{2}+it)|$ is no more than $50P_{j}$, in which case by a Taylor approximation one can approximate $\exp(\alpha{\mathcal{P}}_{j}(\frac{1}{2}+it))$ by ${\mathcal{N}}_{j}(\frac{1}{2}+it;\alpha)$ (see Lemma 1 below). Thus, for most $t$ we shall be able to replace $\zeta(\frac{1}{2}+it)^{\alpha}$ by $\prod_{j\leqslant\ell}{\mathcal{N}}_{j}(\frac{1}{2}+it;\alpha)$, which is a short Dirichlet polynomial (of length $\leqslant T^{1/10}$, say) and thus facilitates computations.

We now state three propositions from which the main theorem will follow, postponing the proofs of the propositions to later sections.

Proposition 1**.**

Let $0\leqslant k\leqslant 2$ be a given real number. Then, for all complex numbers $s$ inside the critical strip $0<\text{Re }s<1$,

[TABLE]

Proposition 2**.**

Let $0\leqslant k\leqslant 2$ real, be given. Then

[TABLE]

and for all $2\leqslant v\leqslant\ell$ and $0\leqslant r\leqslant\lceil 50P_{v}\rceil$,

[TABLE]

Proposition 3**.**

Let $0\leqslant k\leqslant 2$ real, be given. Then

[TABLE]

and for all $2\leqslant v\leqslant\ell$ and $0\leqslant r\leqslant\lceil 50P_{v}\rceil$,

[TABLE]

We quickly deduce Theorem 1 from the above propositions.

Proof of Theorem 1.

Combining the above propositions we find

[TABLE]

A quick calculation shows that the above is

[TABLE]

∎

3. Proof of Proposition 1

Lemma 1**.**

Let $|\alpha|\leqslant 2$ be a real number, $T$ be sufficiently large, and $s$ be a complex number. For all $2\leqslant j\leqslant\ell$, if $|\mathcal{P}_{j}(s)|\leqslant 50P_{j}$ then

[TABLE]

Proof.

Expanding $\exp(\alpha{\mathcal{P}}_{j}(s))$ using a Taylor series, and using the assumption $|\mathcal{P}_{j}(s)|\leqslant 50P_{j}$, we find that

[TABLE]

The last term is $\leqslant e^{-250P_{j}}$, while $|\exp(\alpha{\mathcal{P}}_{j}(s))|\geqslant\exp(-|\alpha|50P_{j})\geqslant\exp(-100P_{j})$. Therefore, since $P_{j}\geqslant 10^{4}$, we may easily conclude that

[TABLE]

Since

[TABLE]

the proposition follows. ∎

Proof of Proposition 1.

This proposition is an analogue of Lemma 2 of [10], and is proved similarly. We make use of Young’s inequality $ab\leqslant a^{p}/p+b^{q}/q$ for any non-negative real numbers $a$ and $b$, and non-negative $p$ and $q$ with $1/p+1/q=1$.

If $|\mathcal{P}_{j}(s)|\leqslant 50P_{j}$ for all $2\leqslant j\leqslant\ell$ then using Young’s inequality with $p=4/2k$ and $q=4/(4-2k)$ we have

[TABLE]

By Lemma 1 the right hand side is

[TABLE]

Since $\prod_{2\leqslant j\leqslant\ell}(1-e^{-P_{j}})^{-1}\leqslant 2$, this contribution is bounded by the first two terms in the proposition.

Now suppose that there exists an integer $2\leqslant v\leqslant\ell$ for which $|\mathcal{P}_{j}(s)|\leqslant 50P_{j}$ whenever $2\leqslant j<v$, but with $|\mathcal{P}_{v}(s)|>50P_{v}$. Then applying Young’s inequality and Lemma 1 as before, and noting that $|\mathcal{P}_{v}(s)|/(50P_{v})\geqslant 1$, we find

[TABLE]

Summing this over all $2\leqslant v\leqslant\ell$, we obtain Proposition 1. ∎

4. Proof of Proposition 2

We give a proof of the second assertion of the proposition, the first statement being similar. Since $\prod_{2\leqslant j<v}\mathcal{N}_{j}(s;k)\mathcal{P}_{v}(s)^{r}$ is a Dirichlet polynomial of length $\leqslant T^{1/10}$, using the familiar mean value estimate for Dirichlet polynomials, we find that

[TABLE]

Now note that,

[TABLE]

where we used that $p\geqslant T_{1}\geqslant e^{2}>k^{2}$ so that the convergence of $\sum_{r=0}^{\infty}k^{2r}/p^{r}$ is assured. Further, since $g(n)\leqslant 1$ always,

[TABLE]

The second assertion of the proposition follows.

5. Twisted fourth moments

In order to establish Proposition 3 we shall require a formula for the twisted fourth moment,

[TABLE]

where $\Phi$ is a smooth non-negative function such that $\Phi(x)\geqslant 1$ for $1\leqslant x\leqslant 2$. Such mean values have been considered by many authors (for example see [7]), and we shall make use of the asymptotic established in [1].

To state the asymptotic formula, we introduce some notation. Put

[TABLE]

and

[TABLE]

where $\sigma_{z_{1},z_{2}}(n)=\sum_{n_{1}n_{2}=n}n_{1}^{-z_{1}}n_{2}^{-z_{2}}$ and $n_{p}$ is the highest power of $p$ dividing $n$. Finally, define

[TABLE]

Note that $F$ depends on the coefficients of the Dirichlet polynomial twisting the fourth moment.

Proposition 4**.**

Let $T\geqslant 2$ and let $\Phi(x)$ be a smooth function supported on $[1/2,4]$ satisfying $\Phi^{(j)}(x)\ll_{\varepsilon}T^{\varepsilon}$ for any $j\geqslant 0$ and all $\varepsilon>0$. Let $a(n)$ be a sequence of complex numbers obeying the bound $|a(n)|\ll_{\varepsilon}n^{\varepsilon}$ for all $n\geqslant 1$ and all $\varepsilon>0$. Then, for $\theta<\tfrac{1}{4}$, we have

[TABLE]

where

[TABLE]

denotes the Vandermonde determinant.

Proof.

Theorem 1 in [1] gives an asymptotic formula for

[TABLE]

with $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ complex numbers of modulus $\ll(\log T)^{-1}$. We apply Lemma 2.5.1 of [3] to express that formula in terms of a multiple contour integral. Setting all the shifts $\alpha_{j}$ equal to zero then gives the claim. ∎

6. Proof of Proposition 3

Again we confine ourselves to proving the second assertion of the proposition; the first statement follows similarly. We apply Proposition 4 with coefficients $a(n)$ given by

[TABLE]

and taking $\Phi$ to be a non-negative smooth function supported on $[1/2,4]$ with $\Phi(x)=1$ on $[1,2]$. On the circles $|z_{j}|=3^{j}/\log T$ (for $1\leqslant j\leqslant 4$) we note that

[TABLE]

and that

[TABLE]

Therefore by Proposition 4 we conclude that

[TABLE]

where

[TABLE]

The estimate in Proposition 3 will now follow once we establish the bound

[TABLE]

when $|z_{j}|=3^{j}/\log T$ for $1\leqslant j\leqslant 4$.

From the multiplicative nature of the coefficients $a$, and $B_{z_{1},z_{2},z_{3},z_{4}}$, we may express $G(z_{1},z_{2},z_{3},z_{4})$ as the product of

[TABLE]

and

[TABLE]

We now estimate the quantities in (5) and (6). To do this, it is helpful to note that from the definition (2) one has for $p\leqslant T^{10^{-8}}$ and $|z_{j}|=3^{j}/\log T$

[TABLE]

from which we may deduce that

[TABLE]

for integers $n$ composed only of primes below $T^{10^{-8}}$, and where $d_{3}$ denotes the $3$–divisor function.

Consider first the expression in (6). Using (8) we have $|B_{z_{1},z_{2},z_{3},z_{4}}(n/(n,m))|\ll 3^{r}$ and $|B_{z_{3},z_{4},z_{1},z_{2}}(m/(n,m))|\ll 3^{r}$, and so the quantity in (6) is

[TABLE]

Since $g(nd)\leqslant g(n)$, the above may be bounded by

[TABLE]

upon noting that $\binom{r}{j}\leqslant 2^{r}$ and $\sum_{j=0}^{r}P_{v}^{r-j}/(r-j)!\leqslant\exp(P_{v})$.

Now we turn to the expression in (5), treating the contribution for a given $j$ in the range $2\leqslant j<v$. First we show that the constraints $\Omega(n)$ and $\Omega(m)\leqslant 500P_{j}$ may be dropped from the expression there with negligible error. We bound these terms using Rankin’s trick, in the form $\exp(\Omega(m)+\Omega(n)-500P_{j})\geqslant 1$ if either $\Omega(m)$ or $\Omega(n)$ exceeds $500P_{j}$. By (8) and since $|k-2|\leqslant 2$, the error induced in dropping the constraint on $\Omega(m)$ and $\Omega(n)$ is

[TABLE]

After discarding the constraint on $\Omega(m)$ and $\Omega(n)$, the contribution of the term in (5) is

[TABLE]

Upon using (7), we see that only the terms $a,b=0$, or $1$ are relevant and the total contribution is

[TABLE]

since $\sigma_{z_{1},z_{2}}(p)=p^{-z_{1}}+p^{-z_{2}}=2+O(\log p/\log T)$, and similarly for $\sigma_{z_{3},z_{4}}(p)$. We conclude that the expression in (5) equals

[TABLE]

Combining this estimate with (9), the bound (4) follows, and with it the proof of Proposition 3 is complete.