We establish sharp upper bounds for the 2kth moment of the Riemann zeta function on the critical line, for all real 0 ⩽ k ⩽ 2⁠. This improves on earlier work of Ramachandra, Heath-Brown and Bettin–Chandee–Radziwiłł

Heap, Winston

Radziwiłł, Maksym

Soundararajan, K.

English

Caltech Authors - Main

ar
X
iv
:1
90
1.
08
42
3v
1 
 [m
ath
.N
T]
  2
4 J
an
 20
19
SHARP UPPER BOUNDS FOR FRACTIONAL MOMENTS OF
THE RIEMANN ZETA FUNCTION
WINSTON HEAP, MAKSYM RADZIWI L L, AND K. SOUNDARARAJAN
Abstract. We establish sharp upper bounds for the 2kth moment of the Riemann
zeta function on the critical line, for all real 0 6 k 6 2. This improves on earlier
work of Ramachandra, Heath-Brown and Bettin-Chandee-Radziwi l l.
1. Introduction
This paper is concerned with the moments of the Riemann zeta function on the
critical line: namely, with the quantity
Ik(T ) =
∫ 2T
T
|ζ(1
2
+ it)|2kdt,
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 Ik(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 Ik(T ) ≫k T (log T )k
2
are established for all k > 1 in
Radziwi l l and Soundararajan [9] unconditionally, and for all k > 0 conditionally on
the Riemann Hypothesis in papers of Heath-Brown and Ramachandra, see [11, 12,
5]. Upper bounds of the form Ik(T ) ≪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 Radziwi l l [2]). Conditionally on the
Riemann Hypothesis, the work of Harper [4], refining earlier work of Soundararajan
[13], establishes that Ik(T ) ≪k T (log T )k
2
for all k > 0. This paper adds to our
knowledge on moments by establishing a sharp upper bound for Ik(T ) for all real
0 6 k 6 2.
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
2 WINSTON HEAP, MAKSYM RADZIWI L L, AND K. SOUNDARARAJAN
Theorem 1. Let 0 6 k 6 2. Then, for T > 10,
Ik(T )≪ T (log T )
k2.
The proof of the theorem is based on the method introduced in Radziwi l l 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 ζ(s) twisted by short Dirichlet polynomials (see the work of Hughes and
Young [7], and Betin, Bui, Li, and Radziwi l l [1]).
2. Plan of the Proof of Theorem 1
Throughout, logj will denote the j-fold iterated logarithm. Let T be large, and let
ℓ denote the largest integer such that logℓ T > 10
4. Define a sequence Tj by setting
T1 = e
2, and for 2 6 j 6 ℓ by
Tj := exp
( log T
(logj T )
2
)
.
Note that e2 = T1 < T2 < . . . < Tℓ 6 T
10−8 .
For each 2 6 j 6 ℓ, set
Pj(s) :=
∑
Tj−16p<Tj
1
ps
, and Pj = Pj(1) =
∑
Tj−16p<Tj
1
p
.
Note that for large T ,
Pj ∼ log
log Tj
log Tj−1
= 2 log
( logj−1 T
logj T
)
= 2 logj T − 2 logj+1 T,
so that Pℓ > 10
4, Pℓ−1 > exp(10
4), and so on. Further, define
(1) Nj(s;α) :=
∑
p|n =⇒ Tj−16p<Tj
Ω(n)6500Pj
αΩ(n)g(n)
ns
where g(n) denotes the multiplicative function given on prime powers by g(pr) = 1/r!.
The motivation for these definitions is the following. Typically one might expect
that ζ(1
2
+ it)α is similar to
∏
j6ℓ exp(αPj(
1
2
+ it)). Now most of the time, |Pj(
1
2
+ it)|
is no more than 50Pj, in which case by a Taylor approximation one can approximate
exp(αPj(
1
2
+ it)) by Nj(
1
2
+ it;α) (see Lemma 1 below). Thus, for most t we shall be
able to replace ζ(1
2
+ it)α by
∏
j6ℓNj(
1
2
+ it;α), which is a short Dirichlet polynomial
(of length 6 T 1/10
UPPER BOUNDS FOR MOMENTS OF THE RIEMANN ZETA-FUNCTION 3
We now state three propositions from which the main theorem will follow, post-
poning the proofs of the propositions to later sections.
Proposition 1. Let 0 6 k 6 2 be a given real number. Then, for all complex
numbers s inside the critical strip 0 < Re s < 1,
|ζ(s)|2k 6 k|ζ(s)|4
∏
26j6ℓ
|Nj(s; k − 2)|
2 + (2− k)
∏
26j6ℓ
|Nj(s; k)|
2
+
∑
26v6ℓ
(
k|ζ(s)|4
∏
26j<v
|Nj(s, k − 2)|
2 + (2− k)
∏
26j<v
|Nj(s; k)|
2
)∣∣∣Pv(s)
50Pv
∣∣∣2⌈50Pv⌉.
Proposition 2. Let 0 6 k 6 2 real, be given. Then
∫ 2T
T
∏
26j6ℓ
|Nj(
1
2
+ it; k)|2dt≪ T (log T )k
2
,
and for all 2 6 v 6 ℓ and 0 6 r 6 ⌈50Pv⌉,∫ 2T
T
∏
26j<v
|Nj(
1
2
+ it; k)|2|Pv(
1
2
+ it)|2rdt≪ T (log Tv−1)
k2
(
r!P rv
)
.
Proposition 3. Let 0 6 k 6 2 real, be given. Then
∫ 2T
T
|ζ(1
2
+ it)|4
∏
26j6ℓ
|Nj(
1
2
+ it; k − 2)|2dt≪ T (log T )k
2
,
and for all 2 6 v 6 ℓ and 0 6 r 6 ⌈50Pv⌉,∫ 2T
T
|ζ(1
2
+ it)|4
∏
26j<v
|Nj(
1
2
+ it; k − 2)|2|Pv(
1
2
+ it)|2rdt
≪ T (log T )4(log Tv−1)
k2−4
(
18rr!P rv exp(Pv)
)
.
We quickly deduce Theorem 1 from the above propositions.
Proof of Theorem 1. Combining the above propositions we find
∫ 2T
T
|ζ(1
2
+ it)|2kdt≪ T (log T )k
2
+
∑
26v6ℓ
T (log Tv−1)
k2
×
(⌈50Pv⌉!P ⌈50Pv⌉v
(50Pv)2⌈50Pv⌉
+
( log T
log Tv−1
)4 18⌈50Pv⌉⌈50Pv⌉!P ⌈50Pv⌉v exp(Pv)
(50Pv)2⌈50Pv⌉
)
.
4 WINSTON HEAP, MAKSYM RADZIWI L L, AND K. SOUNDARARAJAN
A quick calculation shows that the above is
≪ T (log T )k
2
(
1 +
∑
26v6ℓ
( log T
log Tv−1
)4
exp(−50Pv)
)
≪ T (log T )k
2
(
1 +
∑
26v6ℓ
(logv−1 T )
8
( logv T
logv−1 T
)100)
≪ T (log T )k
2
.

3. Proof of Proposition 1
Lemma 1. Let |α| 6 2 be a real number, T be sufficiently large, and s be a complex
number. For all 2 6 j 6 ℓ, if |Pj(s)| 6 50Pj then
exp(2αRe Pj(s)) 6
(
1− e−Pj
)−1
|Nj(s;α)|
2.
Proof. Expanding exp(αPj(s)) using a Taylor series, and using the assumption |Pj(s)| 6
50Pj, we find that
| exp(αPj(s))| 6
∣∣∣ ∑
m6500Pj
αmPj(s)m
m!
∣∣∣+ 2 · (100Pj)500Pj
⌈500Pj⌉!
.
The last term is 6 e−250Pj , while | exp(αPj(s))| > exp(−|α|50Pj) > exp(−100Pj).
Therefore, since Pj > 10
4, we may easily conclude that
| exp(αPj(s))|
2 6 (1− e−Pj)−1
∣∣∣ ∑
m6500Pj
αmPj(s)
m
m!
∣∣∣2.
Since
Pj(s)m
m!
=
1
m!
∑
Tj−16p1,...,pm<Tj
1
(p1 . . . pm)s
=
∑
p|n =⇒ Tj−16p<Tj
Ω(n)=m
g(n)
ns
,
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 6 ap/p + bq/q for any non-
negative real numbers a and b, and non-negative p and q with 1/p+ 1/q = 1.
If |Pj(s)| 6 50Pj for all 2 6 j 6 ℓ then using Young’s inequality with p = 4/2k
and q = 4/(4− 2k) we have
|ζ(s)|2k 6
k
2
|ζ(s)|4
∏
26j6ℓ
e(−4+2k)Re Pj(s) +
(
1−
k
2
) ∏
26j6ℓ
e2kRe Pj(s).
UPPER BOUNDS FOR MOMENTS OF THE RIEMANN ZETA-FUNCTION 5
By Lemma 1 the right hand side is
6
∏
26j6ℓ
(1− e−Pj)−1
(k
2
|ζ(s)|4
∏
26j6ℓ
|Nj(s; k − 2)|
2 +
(
1−
k
2
) ∏
26j6ℓ
|Nj(s; k)|
2
)
.
Since
∏
26j6ℓ(1− e
−Pj )−1 6 2, this contribution is bounded by the first two terms in
the proposition.
Now suppose that there exists an integer 2 6 v 6 ℓ for which |Pj(s)| 6 50Pj
whenever 2 6 j < v, but with |Pv(s)| > 50Pv. Then applying Young’s inequality
and Lemma 1 as before, and noting that |Pv(s)|/(50Pv) > 1, we find
|ζ(s)|2k 6
(
k|ζ(s)|4
∏
26j<v
|Nj(s; k − 2)|
2 + (2− k)
∏
26j<v
|Nj(s; k)|
2
)∣∣∣Pv(s)
50Pv
∣∣∣2⌈50Pv⌉.
Summing this over all 2 6 v 6 ℓ, 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
∏
26j<vNj(s; k)Pv(s)
r is a Dirichlet polynomial of length 6 T 1/10,
using the familiar mean value estimate for Dirichlet polynomials, we find that
∫ 2T
T
∏
26j<v
|Nj(s; k)|
2|Pv(s)|
2rdt
≪ T
∏
26j<v
( ∑
p|nj =⇒ Tj−16p<Tj
Ω(nj)6500Pj
k2Ω(nj)
nj
) ∑
p|n =⇒ Tv−16p<Tv
Ω(n)=r
(r!g(n))2
n
.
Now note that,
∑
p|nj =⇒ Tj−16p<Tj
k2Ω(nj )
nj
6
∏
Tj−16p<Tj
(
1 +
k2
p
+
k4
p2
+ . . .
)
≪
( log Tj
log Tj−1
)k2
,
where we used that p > T1 > e
2 > k2 so that the convergence of
∑∞
r=0 k
2r/pr is
assured. Further, since g(n) 6 1 always,
∑
p|n =⇒ Tv−16p<Tv
Ω(n)=r
(r!g(n))2
n
6 r!
∑
p|n =⇒ Tv−16p<Tv
Ω(n)=r
r!g(n)
n
= r!P rv .
The second assertion of the proposition follows.
6 WINSTON HEAP, MAKSYM RADZIWI L L, AND K. SOUNDARARAJAN
5. Twisted fourth moments
In order to establish Proposition 3 we shall require a formula for the twisted fourth
moment, ∫ 2T
T
|ζ(1
2
+ it)|4 ·
∣∣∣ ∑
n6T θ
a(n)
n1/2+it
∣∣∣2Φ( t
T
)
dt,
where Φ is a smooth non-negative function such that Φ(x) > 1 for 1 6 x 6 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
Az1,z2,z3,z4 =
ζ(1 + z1 + z3)ζ(1 + z1 + z4)ζ(1 + z2 + z3)ζ(1 + z2 + z4)
ζ(2 + z1 + z2 + z3 + z4)
,
and
(2) Bz1,z2,z3,z4(n) =
∏
pnp‖n
(∑
j>0
σz1,z2(p
np+j)σz3,z4(p
j)
pj
)(∑
j>0
σz1,z2(p
j)σz3,z4(p
j)
pj
)−1
where σz1,z2(n) =
∑
n1n2=n
n−z11 n
−z2
2 and np is the highest power of p dividing n.
Finally, define
(3)
F (z1, z2, z3, z4) = Az1,z2,z3,z4
∑
m,n
a(n)a(m)
[m,n]
Bz1,z2,z3,z4
( n
(m,n)
)
Bz3,z4,z1,z2
( m
(m,n)
)
.
Note that F depends on the coefficients of the Dirichlet polynomial twisting the
fourth moment.
Proposition 4. Let T > 2 and let Φ(x) be a smooth function supported on [1/2, 4]
satisfying Φ(j)(x) ≪ε T ε for any j > 0 and all ε > 0. Let a(n) be a sequence of
complex numbers obeying the bound |a(n)| ≪ε nε for all n > 1 and all ε > 0. Then,
for θ < 1
4
, we have∫
R
|ζ(1
2
+ it)|4 ·
∣∣∣ ∑
n6T θ
a(n)
n1/2+it
∣∣∣2Φ( t
T
)
dt = O(T 1−ǫ)+
1
4(2πi)4
∫
|zj |=3j/ log T
16j64
F (z1, z2, z3, z4)∆(z1, z2,−z3,−z4)
2
(∫
R
Φ
( t
T
) 4∏
j=1
( t
2π
)zj/2
dt
) 4∏
j=1
dzj
z4j
where
∆(z1, z2, z3, z4) =
∏
16i<j64
(zj − zi)
denotes the Vandermonde determinant.
UPPER BOUNDS FOR MOMENTS OF THE RIEMANN ZETA-FUNCTION 7
Proof. Theorem 1 in [1] gives an asymptotic formula for∫
R
ζ(1
2
+ α1 + it)ζ(
1
2
+ α2 + it)ζ(
1
2
+ α3 − it)ζ(
1
2
+ α4 − it)
∣∣∣ ∑
n6T θ
a(n)
n1/2+it
∣∣∣2Φ( t
T
)
dt,
with α1, α2, α3, α4 complex numbers of modulus≪ (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 αj 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 ∑
n
a(n)
ns
=
( ∏
26j<v
Nj(s; k − 2)
)
Pv(s)
r,
and taking Φ to be a non-negative smooth function supported on [1/2, 4] with Φ(x) =
1 on [1, 2]. On the circles |zj | = 3j/ log T (for 1 6 j 6 4) we note that
∆(z1, z2,−z3,−z4)
2 ≪ (log T )−12, Az1,z2,z3,z4 ≪ (log T )
4,
and that ∫
R
Φ
( t
T
) 4∏
j=1
( t
2π
)zj/2
dt≪ T.
Therefore by Proposition 4 we conclude that
∫ 2T
T
|ζ(1
2
+ it)|4
∣∣∣∑
n
a(n)
n1/2+it
∣∣∣2dt≪ T (log T )4 · max
|zj |=3j/ log T
|G(z1, z2, z3, z4)|
where
G(z1, z2, z3, z4) =
∑
n,m
a(n)a(m)
[n,m]
Bz1,z2,z3,z4
( n
(n,m)
)
Bz3,z4,z1,z2
( m
(n,m)
)
.
The estimate in Proposition 3 will now follow once we establish the bound
(4) G(z1, z2, z3, z4)≪ (log Tv−1)
k2−4
(
18rr!P rv exp(Pv)
)
,
when |zj| = 3j/ log T for 1 6 j 6 4.
8 WINSTON HEAP, MAKSYM RADZIWI L L, AND K. SOUNDARARAJAN
From the multiplicative nature of the coefficients a, and Bz1,z2,z3,z4, we may express
G(z1, z2, z3, z4) as the product of
(5)∏
26j<v
( ∑
p|n,m =⇒ Tj−16p<Tj
Ω(n),Ω(m)6500Pj
(k − 2)Ω(n)+Ω(m)g(n)g(m)
[n,m]
Bz1,z2,z3,z4
( n
(m,n)
)
Bz3,z4,z1,z2
( m
(m,n)
))
,
and
(6)
∑
p|mn =⇒ Tv−16p≤Tv
Ω(m)=Ω(n)=r
r!2g(m)g(n)
[m,n]
Bz1,z2,z3,z4
( n
(n,m)
)
Bz3,z4,z1,z2
( m
(m,n)
)
.
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 6 T 10
−8
and |zj | = 3j/ log T
(7) Bz1,z2,z3,z4(p
u) = σz1,z2(p
u)
(
1 +O
(1
p
))
,
from which we may deduce that
(8) |Bz1,z2,z3,z4(n)| ≪ d3(n) 6 3
Ω(n),
for integers n composed only of primes below T 10
−8
, and where d3 denotes the 3–
divisor function.
Consider first the expression in (6). Using (8) we have |Bz1,z2,z3,z4(n/(n,m))| ≪ 3
r
and |Bz3,z4,z1,z2(m/(n,m))| ≪ 3
r, and so the quantity in (6) is
≪ 9r
∑
p|mn =⇒ Tv−16p<Tv
Ω(m)=Ω(n)=r
r!2g(m)g(n)
[m,n]
6 9rr!2
r∑
j=0
∑
p|d =⇒ Tv−16p<Tv
Ω(d)=j
1
d
( ∑
p|n =⇒ Tv−16p<Tv
Ω(n)=r−j
g(nd)
n
)2
.
Since g(nd) 6 g(n), the above may be bounded by
(9)
6 9rr!2
r∑
j=0
( 1
j!
P jv
)( 1
(r − j)!
P r−jv
)2
= 9rr!P rv
r∑
j=0
(
r
j
)
P r−jv
(r − j)!
6 18rr!P rv exp(Pv),
upon noting that
(
r
j
)
6 2r and
∑r
j=0 P
r−j
v /(r − j)! 6 exp(Pv).
Now we turn to the expression in (5), treating the contribution for a given j in the
range 2 6 j < v. First we show that the constraints Ω(n) and Ω(m) 6 500Pj may
be dropped from the expression there with negligible error. We bound these terms
UPPER BOUNDS FOR MOMENTS OF THE RIEMANN ZETA-FUNCTION 9
using Rankin’s trick, in the form exp(Ω(m) + Ω(n) − 500Pj) > 1 if either Ω(m) or
Ω(n) exceeds 500Pj. By (8) and since |k− 2| 6 2, the error induced in dropping the
constraint on Ω(m) and Ω(n) is
6 e−500Pj
∑
p|m,n =⇒ Tj−16p<Tj
(2e)Ω(m)+Ω(n)
[m,n]
d3(m)d3(n)
≪ e−500Pj
∏
Tj−16p<Tj
(
1 +
6e+ 6e+ (6e)2
p
+O
( 1
p2
))
≪ e−100Pj .
After discarding the constraint on Ω(m) and Ω(n), the contribution of the term in
(5) is
∏
Tj−16p<Tj
( ∞∑
a,b=0
(k − 2)a+bg(pa)g(pb)
pmax(a,b)
Bz1,z2,z3,z4(p
a−min(a,b))Bz3,z4,z1,z2(p
b−min(a,b))
)
.
Upon using (7), we see that only the terms a, b = 0, or 1 are relevant and the total
contribution is∏
Tj−16p<Tj
(
1 +
(k − 2)(σz1,z2(p) + σz3,z4(p)) + (k − 2)
2
p
+O
( 1
p2
))
=
∏
Tj−16p<Tj
(
1 +
k2 − 4
p
+O
( 1
p2
+
log p
p log T
))
,
since σz1,z2(p) = p
−z1 + p−z2 = 2 + O(log p/ log T ), and similarly for σz3,z4(p). We
conclude that the expression in (5) equals
∏
26j<v
( ∏
Tj−16p<Tj
(
1 +
k2 − 4
p
+O
( 1
p2
+
log p
p log T
))
+O(e−100Pj)
)
≪ (log Tv−1)
k2−4.
Combining this estimate with (9), the bound (4) follows, and with it the proof of
Proposition 3 is complete.
References
[1] S. Bettin, H. M. Bui, X. Li, M. Radziwi l l, A quadratic divisor problem and moments of the
Riemann zeta-function, preprint, available at arXiv.1609.02539.
[2] S. Bettin, V. Chandee, M. Radziwi l l, The mean square of the product of ζ(s) with Dirichlet
polynomials, J. Reine Angew. Math, 729 (2017), 51–79.
[3] J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, N. C. Snaith, Integral moments
of L-functions, Proc. London Math. Soc. 91 no. 3 (2005), 33–104
[4] A. Harper, Sharp conditional bounds for moments of the Riemann zeta function. Preprint avail-
able at arXiv.1305.4618.
10 WINSTON HEAP, MAKSYM RADZIWI L L, AND K. SOUNDARARAJAN
[5] D. R. Heath-Brown, Fractional moments of the Riemann zeta function, J. London Math. Soc.,
24, no. 1 (1981), 65–78.
[6] G.H. Hardy, J.E. Littlewood, Contributions to the theory of the Riemann zeta-function and the
theory of the distribution of primes, Acta Arith. 41 (1918), 119–196.
[7] C. P. Hughes, M. P. Young, The twisted fourth moment of the Riemann zeta function, J. Reine
Angew. Math. 641 (2010), 203–236.
[8] A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta function, Proc. London
Math. Soc. 27 (1926), 273–300.
[9] M. Radziwi l l, K. Soundararajan, Continuous lower bounds for moments of zeta and L-functions,
Mathematika, 59 no. 1 (2013), 119–128.
[10] M. Radziwi l l, K. Soundararajan, Moments and distribution of central L-values of quadratic
twists of elliptic curves, Invent. Math. 202 no. 3 (2015) 1029–1068
[11] K. Ramachandra, Some remarks on the mean value of the Riemann zeta function and other
Dirichlet series, Ann. Acad. Sci. Fennicae, 5 (1980), 145–158
[12] K. Ramachandra. Some remarks on the mean value of the Riemann zeta function and other
Dirichlet series. II Hardy-Ramanujan J., 3 (1980), 1-24.
[13] K. Soundararajan, Moments of the Riemann zeta function, Annals of Math. 170 no. 2 (2009),
981–993.
Department of Mathematics, University College London, 25 Gordon Street, Lon-
don WC1H.
E-mail address : winstonheap@gmail.com
Department of Mathematics Caltech, 1200 E California Blvd Pasadena, CA, 91125
E-mail address : maksym.radziwill@gmail.com
Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
E-mail address : ksound@stanford.edu


Sharp Upper Bounds for Fractional Moments of the Riemann Zeta Function

https://authors.library.caltech.edu/101259/1/1901.08423.pdf

Sharp Upper Bounds for Fractional Moments of the Riemann Zeta Function

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main