A CONJECTURE FOR THE SIXTH POWER MOMENT OF THE RIEMANN ZETA-FUNCTION

J. B. Conrey A. Ghosh

In 1918 Hardy and Littlewood [2] proved that

∫_{1}^{T} |ζ(1/2 + it)|^{2} dt ∼ T log T

and in 1926 Ingham [4] showed that

∫_{1}^{T} |ζ(1/2 + it)|^{4} dt ∼ \frac{1}{2π^{2}} T log^{4} T.

In general, it is conjectured that if k > 0, then there exists a c_{k} > 0 such that

∫_{1}^{T} |ζ(1/2 + it)|^{2k} dt ∼ c_{k}T log^{k^{2}} T.

No value has been suggested for c_{k} if k is different from 0,1, or 2.

In this paper, we present evidence to support the Conjecture. As T → ∞,

∫_{1}^{T} |ζ(1/2 + it)|^{6} dt ∼ \frac{42}{9!} ∏_{p} \{(1 − \frac{1}{p})^{4} (1 + \frac{4}{p} + \frac{1}{p^{2}} )\} T log^{9} T.

SKETCH OF BASIC ARGUMENT

We recall the functional equation of the zeta-function:

ζ(s) = χ(s)ζ(1 − s)

where

χ(1 − s) = χ(s)^{−1} = 2(2π)^{−s}Γ(s) cos \frac{πs}{2}.

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Research of the first author supported by the American Institute of Mathematics. Research of both authors supported in part by a grant from the NSF.

We also require an “approximate” functional equation for ζ(s)^{2}:

ζ(s)^{2} = D(s) + χ(s)^{2}D(1 − s)

where

D(s) = ∑_{n≤ \frac{|t|}{2π}} \frac{d(n_{)}}{n^{s}} + E(s)

with d(n) the usual divisor function and with E(s) a suitable error-term. The estimate

E(1/2 + it) ≪ log(2 + |t|)

is known to hold (see [3]). However, for the purposes of this paper we will not be concerned with E(s).

The beginnings of our argument are as follows.

∫_{0}^{T} |ζ(1/2 + it)|^{6} dt = \frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} ζ(s)^{3}ζ(1 − s)^{3} ds

= \frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} χ(1 − s)ζ(s)^{4}ζ(1 − s)^{2} ds

= \frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} χ(1 − s)ζ(s)^{4} (D(1 − s) + χ(1 − s)^{2}D(s)) ds

= \frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} χ(1 − s)ζ(s)^{4}D(1 − s) ds

+ \frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} χ(1 − s)^{3}ζ(s)^{4}D(s) ds

= I_{1} + I_{2},

say. Now,

I_{2} = \frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} χ(1 − s)^{3}χ(s)^{4}ζ(1 − s)^{4}D(s) ds

= \frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} χ(s)ζ(1 − s)^{4}D(s) ds

= \frac{1}{i} ∫\frac{1}{2} −iT^{\frac{1}{2}} χ(1 − s)ζ(s)^{4}D(1 − s) ds

= \overline{I_{1}}.

Thus,

∫_{1}^{T} |ζ(1/2 + it)|^{6} dt = 2ℜ\frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} χ(1 − s)ζ(s)^{4}D(1 − s) ds.

THEOREM FROM “MEAN-VALUES III”

We appeal to Theorem 2 of Conrey - Ghosh [1] to evalute this integral. We first set up some notation so that we can state a special case of that Theorem. Define D_{N}(s, P) by

D_{N}(s, P) = ∑_{n≤N} \frac{d(n)}{n^{s}}P (log n/ log N)

where P is any real polynomial. Let

\begin{aligned}K_{N}(T) & = ∫_{1}^{T} |ζ(1/2 + it)|^{2}ζ(1/2 + it)^{2}D_{N}(1/2 − it, P) dt \\ & = \frac{1}{i} ∫ \frac{1}{2} +iT_{\frac{1}{2}} χ(1 − s)ζ(s)^{4}D_{N}(1 − s, P) ds. \end{aligned}

Theorem. If N = T^{θ} with 0 < θ < 1/2, then

K_{N}(T) ∼ T (log N)^{9} \frac{a_{3}}{720θ^{3}} ∫_{0}^{1} P (α)α^{5}_{2}F_{1}(−2, −3, 6, −αθ) dα

as T → ∞ where _{2}F_{1} is the usual hypergeometric function and

a_{3} = ∏_{p} \{(1 − \frac{1}{p})^{4} (1 + \frac{4}{p} + \frac{1}{p^{2}} )\} .

The hypergeometric function simplifies to

1 − αθ + (αθ)^{2}/7.

We are interested in the case where P = 1. We find in this case that

\begin{aligned}K_{N}(T) & ∼ \frac{a_{3}}{θ^{3}3!5!} ∫_{0}^{1} α^{5} (1 − αθ + \frac{(αθ)^{2}}{7} ) dαT log^{9} N \\ & = \frac{θ^{6}a_{3}}{720} (\frac{1}{6} − \frac{θ}{7} + \frac{θ^{2}}{56}) T log^{9} T. \end{aligned}

This formula is valid for θ < 1/2. To apply it to our formula from the last section, we would need it to hold for θ = 1. In our paper [1], we expressed the belief that the Theorem from which the above is taken is actually valid for all θ ≤ 1.

If we assume that we can take θ = 1 in this Theorem, then we are immediately led to

∫_{0}^{T} |ζ(1/2 + it)|^{6} dt ∼ 42\frac{a_{3}}{9!} T log^{9} T.

SKETCH OF ANOTHER METHOD

We remark that we can arrive at the same conclusion by another method, which we briefly sketch.

With s = 1/2 + it, we have

∫_{0}^{T} |ζ(1/2 + it)|^{6} dt = ∫_{0}^{T} |ζ(s)|^{2} |ζ(s)^{2}|^{2} dt
= ∫_{0}^{T} |ζ(s)|^{2} |D(s) + χ(s)^{2}D(1 − s)|^{2} dt

= 2 ∫_{0}^{T} |ζ(s)|^{2}|D(s)|^{2} dt + 2ℜ ∫_{0}^{T} |ζ(s)|^{2}χ(1 − s)^{2}D(s)^{2} dt

since |χ(1/2 − it)| = 1.

To evaluate the first integral here we appeal to a special case of Theorem 1 of Conrey - Ghosh [1].

Theorem. Let

J_{N}(T) = ∫_{1}^{T} |ζ(1/2 + it)|^{2}|D_{N}(1/2 + it, P)|^{2}dt

If N = T^{θ} for some θ with 0 < θ < 1/2, and if P is a real polynomial, then

J_{N}(T) ∼ T (log N)^{9} \frac{a_{3}}{24} ∫_{0}^{1} α^{3} (\frac{1}{θ}h′(α)^{2} + 4h(α)h′(α)) dα

as T → ∞ where

h(α) = ∫_{α}^{1}(β − α)^{2}P (β) dβ.

Again this Theorem can be proven for θ < 1/2 and again we expressed the belief in [1] that it actually holds true for θ ≤ 1. Assuming the formula for θ = 1 leads to

2 ∫_{0}^{T} |ζ(s)|^{2}|D(s)|^{2} dt ∼ 28\frac{a_{3}}{9!} T log^{9} T dt.

To handle the second integral we appeal again to the approximate functional equation for ζ(s)^{2}. We find that the second term above is

= 2ℜ ∫_{0}^{T} χ(1 − s)^{3}ζ(s)^{2}D(s)^{2} ds
= 2ℜ ∫_{0}^{T} χ(1 − s)^{3} (D(s) + χ(s)^{2}D(1 − s)) D(s)^{2} ds
= 2ℜ ∫_{0}^{T} χ(1 − s)^{3}D(s)^{3} ds + 2ℜ ∫_{0}^{T} χ(1 − s)D(1 − s)D(s)^{2} ds.

The first integral here is not expected to contribute to the main term, essentially because

χ(1/2 − it)^{3} = exp (3it log \frac{t}{2πe})

is “spinning” too fast. To evaluate the second integral, we proceed as in the proof in [1] of Theorem 2. If the D(1 − s) were replaced by D_{N}(1 − s, 1) with N = T^{θ} and θ < 1/2 then, in a way similar to the proof of Theorem 2, we could obtain an asymptotic evaluation. Again we assume that this asymptotic evaluation is actually correct for all θ ≤ 1. In this way we obtain

2ℜ ∫_{0}^{T} χ(1 − s)D(1 − s)D(s)^{2} ds ∼ 14\frac{a_{3}}{9!} T log^{9} T.

This argument again leads to the conjecture of this paper.

FINAL REMARKS

Conrey and Gonek, in work in progress, have arrived at exactly the same conjecture using a third method. Their method is to consider the asymptotic behavior of “long” Dirichlet polynomials, based on techniques developed by Goldston and Gonek. Their method supposes that asymptotic formulae exist for sums

∑_{n≤x} d_{3}(n)d_{3}(n + h),

where d_{3} can be defined by

ζ(s)^{3} = ∑_{n=1}^{∞} \frac{d_{3}(n)}{n^{s}},

and that the asymptotic formulae for these sums have smooth main-terms and error terms which are bounded on average over h by x^{1/2+ϵ} for h up to x^{1/2+ϵ}.

Finally, we mention that another possibility for testing our conjecture would be through the method recently developed in the thesis of Jose Gaggero Jara (under the direction of S. M. Gonek at the University of Rochester). In that work Jara develops an asymptotic formula for

∫_{0}^{T} |ζ(1/2 + it)|^{4} |∑_{n=1}^{N} a_{n}n^{it}|^{2} dt

for arbitrary positive coefficients a_{n} provided that N = T^{θ} with θ < 4/589. It is probably the case that the formula should actually hold for all θ < 1/2 and possibly even for all θ < 1. With θ = 1/2 and a_{n} = 1, the result should give one-half of the sixth moment. With θ = 1 and a_{n} = 1 it should give all of the sixth moment.

Generally, by taking an approximate functional equation for ζ(1/2 + it) with “uneven” lengths t^{θ} and t^{1−θ} the sum of the results of Jara’s theorem with a_{n} = 1 and N = T^{θ} and N = T^{1−θ} (for any θ < 1) should also give the sixth moment.

References

1. J. B. Conrey and A. Ghosh, Mean values of the Riemann zeta-function, III, Proceedings of the Amalfi Conference on Analytic Number Theory, Università di Salerno, 1992.

2. G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Mathematica 41 (1918), 119 - 196.

3. G. H. Hardy and J. E. Littlewood, The approximate functional equations for ζ(s) and ζ^{2}(s), Proceedings of the London Mathematical Society (2) 29 (1929), 81–97.

4. A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proceedings of the London Mathematical Society 92) 27 (1926), 273–300.

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