PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114)) (2016), 229–241 DOI: 10.2298/PIM1614229K

THE FOURTH POWER MOMENT OF THE DOUBLE ZETA-FUNCTION

Isao Kiuchi

Abstract. We prove the fourth power moment of the Euler–Zagier type double zeta-function ζ_{2}(s_{1}, s_{2}) and provide an improvement on the Ω results of Kiuchi, Tanigawa, and Zhai. We also calculate the double integral under certain conditions.

1. Introduction

Let s_{j} = σ_{j} + it_{j} (σ_{j} , t_{j} ∈ ℝ, j = 1, 2) be complex variables, and let ζ(s) be

the Riemann zeta function, which is defined as ∑ _{n=1}^{∞} n^{−s} for Re s > 1. The double zeta-function of the Euler–Zagier type is defined by

ζ_{2}(s_{1}, s_{2}) = ∑_{1⩽m<n} \frac{1}{m^{s_{1}} n^{s_{2}}} , (1.1)

which is absolutely convergent for σ_{2} > 1 and σ_{1} + σ_{2} > 2. The proof of Atkinson’s formula for the mean value theorem of the Riemann zeta-function ζ(s) (see Atkinson [2] or Ivić [7]) is applied applies to series (1.1). Double zeta-function (1.1) has many applications to mathematical physics. In particular, some algebraic relations among the values of double zeta-function (1.1) at positive integers have been extensively studied by Ohno [17]. Some analytic properties of this function have been obtained by Akiyama, Egami, and Tanigawa [1], Ishikawa and Matsumoto [6], Kiuchi and Tanigawa [12], Kiuchi, Tanigawa, and Zhai [13], Matsumoto [14, 15], Zhao [19], and others.

1.1. Mean square formula. Matsumoto and Tsumura [16] were the first to study a new type of the mean value formula for ∫_{2}^{T} |ζ(s_{1}, s_{2})|^{2}dt_{2} with a fixed complex number s_{1}. They conjectured that when σ_{1} + σ_{2} = \frac{3}{2}, the form of the main term of the mean square formula would not be CT with a constant C, and that, most probably, some log-factor would appear. Their results were considered by Ikeda, Matsuoka, and Nagata [5], who showed that

2010 Mathematics Subject Classification: Primary 11M41; Secondary 11M06.

Key words and phrases: double zeta-functions, fourth power moment, Riemann zeta-function.

Communicated by Žarko Mijajlović.

∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{2}dt_{1} = ( ∑_{m=1}^{∞} \frac{1}{m^{2σ_{1}}} |ζ(s_{2}) − ∑_{n=1}^{m} \frac{1}{n^{s_{2}}} |^{2}) T

+ \{ \begin{matrix}O(log^{2} T) & if σ_{1} + σ_{2} = 2, \\ O(T^{4−2σ_{1} −2σ_{2}} ) & if \frac{3}{2} < σ_{1} + σ_{2} < 2. \end{matrix} (1.2)

Here, the coefficient of the main term on the right-hand side of (1.2) converges if σ_{1} + σ_{2} > \frac{3}{2}. Ikeda, Matsuoka, and Nagata deduced that the asymptotic formula

(1.3)  ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{2}dt_{1} = \frac{1}{|s_{2} − 1|^{2}} T logT + O(T)

holds on the line σ_{1}+σ_{2} = \frac{3}{2}. This result implied that the conjecture of Matsumoto and Tsumura on the line σ_{1} +σ_{2} = \frac{3}{2} was true. Ikeda, Matsuoka, and Nagata made use of the mean value theorem for Dirichlet polynomials and suitable approximations to the Euler–Maclaurin summation formula to obtain formulas (1.2) and (1.3). Assuming that 2 ⩽ t_{1} ⩽ T, 0 < σ_{1} < 1, 0 < σ_{2} < 1 and 0 < σ_{1} + σ_{2} < \frac{3}{2}, Kiuchi and Minamide [11] recently considered five formulas for the mean square of the double zeta-function ζ_{2}(s_{1}, s_{2}) concerning the variable t_{1} and showed that, for any sufficiently large positive number T > 2,

∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{2}dt_{1} = (2π)^{2σ_{1} +2σ_{2} −3} \frac{ζ(4 − 2σ_{1} − 2σ_{2})}{(4 − 2σ_{1} − 2σ_{2})|s_{2} − 1|^{2}} T^{4−2σ_{1} −2σ_{2}}

+ O(t_{2}^{− \frac{1}{2}} T^{\frac{5}{2} −σ_{1} −σ_{2}} ) (1.4)

with 1 < σ_{1} + σ_{2} < \frac{3}{2} and 2 ⩽ t_{2} ⩽ T^{1− \frac{2}{3} (σ_{1} +σ_{2} )} ,

(1.5)  ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{2}dt_{1} = \frac{ζ(2)}{4π|s_{2} − 1|^{2}} T^{2} + O(t_{2}^{− \frac{1}{2}} (logt_{2})T^{\frac{3}{2}} )

with σ_{1} + σ_{2} = 1 and 2 ⩽ t_{2} ⩽ \frac{T ^{\frac{1}{3}}}{log T},

(1.6) ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{2}dt_{1} = (2π)^{2σ_{1} +2σ_{2} −3} \frac{ζ(4 − 2σ_{1} − 2σ_{2})}{(4 − 2σ_{1} − 2σ_{2})|s_{2} − 1|^{2}} T^{4−2σ_{1} −2σ_{2}}
+ O(t_{2}^{\frac{1}{2} −σ_{1} −σ_{2}} T^{\frac{5}{2} −σ_{1} −σ_{2}} )

with \frac{1}{2} < σ_{1} + σ_{2} < 1, and 2 ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} , and

∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{2}dt_{1} = \frac{ζ(3)}{12π^{2}|s_{2} − 1|^{2}} T^{3} (1.7)

+ \{ \begin{matrix}O(T^{2}) & if σ_{1} + σ_{2} = \frac{1}{2} and \sqrt{logT} ⩽ t_{2} ⩽ T^{\frac{1}{2}} , \\ O(t_{2}^{−1} T^{2} \sqrt{logT}) & if σ_{1} + σ_{2} = \frac{1}{2} and 2 ⩽ t_{2} ⩽ \sqrt{logT}. \end{matrix}

Furthermore, they derived the formula

∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{2}dt_{1} = (2π)^{2σ_{1} +2σ_{2} −3} \frac{ζ(4 − 2σ_{1} − 2σ_{2})}{(4 − 2σ_{1} − 2σ_{2})|s_{2} − 1|^{2}} T^{4−2σ_{1} −2σ_{2}} (1.8)

+ \{ \begin{matrix}O(t_{2}^{\frac{1}{2} −σ_{1} −σ_{2}} T^{\frac{5}{2} −σ_{1} −σ_{2}} ) & if T^{\frac{1−2σ_{1} −2σ_{2}}{3−2σ_{1} −2σ_{2}}} ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} , \\ O(t_{2}^{−1} T^{3−2σ_{1} −2σ_{2}} ) & if 2 ⩽ t_{2} ⩽ T^{\frac{1−2σ_{1} −2σ_{2}}{3−2σ_{1} −2σ_{2}}} \end{matrix}

for 0 < σ_{1} + σ_{2} < \frac{1}{2}. They used the mean value formula for |ζ(σ + it)| and a weak form of Kiuchi, Tanigawa, and Zhai’s approximate formula for ζ_{2}(s_{1}, s_{2}) to obtain formulas (1.4)–(1.8). Kiuchi and Minamide also showed that

(1.9)  ζ_{2}(σ_{1} + it_{1}, σ_{2} + it_{2}) = Ω( \frac{t_{1}^{\frac{3}{2} −σ_{1} −σ_{2}}}{t_{2}} )

for 1 ⩽ σ_{1} + σ_{2} < \frac{3}{2}, 2 ⩽ t_{1} ⩽ T and 2 ⩽ t_{2} ⩽ T^{1− \frac{2}{3} σ_{1} − \frac{2}{3} σ_{2} −ε} , and

(1.10)  ζ_{2}(σ_{1} + it_{1}, σ_{2} + it_{2}) = Ω( \frac{t_{1}^{\frac{3}{2} −σ_{1} −σ_{2}}}{t_{2}} )

for 0 < σ_{1} + σ_{2} < 1, 2 ⩽ t_{1} ⩽ T, and 2 ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}} −ε} , with ε being any small positive constant. Formulas (1.9) and (1.10) provide a certain improvement on the Ω-results of Kiuchi, Tanigawa, and Zhai [13].

1.2. Fourth power moment. Before the introduction of our theorems, let us recall the 2k-th power moment of the Riemann zeta-function ζ(σ + it). It is well known that

∫_{2}^{T} |ζ(σ + it)|^{2k}dt = ( ∑_{n=1}^{∞} \frac{d_{k}^{2}(n)}{n^{2σ}}) T + O(T^{2−σ+ε} ) + O(1)

holds, when σ > 1 is fixed and k ⩾ 1 is a fixed integer (see Theorem 1.10 in [7], or [8]), where the divisor function d_{k}(n) is the number of ways n can be written as a product of k fixed factors. Then, in the case k = 2, we have

(1.11)  ∫_{2}^{T} |ζ(σ + it)|^{4}dt = \frac{ζ^{4}(2σ)}{ζ(4σ)}T + O(T^{2−σ+ε} ) + O(1)

for σ > 1 and any small positive number ε. The higher power moment for the Riemann zeta-function ζ(1 + it) was derived by Balasubramanian, Ivić, and Ramachandra (see Theorem 1 in [3]), who showed that

∫_{2}^{T} |(ζ(1 + it))^{k}|^{2}dt = ( ∑_{n=1}^{∞} \frac{|d_{k}(n)|^{2}}{n^{2}} ) T + O((log T)^{|k^{2} |})

holds for any complex number k and (ζ(s))^{k} = ∑ _{n=1}^{∞} \frac{d_{k} (n)}{n^{s}} in σ ⩾ 2. For the case of k = 2 in the above expression, we have

(1.12)  ∫_{2}^{T} |ζ(1 + it)|^{4}dt = \frac{ζ^{4}(2)}{ζ(4)}T + O(log^{4} T).

Ivić [9, 10] obtained the formula

(1.13)  ∫_{2}^{T} |ζ(σ + it)|^{4}dt = \frac{ζ^{4}(2σ)}{ζ(4σ)}T + O(T^{2−2σ} log^{3} T)

for \frac{1}{2} < σ < 1. The fourth power moment for the Riemann zeta-function ζ(\frac{1}{2} + it) in the critical line was obtained by Heath-Brown [4], who showed that

(1.14)  ∫_{2}^{T} |ζ( \frac{1}{2} + it) |^{4}dt = T ∑_{k=0}^{4} c_{k}(logT)^{4−k} + O(T^{\frac{7}{8} +ε} )

holds with c_{0} = \frac{1}{2π^{2}} , and the other constants c_{k} being computable.

The main purpose of this paper is to prove the fourth power moment for the double zeta-function ζ_{2}(s_{1}, s_{2}) within the region 0 < σ_{1} < 1, 0 < σ_{2} < 1, and 0 < σ_{1} + σ_{2} < 2 without using the mean square formulas (1.4)–(1.8). We use formulas (1.11)–(1.14) for the fourth power moment of |ζ(σ + it)| and Lemma 3 below, as well as Lemma 2 below, which was derived from a weak form of the approximate formula of Kiuchi and Minamide [11] for ζ_{2}(s_{1}, s_{2}) to obtain the following formula.

Theorem 1.1. Suppose that 2 ⩽ t_{1} ⩽ T, 0 < σ_{1} < 1, 0 < σ_{2} < 1 and

σ_{1} + σ_{2} = 1. Then, for any sufficiently large positive number T > 2, we have

∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} = \frac{ζ^{4}(2)}{12π^{2}ζ(4)} \frac{T^{3}}{|s_{2} − 1|^{4}} + O(t_{2}^{− \frac{5}{2}} (logt_{2}T^{\frac{5}{2}} ) (1.15)

with 2 ⩽ t_{2} ⩽ \frac{T ^{\frac{1}{3}}}{log T}.

Remark 1.1. Inserting t_{2} = \frac{T ^{\frac{1}{3}}}{log T} into (1.15), the right-hand side of the formula

(1.15) is estimated by T^{\frac{5}{3}} log^{4} T, but if we can take t_{2} = \frac{T ^{\frac{1}{2}}}{log T}, we can estimate that T log^{4} T. The main term of this theorem is not T log^{A} T (A > 0), but T^{3} since the analytic behavior of the double zeta-function ζ_{2}(s_{1}, s_{2}) depends on both s_{1} and s_{2}. This observation in fact supports the result of Kiuchi, Tanigawa, and Zhai.

As an application of (1.15), we consider the evaluation of the double integral

∫_{2}^{N} ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1}dt_{2},

and then we deduce the following.

Corollary 1.1. Let 0 < σ_{1} < 1, 0 < σ_{2} < 1 and σ_{1} + σ_{2} = 1. Within the

region 2 ⩽ N ⩽ \frac{T ^{\frac{1}{3}}}{log T}, we obtain

\frac{1}{T^{3}} ∫_{2}^{N} ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1}dt_{2} → \frac{5π^{2}}{1728σ_{1}^{3}} ( \frac{π}{2} − Tan^{−1} \frac{2}{σ_{1}} − \frac{2σ_{1}}{4 + σ_{1}^{2}} ) + O( \frac{1}{N} )

as T → ∞.

Hence, this observation may be regarded as an average order of magnitude for the double zeta-function, which is

\frac{5π^{2}}{1728σ_{1}^{3}} ( \frac{π}{2} − Tan^{−1} \frac{2}{σ_{1}} − \frac{2σ_{1}}{4 + σ_{1}^{2}} )

if 0 < σ_{1} < 1, 0 < σ_{2} < 1, σ_{1} + σ_{2} = 1, and 2 ⩽ N ⩽ \frac{T ^{\frac{1}{3}}}{log T}.

Theorem 1.2. Suppose that 2 ⩽ t_{1} ⩽ T, 0 < σ_{1} < 1, 0 < σ_{2} < 1, \frac{1}{2} ⩽

σ_{1} + σ_{2} < \frac{3}{2}, and σ_{1} + σ_{2} ≠ 1. Then, for any sufficiently large positive number T > 2, we have

(1.16)  ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} = \frac{(2π)^{4σ_{1} +4σ_{2} −6}}{7 − 4σ_{1} − 4σ_{2}} \frac{ζ^{4}(4 − 2σ_{1} − 2σ_{2})}{ζ(8 − 4σ_{1} − 4σ_{2})} \frac{T^{7−4σ_{1} −4σ_{2}}}{|s_{2} − 1|^{4}}
+ O(t_{2}^{− \frac{5}{2}} T^{\frac{11}{2} −3σ_{1} −3σ_{2}} )

for 2 ⩽ t_{2} ⩽ T^{1− \frac{2}{3} (σ_{1} +σ_{2} )} and 1 < σ_{1} + σ_{2} < \frac{3}{2}, and

(1.17) ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} = \frac{(2π)^{4σ_{1} +4σ_{2} −6}}{7 − 4σ_{1} − 4σ_{2}} \frac{ζ^{4}(4 − 2σ_{1} − 2σ_{2})}{ζ(8 − 4σ_{1} − 4σ_{2})} \frac{T^{7−4σ_{1} −4σ_{2}}}{|s_{2} − 1|^{4}}

+ \{ \begin{matrix}O(t_{2}^{−4} T^{6−3σ_{1} −3σ_{2} +ε} ) & if 2 ⩽ t_{2} ⩽ T^{\frac{1}{5−2σ_{1} −2σ_{2}}  +ε} , \\ O(t_{2}^{− \frac{3}{2} −σ_{1} −σ_{2}} T^{\frac{11}{2} −3σ_{1} −3σ_{2}} ) & if T^{\frac{1}{5−2σ_{1} −2σ_{2}}  +ε} ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} \end{matrix}

for \frac{1}{2} ⩽ σ_{1} + σ_{2} < 1.

Theorem 1.3. Suppose that 2 ⩽ t_{1} ⩽ T, 0 < σ_{1} < 1, 0 < σ_{2} < 1, and

0 < σ_{1} + σ_{2} < \frac{1}{2}. Then, for any sufficiently large positive number T > 2, we have,

(1.18) ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} = \frac{(2π)^{4σ_{1} +4σ_{2} −6} ζ^{4}(4 − 2σ_{1} − 2σ_{2})}{(7 − 4σ_{1} − 4σ_{2})ζ(8 − 4σ_{1} − 4σ_{2})} \frac{T^{7−4σ_{1} −4σ_{2}}}{|s_{2} − 1|^{4}}
+ O(t_{2}^{6−4σ_{1} −4σ_{2}} T) + O(t_{2}^{−3} T^{6−4σ_{1} −4σ_{2}} )
+ O(t_{2}^{− \frac{3}{2} −σ_{1} −σ_{2}} T^{\frac{11}{2} −3σ_{1} −3σ_{2}} ) + O(t_{2}^{−4} T^{6−3σ_{1} −3σ_{2} +ε} )

for 2 ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} .

Remark 1.2. An average order of magnitude for double zeta-function (1.1) derived from the asymptotic behavior of the integrals of the double zeta-function with (1.15)–(1.18) holds when the ratio of the order of t_{1} to that of t_{2} is small. However, it is often difficult to determine their analytic behavior for a general ratio of the order of t_{1} to the order of t_{2}.

From Theorems 1.1, 1.2, and 1.3, we are able to determine an additional proof for the Ω-result of Kiuchi, Tanigawa, and Zhai [13], and Kiuchi and Minamide [11].

Corollary 1.2. Let 0 < σ_{1} < 1, 0 < σ_{2} < 1, and 2 ⩽ t_{1} ⩽ T. Then, we have
(1.19)  ζ_{2}(σ_{1} + it_{1}, σ_{2} + it_{2}) = Ω( \frac{t_{1}^{\frac{3}{2} −σ_{1} −σ_{2}}}{t_{2}} )

for 1 ⩽ σ_{1} + σ_{2} < \frac{3}{2} and 2 ⩽ t_{2} ⩽ T^{1− \frac{2}{3} (σ_{1} +σ_{2} )−ε} , and

(1.20)  ζ_{2}(σ_{1} + it_{1}, σ_{2} + it_{2}) = Ω( \frac{t_{1}^{\frac{3}{2} −σ_{1} −σ_{2}}}{t_{2}} )

for 0 < σ_{1} + σ_{2} < 1 and 2 ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}} −ε} , where ε is any small positive constant.

Formulas (1.19) and (1.20) are equivalent to (1.9) and (1.10), respectively, and Corollary 1.2 provides an improvement upon the Ω-result of Kiuchi, Tanigawa, and Zhai [13].

Theorem 1.4. Suppose that 2 ⩽ t_{1} ⩽ T, 0 < σ_{1} < 1, 0 < σ_{2} < 1, and \frac{3}{2} ⩽ σ_{1} + σ_{2} < 2. Then, for any sufficiently large positive number T > 2, we have

(1.21)  ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} = O(t_{2}^{2}T)

for \frac{3}{2} < σ_{1} + σ_{2} < 2, and

(1.22) ∫_{2}^{T}|ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} =\{ \begin{matrix}\frac{1}{2π^{2}} \frac{T log^{4} T}{|s_{2} −1|^{4}} + O(t_{2}^{− \frac{5}{2}} T log^{3} T) & if 2 ⩽ t_{2} ⩽ (logT)^{\frac{2}{3}} , \\ O(t_{2}^{2}T) & if t_{2} ⩾ (logT)^{\frac{2}{3}} \end{matrix}

for σ_{1} + σ_{2} = \frac{3}{2}.

To improve our theorems, we must obtain a sharper estimate for the integral ∫_{2}^{T} |E(s_{1}, s_{2})|^{4}dt_{1} in Lemma 2.2 below, but this is very difficult.

Notations. When g(x) is a positive function of x for x ⩾ x_{0}, f(x) = Ω(g(x)) means that f(x) = o(g(x)) does not hold as x → ∞. In what follows, ε denotes any arbitrarily small positive number, not necessarily the same ones at each occurrence.

2. Some lemmas

Let a be any complex number and χ(s_{2}) = 2(2π)^{s_{2} −1} sin(\frac{π}{2} s_{2})Γ(1 − s_{2}). The generalized divisor function σ_{a}(n) is defined by ∑ _{d|n} d^{a}. We use a weak form of the approximate formula of Kiuchi, Tanigawa, and Zhai [13] to prove our theorems. For our purpose, it is enough to quote the following weak form, which follows from Lemma 1 of Kiuchi and Minamide [11].

Lemma 2.1 ([11]). Suppose that 0 < σ_{1} < 1 and 0 < σ_{2} < 1. Then, we have

(2.1)  ζ_{2}(s_{1}, s_{2}) = \frac{ζ(s_{1} + s_{2} − 1)}{s_{2} − 1} − \frac{1}{2}ζ(s_{1} + s_{2}) + E(s_{1}, s_{2}),

where the error term E(s_{1}, s_{2}) is estimated as

(2.2)  E(s_{1}, s_{2}) ≪ \{ \begin{matrix}|t_{2}|^{\frac{3}{2} −σ_{1} −σ_{2}} & if 0 < σ_{1} + σ_{2} < 1, \\ |t_{2}|^{\frac{1}{2}} log|t_{2}| & if σ_{1} + σ_{2} = 1, \\ |t_{2}|^{\frac{1}{2}} & if σ_{1} + σ_{2} > 1. \end{matrix}

Note that the error term in this lemma is independent of t_{1}. We establish a formula for the fourth power moment of ζ_{2}(s_{1}, s_{2}). Using (2.1), (2.2), and Hölder’s inequality, we deduce the following:

Lemma 2.2. For 0 < σ_{1} < 1, 0 < σ_{2} < 1 and any sufficiently large number T > 2, we have

∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} (2.3)

= I_{1} + I_{2} + I_{3} + O(I1^{\frac{3}{4}} I2^{\frac{1}{4}} + I1^{\frac{3}{4}} I3^{\frac{1}{4}} + I2^{\frac{3}{4}} I3^{\frac{1}{4}} + I2^{\frac{3}{4}} I1^{\frac{1}{4}} + I3^{\frac{3}{4}} I1^{\frac{1}{4}} + I3^{\frac{3}{4}} I2^{\frac{1}{4}} ),

where

(2.4)  I_{1} = \frac{1}{|s_{2} − 1|^{4}} ∫_{2}^{T} |ζ(s_{1} + s_{2} − 1)|^{4}dt_{1},
(2.5)  I_{2} = \frac{1}{16} ∫_{2}^{T} |ζ(s_{1} + s_{2})|^{4}dt_{1}

(2.6)  I_{3} = ∫_{2}^{T} |E(s_{1} , s_{2})|^{4}dt_{1} ≪, \{ \begin{matrix}T |t_{2}|^{6−4σ_{1} −4σ_{2}} & if 0 < σ_{1} + σ_{2} < 1, \\ T |t_{2}|^{2} log^{4} |t_{2}| & if σ_{1} + σ_{2} = 1, \\ T |t_{2}|^{2} & if σ_{1} + σ_{2} > 1. \end{matrix}

Kiuchi and Minamide [11] suggested that

(2.7)  E(s_{1}, s_{2}) = χ(s_{2}) ∑_{n⩽ \frac{|t_{2} |}{2π}} \frac{(1 ∗ id^{1−s_{1} −s_{2}} )(n)}{n^{1−s_{2}}} + O(|t_{2}|^{max (0,1−σ_{1} −σ_{2} )+ε} )

= χ(s_{2}) ∑_{ml⩽ \frac{|t_{2} |}{2π}} \frac{1}{m^{1−s_{2}}} · \frac{1}{l^{s_{1}}} + O(|t_{2}|^{max (0,1−σ_{1} −σ_{2} )+ε} )

holds, where ∗ denotes the Dirichlet convolution. The first term on the right-hand side of (2.7) can be written as

χ(s_{2}) ∑_{m⩽M} \frac{1}{m^{1−s_{2}}} ∑_{l⩽L} \frac{1}{l^{s_{1}}} + O(⋯)

with M ⩾ 1, L ⩾ 1 and ML = \frac{|t_{2} |}{2π}. We use the approximate functional equation of the Riemann zeta-function [18, Theorem 4.13] and the simplest form of the approximation to the Riemann zeta-function [18, Theorem 4.11] to obtain

χ(s_{2}) ∑_{m⩽M} \frac{1}{m^{1−s_{2}}} = ζ(s_{2}) − ∑_{l⩽L} \frac{1}{l^{s_{2}}} + O(L^{−σ_{2}} log|t_{2}|) + O(|t_{2}|^{\frac{1}{2} −σ_{2}} M^{σ_{2} −1} )

with 0 < σ_{2} < 1, and

∑_{l⩽L} \frac{1}{l^{s_{1}}} = ζ(s_{1}) + \frac{L^{1−s_{1}}}{1 − s_{1}} + O(L^{−σ_{1}} )

with 0 < σ_{1} < 1, respectively. Taking the product of the above formulas and using (2.7), we see that the function E(s_{1}, s_{2}) is given by

E(s_{1}, s_{2}) = ζ(s_{2})ζ(s_{1}) − ζ(s_{1}) ∑_{l⩽L} \frac{1}{l^{s_{2}}} + \frac{L^{1−s_{1}}}{1 − s_{1}} ζ(s_{2}) + \frac{L^{1−s_{1}}}{1 − s_{1}} ∑_{l⩽L} \frac{1}{l^{s_{2}}} + O(⋯).

Roughly speaking, in the case where |t_{1}| ≍ |t_{2}|, the true order of magnitude of the function E(s_{1}, s_{2}) may be regarded as |E(s_{1}, s_{2})| ≍ |ζ(s_{1})||ζ(s_{2})|. Thus, it follows from the above and Lemma 2.1 that |ζ_{2}(s_{1}, s_{2})| ≍ |ζ(s_{1})||ζ(s_{2})|. However, in the case where |t_{2}| ≪ |t_{1}|^{α} (0 < α < \frac{1}{3}), the order of magnitude of the function E(s_{1}, s_{2}) is smaller than that of the first term on the right-hand side of (2.1); hence, (2.7) implies the error term of (2.1).

To deal with the integrals I_{j} (j = 1, 2, 3), we shall use Lemma 2.3 and formulas (1.11)–(1.14).

Lemma 2.3. For any sufficiently large positive number T > 2, we have

(2.8)  ∫_{2}^{T} |ζ(σ + it)|^{4}dt = \frac{(2π)^{4σ−2}}{3 − 4σ} \frac{ζ^{4}(2 − 2σ)}{ζ(4 − 4σ)} T^{3−4σ} + O(T^{2−2σ} log^{3} T)

with 0 < σ < \frac{1}{2},

(2.9)  ∫_{2}^{T} |ζ(it)|^{4}dt = \frac{ζ^{4}(2)}{12π^{2}ζ(4)}T^{3} + O(T^{2} log^{4} T)

with σ = 0, and

(2.10)  ∫_{2}^{T} |ζ(σ + it)|^{4}dt = \frac{(2π)^{4σ−2}}{3 − 4σ} \frac{ζ^{4}(2 − 2σ)}{ζ(4 − 4σ)} T^{3−4σ} + O(T^{3−3σ+ε} )

with −1 < σ < 0.

Proof. Since the functional equation is ζ(σ + it) = χ(σ + it)ζ(1 − σ − it) (see Titchmarsh [18] or Ivić [7]), the equality |ζ(1 − σ − it)| = |ζ(1 − σ + it)| and the formula

|χ(σ + it)|^{4} = ( \frac{t}{2π} )^{2−4σ} + O(t^{1−4σ} ) (t ⩾ t_{0} > 0),

we can integrate by parts and use (1.13) to obtain

∫_{2}^{T} |ζ(σ + it)|^{4}dt = ∫_{2}^{T} |χ(σ + it)|^{4}|ζ(1 − σ + it)|^{4}dt

= ( \frac{1}{2π} )^{2−4σ} ∫_{2}^{T} t^{2−4σ} |ζ(1 − σ + it)|^{4}dt + O( ∫_{2}^{T} t^{1−4σ} |ζ(1 − σ + it)|^{4}dt)

= \frac{(2π)^{4σ−2}}{3 − 4σ} \frac{ζ^{4}(2 − 2σ)}{ζ(4 − 4σ)} T^{3−4σ} + O(T^{2−2σ} log^{3} T)

for 0 < σ < \frac{1}{2}. Similarly in the case of σ = 0, we have, by integrating by parts and using (1.12)

∫_{2}^{T} |ζ(it)|^{4}dt= ∫_{2}^{T} |χ(it)|^{4}|ζ(1 + it)|^{4}dt

= \frac{1}{4π^{2}} ∫_{2}^{T} t^{2}|ζ(1 + it)|^{4}dt + O( ∫_{2}^{T} t|ζ(1 + it)|^{4}dt)

= \frac{ζ^{4}(2)}{12π^{2}ζ(4)}T^{3} + O(T^{2} log^{4} T).

In a similar manner, we deduce from (1.11) that formula (2.10) holds. □

3. Proofs

3.1. Proofs of Theorem 1.1 and Corollary 1.1. We set 2 ⩽ t_{1} ⩽ T and 2 ⩽ t_{2} ⩽ \frac{T ^{\frac{1}{3}}}{log T}. We shall evaluate the integral ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} under the condition 0 < σ_{1} < 1, 0 < σ_{2} < 1, and σ_{1} + σ_{2} = 1. From (1.12), we have

(3.1)  I_{2} = \frac{1}{16}\{ ∫_{2}^{T +t_{2}} |ζ(1 + it)|^{4}dt − ∫_{2}^{2+t_{2}} |ζ(1 + it)|^{4}dt\} = O(T).

Similarly,

(3.2)  I_{1} = \frac{1}{|s_{2} − 1|^{4}} \{ ∫_{2}^{T +t_{2}} |ζ(it)|^{4}dt − ∫_{2}^{2+t_{2}} |ζ(it)|^{4}dt\} .

Inserting (2.9) into (3.2), we obtain

(3.3)  I_{1} = \frac{ζ^{4}(2)}{12π^{2}ζ(4)} \frac{(T + t_{2})^{3} − (t_{2} + 2)^{3}}{|s_{2} − 1|^{4}} + O( \frac{(T + t_{2})^{2}}{|s_{2} − 1|^{4}} log^{4} T )

= \frac{ζ^{4}(2)}{12π^{2}ζ(4)|s_{2} − 1|^{4}} T^{3} + O(t_{2}^{−3} T^{2}) + O(t_{2}^{−4} T^{2} log^{4} T).

From (2.6), we have

(3.4)  I_{3} = O((t_{2}^{2} log^{4} t_{2})T).

Substituting (3.1), (3.3), and (3.4) into (2.3), we observe that all error terms on the right-hand side of (2.3) are absorbed into O(t_{2}^{− \frac{5}{2}} (log t_{2})T^{\frac{5}{2}} ), completing the proof of (1.15).

As an application of (1.15), we shall evaluate the double integral for the double zeta-function ζ_{2}(s_{1}, s_{2}):

∫_{2}^{N} ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1}dt_{2}.

From (1.15), we have

∫_{2}^{N} ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1}dt_{2}

= \frac{ζ^{4}(2)}{12π^{2}ζ(4)}( ∫_{2}^{N} \frac{1}{|s_{2} − 1|^{4}} dt_{2}) T^{3} + O( T^{\frac{5}{2}} ∫_{2}^{N} t_{2}^{− \frac{5}{2}} logt_{2}dt_{2})

238 KIUCHI

for 2 ⩽ N ⩽ \frac{T ^{\frac{1}{3}}}{log T}. It follows that

∫_{2}^{N} \frac{1}{|s_{2} − 1|^{4}} dt_{2} = \frac{1}{2(1 − σ_{2})^{3}} ( \frac{π}{2} − Tan^{−1} \frac{2}{1 − σ_{2}} − \frac{2(1 − σ_{2})}{4 + (1 − σ_{2})^{2}} ) + O( \frac{1}{N} )

and

∫_{2}^{N} t_{2}^{− \frac{5}{2}} logt_{2}dt_{2} = ∫_{2}^{∞} t_{2}^{− \frac{5}{2}} logt_{2}dt_{2} + O(N^{− \frac{3}{2}} logN).

Then, we easily see that

∫_{2}^{N} ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1}dt_{2}

= \frac{5π^{2}}{1728(1 − σ_{2})^{3}} ( \frac{π}{2} − Tan^{−1} \frac{2}{1 − σ_{2}} − \frac{2(1 − σ_{2})}{4 + (1 − σ_{2})^{2}} ) T^{3}

+ O(T^{\frac{5}{2}} N^{− \frac{3}{2}} logN) + O(T^{\frac{5}{2}} ) + O(T^{3}N^{−1} ).

Hence, for 0 < σ_{1} < 1, 0 < σ_{2} < 1, σ_{1} + σ_{2} = 1, and 2 ⩽ N ⩽ \frac{T ^{\frac{1}{3}}}{log T}, we obtain, as T → ∞,

\frac{1}{T^{3}} ∫_{2}^{N} ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1}dt_{2} → \frac{5π^{2}}{1728σ_{1}^{3}} ( \frac{π}{2} − Tan^{−1} \frac{2}{σ_{1}} − \frac{2σ_{1}}{4 + σ_{1}^{2}} ) + O( \frac{1}{N} ) .

Therefore, we obtain the assertion of Corollary 2.1.

3.2. Proof of Theorem 1.2. Throughout this section, we assume that 0 < σ_{1} < 1 and 0 < σ_{2} < 1. As in the proof of Theorem 1.1, we shall calculate the integral ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} under the condition that 1 < σ_{1} + σ_{2} < \frac{3}{2} with 2 ⩽ t_{2} ⩽ T^{1− \frac{2}{3} (σ_{1} +σ_{2} )} . From (1.11) and (2.5), we have

(3.5) I_{2} = \frac{1}{16}\{ ∫_{2}^{T +t_{2}} |ζ(σ_{1} + σ_{2} + it)|^{4}dt − ∫_{2}^{2+t_{2}} |ζ(σ_{1} + σ_{2} + it)|^{4}dt\} = O(T).

From (2.4) and (2.8), we have

(3.6) I_{1} = \frac{(2π)^{4σ_{1} +4σ_{2} −6}}{7 − 4σ_{1} − 4σ_{2}} \frac{ζ^{4}(4 − 2σ_{1} − 2σ_{2})}{ζ(8 − 4σ_{1} − 4σ_{2})} \frac{(T + t_{2})^{7−4σ_{1} −4σ_{2}} − (t_{2} + 2)^{7−4σ_{1} −4σ_{2}}}{|s_{2} − 1|^{4}}

+ O( \frac{(T + t_{2})^{4−2σ_{1} −2σ_{2}}}{|s_{2} − 1|^{4}} log^{3} T )

= \frac{(2π)^{4σ_{1} +4σ_{2} −6}}{7 − 4σ_{1} − 4σ_{2}} \frac{ζ^{4}(4 − 2σ_{1} − 2σ_{2})}{ζ(8 − 4σ_{1} − 4σ_{2})} \frac{1}{|s_{2} − 1|^{4}} T^{7−4σ_{1} −4σ_{2}}

+ O(t_{2}^{−3} T^{6−4σ_{1} −4σ_{2}} ) + O(t_{2}^{−4} T^{4−2σ_{1} −2σ_{2}} log^{3} T).

From (2.6), we have

(3.7)  I_{3} = O(t_{2}^{2}T).

Substituting (3.5), (3.6), and (3.7) into (2.3), we observe that all error terms on the right-hand side of (2.3) are absorbed into O(t_{2}^{− \frac{5}{2}} T^{\frac{11}{2} −3σ_{1} −3σ_{2}} ). Hence, we derive formula (1.16) with 2 ⩽ t_{2} ⩽ T^{1− \frac{2}{3} (σ_{1} +σ_{2} )} . In a similar manner as above, we

THE FOURTH POWER MOMENT OF THE DOUBLE ZETA-FUNCTION 239

shall calculate the integral ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} in the case of \frac{1}{2} < σ_{1} + σ_{2} < 1 with 2 ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} . By (1.13) and (2.5), we have

(3.8)  I_{2} = O(T),

and by (2.6)

(3.9)  I_{3} = O(t_{2}^{6−4σ_{1} −4σ_{2}} T).

By (2.4) and (2.10) we have

(3.10)  I_{1} = \frac{1}{|s_{2} − 1|^{4}} ∫_{2}^{T} |ζ(σ_{1} + σ_{2} − 1 + i(t_{1} + t_{2}))|^{4}dt_{1}

= (2π)^{4σ_{1} +4σ_{2} −6} \frac{ζ^{4}(4 − 2σ_{1} − 2σ_{2})}{(7 − 4σ_{1} − 4σ_{2})ζ(8 − 4σ_{1} − 4σ_{2})} \frac{T^{7−4σ_{1} −4σ_{2}}}{|s_{2} − 1|^{4}}

+ O(t_{2}^{−3} T^{6−4σ_{1} −4σ_{2}} ) + O(t_{2}^{−4} T^{6−3σ_{1} −3σ_{2} +ε} ).

Substituting (3.8), (3.9), and (3.10) into (2.3), we observe that all error terms on the right-hand side of (2.3) are absorbed into O(t_{2}^{−4} T^{6−3σ_{1} −3σ_{2} +ε} ) if 2 ⩽ t_{2} ⩽ T^{\frac{1}{5−2σ_{1} −2σ_{2}} +ε} , or into O(t_{2}^{− \frac{3}{2} −σ_{1} −σ_{2}} T^{\frac{11}{2} −3σ_{1} −3σ_{2}} ) if T^{\frac{1}{5−2σ_{1} −2σ_{2}} +ε} ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} . We derive formula (1.17) with 2 ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} . Next, we shall calculate the integral ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} in the case where σ_{1} + σ_{2} = \frac{1}{2} under the assumption that 2 ⩽ t_{2} ⩽ T^{\frac{1}{2}} . By (1.14) and (2.5), we have

(3.11)  I_{2} = O(T log^{4} T),

and by (2.6), we have

(3.12)  I_{3} = O(t_{2}^{4}T).

By (2.4) and (2.10), we have

(3.13)  I_{1} = \frac{1}{|s_{2} − 1|^{4}} ∫_{2}^{T} |ζ( − \frac{1}{2} + i(t_{1} + t_{2})) |^{4}dt_{1}

= \frac{ζ^{4}(3)}{80π^{4}ζ(6)} \frac{T^{5}}{|s_{2} − 1|^{4}} + O(t_{2}^{−3} T^{4}) + O(t_{2}^{−4} T^{\frac{9}{2} +ε} ).

Substituting (3.11), (3.12), and (3.13) into (2.3), we observe that all error terms on the right-hand side of (2.3) are absorbed into O(t_{2}^{−4} T^{\frac{9}{2} +ε} ) if 2 ⩽ t_{2} ⩽ T^{\frac{1}{4} +ε} and O(t_{2}^{−2} T^{4}) if T^{\frac{1}{4} +ε} ⩽ t_{2} ⩽ T^{\frac{1}{2}} . We derive the formula (1.17) with σ_{1} + σ_{2} = \frac{1}{2} and 2 ⩽ t_{2} ⩽ T^{\frac{1}{2}} . Thus, we obtain the assertions of Theorem 1.2.

3.3. Proof of Theorem 1.3. Assuming that 0 < σ_{1} < 1, 0 < σ_{2} < 1, and 0 < σ_{1} + σ_{2} < \frac{1}{2}, we shall calculate the integral ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} with 2 ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} . Similar to the above theorems, (2.5) and (2.8) provide us

(3.14)  I_{2} = O(T^{3−4σ_{1} −4σ_{2}} ),

and by (2.6)

(3.15)  I_{3} = O(t_{2}^{6−4σ_{1} −4σ_{2}} T).

By (2.4) and (2.10), we have

(3.16)  I_{1} = (2π)^{4σ_{1} +4σ_{2} −6} \frac{ζ^{4}(4 − 2σ_{1} − 2σ_{2})}{(7 − 4σ_{1} − 4σ_{2})ζ(8 − 4σ_{1} − 4σ_{2})} \frac{T^{7−4σ_{1} −4σ_{2}}}{|s_{2} − 1|^{4}}
+ O(t_{2}^{−3} T^{6−4σ_{1} −4σ_{2}} ) + O(t_{2}^{−4} T^{6−3σ_{1} −3σ_{2} +ε} ).

Substituting (3.14), (3.15), and (3.16) into (2.3), we observe that all error terms on the right-hand side of (2.3) are given by

O(t_{2}^{− \frac{3}{2} −σ_{1} −σ_{2}} T^{\frac{11}{2} −3σ_{1} −3σ_{2}} )+ O(t_{2}^{6−4σ_{1} −4σ_{2}} T)
+ O(t_{2}^{−3} T^{6−4σ_{1} −4σ_{2}} ) + O(t_{2}^{−4} T^{6−3σ_{1} −3σ_{2} +ε} )

for 2 ⩽ t_{2} ⩽ T^{\frac{3−2σ_{1} −2σ_{2}}{5−2σ_{1} −2σ_{2}}} . Hence, this completes the proof of Theorem 1.3.

3.4. Proof of Theorem 1.4. Throughout this section, we shall assume that 0 < σ_{1} < 1 and 0 < σ_{2} < 1. We shall calculate the integral ∫_{2}^{T} |ζ_{2}(s_{1}, s_{2})|^{4}dt_{1} with \frac{3}{2} ⩽ σ_{1} + σ_{2} < 2. When \frac{3}{2} < σ_{1} + σ_{2} < 2, (1.11), (1.13), and (2.4)–(2.6) tell us that

(3.17)  I_{2} = O(T), I_{1} = O(t_{2}^{−4} T), I_{3} = O(t_{2}^{2}T).

Substituting (3.17) into (2.3), we derive formula (1.21). Similarly, in the case of σ_{1} + σ_{2} = \frac{3}{2}, (1.11), (1.14), and (2.4)–(2.6) tell us that

(3.18) I_{2} = O(T), I_{3} = O(t_{2}^{2}T), I_{1} = \frac{1}{2π^{2}|s_{2} − 1|^{4}} T log^{4} T + O(t_{2}^{−4} T log^{3} T).

Substituting (3.18) into (2.3), we observe that all error terms on the right-hand side of (2.3) are absorbed into O(t_{2}^{− \frac{5}{2}} T log^{3} T) if 2 ⩽ t_{2} ⩽ (logT)^{\frac{2}{3}} , and that the right-hand side of (2.3) yields O(t_{2}^{2}T) if t_{2} ⩾ (logT)^{\frac{2}{3}} . Hence, we derive formula (1.22).

Acknowledgements. The author would like to thank the referee for his/her careful reading of the earlier version of this paper and giving many valuable suggestions.

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Department of Mathematical Sciences (Received 25 10 2014) Yamaguchi University (Revised 05 01 2015) Yamaguchi Yoshida Japan

 kiuchi@yamaguchi-u.ac.jp