Bulletin T.CXXVII de l’Académie Serbe des Sciences et des Arts - 2003 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 28

ON SOME MEAN VALUE RESULTS INVOLVING |ζ(\frac{1}{2} + it)|

A. IVIĆ

(Presented at the 1st Meeting, held on February 28, 2003)

A b s t r a c t. Several problems involving E(T) and E_{2}(T), the error terms in the mean square and mean fourth moment formula for |ζ(\frac{1}{2} + it)|, are discussed. In particular it is proved that

∫_{0}^{T} E(t)E_{2}(T)dt ≪ T^{7/4}(log T)^{7/2} log log T.

AMS Mathematics Subject Classification (2000): Primary 11M06, Secondary 11F72, 11F66, 11M41

Key Words: Riemann zeta-function, mean square, mean fourth power, Hecke series

1. Introduction and statement of results

Let, as usual (γ = 0.5772157... is Euler’s constant),

E(T) = ∫_{0}^{T} |ζ(\frac{1}{2} + it)|^{2}dt − T (log (\frac{T}{2π} ) + 2γ − 1)

denote the error term in the mean square formula for |ζ(\frac{1}{2} + it)|, and let

E_{2}(T) = ∫_{0}^{T} |ζ(\frac{1}{2} + it)|^{4}dt − T P_{4}(log T)

denote the error term in the asymptotic formula for the fourth moment of |ζ(\frac{1}{2} + it)|. Here P_{4}(x) is a polynomial of degree four in x with leading coefficient 1/(2π^{2}) (see [5] for the explicit evaluation of all the coefficients). Both of these functions play an important rôle in the theory of the Riemann zeta-function ζ(s), and the aim of this note is to discuss several problems involving their mean values. Especially interesting seems the evaluation of the integral

∫_{0}^{T} E(t)E_{2}(t)dt,

or (which is technically more convenient)

I(T) := ∫_{T}^{2T} E(t)E_{2}(t)dt,  (1.1)

since this integral exhibits the superpositions of oscillations of the functions E(t) and E_{2}(t). Namely both functions are oscillating, and we have E(t) = Ω_{±}(t^{1/4}) (see [1], [4]) and E_{2}(t) = Ω_{±}(t^{1/2}) (see [4], [6] and [12]). As usual, f = Ω_{±}(g) means that lim sup f/g > 0 and lim inf f/g < 0. We also have (see [2] and [4])

∫_{0}^{T} E^{2}(t)dt = DT^{3/2} + O(T log^{4} T)  (1.2)

with D = 2(2π)^{−1/2}ζ^{4}(3/2)/(3ζ(3)), and (see [7], [12])

∫_{T}^{2T} E_{2}^{2}(t)dt ≪ T^{2} log^{22} T.  (1.3)

As usual, f ≪ g (same as f = O(g)) means that |f(x)| ≤ Cg(x) for some C > 0 and x ≥ x_{0}). Hence by (1.2), (1.3), and the Cauchy-Schwarz inequality for integrals one obtains

I(T) ≪ T^{7/4} log^{11} T,  (1.4)

and one naturally asks whether (1.4) can be improved. This is indeed so, as shown by the following

Theorem 1. We have

I(T) = ∫_{T}^{2T} E(t)E_{2}(t)dt ≪ T^{7/4}(log T)^{7/2} log log T.  (1.5)

On some mean value results involving |ζ( \frac{1}{2} + it)| 19

We remark here that an analogous formula to (1.5) holds if E(t) is replaced (d(n) is the number of divisors of n) by

Δ(x) = ∑_{n≤x} d(n) − x(log x + 2γ − 1),

the error term in the classical divisor problem. Namely the analogues of (2.4) and (2.5) will hold for Δ(x) by the Voronoi explicit formula for Δ(x) (see [2, Chapter 3]). Thus, following the proof of Theorem 1, we shall obtain

∫_{T}^{2T} Δ(t)E_{2}(t)dt ≪ T^{7/4}(log T)^{7/2} log log T.

In the course of the proof of Theorem 1 we shall encounter the function

g(t) := \frac{1}{2} (\frac{2}{π} )^{3/4} ∑_{n=1}^{∞}(−1)^{n}d(n)n^{−5/4} sin(\sqrt{8πnt} − \frac{1}{4}π),  (1.6)

which appears in the relation

∫_{T}^{2T} E(t)E_{2}(t)dt = − ∫_{T}^{2T} t^{3/4}g(t)|ζ(\frac{1}{2} + it)|^{4}dt + O(T^{3/2} log^{10} T). (1.7)

It is thus seen that the natural question of the true order of magnitude of I(T) involves the evaluation of the integral on the right-hand side of (1.7), which contains the oscillatory function g(t) (it is both O(1) and Ω_{±}(1); see [1] and [4]). Although it appears to the author that the true order of I(T) is T^{3/2+o(1)}, this is certainly hard to prove. However if we set

g_{+}(t) = max(g(t), 0),  g_{−}(t) = min(g(t), 0),  (1.8)

then we have

Theorem 2. If g_{±}(t) is given by (1.8), then for k = 1, 2 we have

∫_{T}^{2T} t^{3/4}g_{+}(t)|ζ(\frac{1}{2} + it)|^{2k}dt ≪ T^{7/4}(log T)^{k^{2}},

(1.9)

∫_{T}^{2T} t^{3/4}g_{+}(t)|ζ(\frac{1}{2} + it)|^{2k}dt ≫ T^{7/4}(log T)^{k^{2}},

and likewise

∫_{T}^{2T} t^{3/4}g_{−}(t)|ζ(\frac{1}{2} + it)|^{2k}dt ≪ T^{7/4}(log T)^{k^{2}},

(1.10)

∫_{T}^{2T} t^{3/4}g_{−}(t)|ζ(\frac{1}{2} + it)|^{2k}dt ≫ T^{7/4}(log T)^{k^{2}}.

Comparing Theorem 1 and Theorem 2 we see that there must be some cancellation when we deal with g(t) = g_{+}(t) + g_{−}(t) instead of only g_{+}(t) or g_{−}(t). We note that, similarly to (1.7), we obtain (with suitable B > 0, see the remark after (2.2))

∫_{T}^{2T} t^{3/4}g(t)|ζ(\frac{1}{2} + it)|^{2}dt = −BT^{3/2} + O(T^{5/4} log T).  (1.11)

The integrals in (1.7) and (1.11) containing the function g(t) are similar, which is why there is reason to think that I(T) is also of the the order T^{3/2+o(1)}. The original motivation for the study of I(T) was to try to obtain a lower bound for the integral on the right-hand side of (1.7). This would in turn, by the Cauchy-Schwarz inequality, provide a lower bound for the mean square integral of E_{2}(T). The author proved in [7] the lower bound

∫_{T}^{2T} E_{2}^{2}(t)dt ≫ T^{2},  (1.12)

which complements (1.3). However, in view of Theorem 1, it does not appear likely that this procedure can shed some new light on the behaviour of the integral in (1.12).

Since the function g(t) is Ω_{±}(1), it means that it takes positive and negative values for some arbitrarily large values of t. Thus it seems of interest to characterize the sets where g(t) > 0 and g(t) < 0. In this direction we have (μ(·) denotes measure) the following result, which will be used in proving Theorem 2.

Theorem 3. There is a number η > 0 such that [T, 2T] contains a subset A(T) in which g(t) > η and

μ(A(T)) ≫_{η} T,  (1.13)

and a subset B(T) in which g(t) < −η and

μ(B(T)) ≫_{η} T.  (1.14)

2. Proof of Theorem 1

Let us define

G(T) := ∫_{0}^{T} (E(t) − π)dt.  (2.1)

If P_{4}(x) is the polynomial appearing in the definition of E_{2}(T) and Q_{4}(x) := P_{4}(x) + P′_{4}(x), then integrating by parts we have

∫_{T}^{2T} E(t)E_{2}(t)dt = ∫_{T}^{2T} (E(t) − π + π)E_{2}(t)dt
= O(T^{3/2}) − ∫_{T}^{2T} G(t)E′_{2}(t)dt
= − ∫_{T}^{2T} G(t)(|ζ(\frac{1}{2} + it)|^{4} − Q_{4}(log t))dt + O(T^{3/2})
= − ∫_{T}^{2T} G(t)|ζ(\frac{1}{2} + it)|^{4}dt + O(T^{3/2}).

(2.2)

Proceeding with E(t) in place of E_{2}(t), we obtain (1.11). Here we used the facts that

E_{2}(T) = O(T^{2/3} log^{8} T), ∫_{0}^{T} E_{2}(t)dt = O(T^{3/2}),  (2.3)

and

G(T) = O(T^{3/4}), ∫_{0}^{T} G(t)dt = O(T^{5/4}).  (2.4)

For a proof of the bounds in (2.3), see [4] or [12]. The bounds in (2.4) follow from the explicit formula of Hafner–Ivić [1], namely

G(t) = 2^{−3/2} ∑_{n≤t}(−1)^{n}d(n)n^{−1/2} (\frac{t}{2πn} + \frac{1}{4})^{−1/4} (ar sinh \sqrt{\frac{πn}{2t}} )^{−2} sin f(t, n)

−2 ∑_{n≤c_{0}t} d(n)n^{−1/2} (log \frac{t}{2πn})^{−2} sin (t log \frac{t}{2πn} − t − \frac{π}{4} ) + O(t^{1/4}),

(2.5)

where c_{0} = 1/(2π) + 1/2 − \sqrt{1/4 + 1/(2π)} = 0.019502 … and

f(t, k) = 2t ar sinh \sqrt{\frac{πk}{2t}} +\sqrt{2πkt + π^{2}k^{2}}− \frac{1}{4}π, ar sinh x = log(x+\sqrt{1 + x^{2}}).

By using the Cauchy–Schwarz inequality for integrals, the bound (e.g., see [2] for a proof)

∫_{0}^{T} |ζ(\frac{1}{2} + it)|^{8}dt ≪ T^{3/2} log^{21/2} T  (2.6)

and the mean theorem for Dirichlet polynomials (e.g., see [2, Chapter 5]), it is seen that the contribution of ∑_{n≤c_{0}t} in (2.5) to the right-hand side of (1.7) is ≪ T^{3/2} log^{10} T (the exponent of the logarithm is not optimal, but it is unimportant). Simplifying the first sum in (2.5) by Taylor’s formula (truncating it at n = \sqrt{T}), we obtain from (2.2) the asympottic formula (1.7), namely

I(T) = − ∫_{T}^{2T} t^{3/4}g(t)|ζ(\frac{1}{2} + it)|^{4}dt + O(T^{3/2} log^{10} T)  (2.7)

with g(t) given by (1.6). Since clearly

|g(t)| ≤ \frac{1}{2} (\frac{2}{π} )^{3/4} ∑_{n=1}^{∞} d(n)n^{−5/4} = \frac{1}{2} (\frac{2}{π} )^{3/4} ζ^{2}(\frac{5}{4}) = O(1),

one obtains easily from (2.7) and the weak bound

∫_{0}^{T} |ζ(\frac{1}{2} + it)|^{4}dt ≪ T log^{4} T,  (2.8)

the upper bound

I(T) = O(T^{7/4} log^{4} T).  (2.9)

Although (2.9) improves (1.4), it is poorer than (1.5) of Theorem 1, so that we must use different tools to obtain the assertion of Theorem 1. To this end we appeal to the following explicit formula of Ivić–Motohashi (see [4] and [12]): For V^{1/2} log^{−A} V ≤ Δ ≤ V^{3/4} (A > 0 is an arbitrary, but fixed constant)

∫_{0}^{V} I(T, Δ)dt = V P_{4}(log V) + O(Δ log^{5} V)

+π\sqrt{\frac{1}{2}V} ∑_{j=1}^{∞} α_{j}H_{j}^{3}(\frac{1}{2})c_{j} cos(κ_{j} log \frac{κ_{j}}{4eV} )e^{−(κ_{j}Δ/2V)^{2}}, (2.10)

where c_{j} ∼ κ_{j}^{−3/2} as κ_{j} → ∞ and

I(T, Δ) := \frac{1}{Δ\sqrt{π}} ∫_{−∞}^{∞} |ζ(\frac{1}{2} + iT + iu)|^{4}e^{−(u/Δ)^{2}}du.  (2.11)

For the definitions and properties of the spectral quantities α_{j}, κ_{j} and H_{j}(\frac{1}{2}), see Y. Motohashi’s monograph [13]. What will be needed here, besides (2.10) and (2.11), is essentially the bound (cf. [13])

∑_{κ_{j}≤K} α_{j}H_{j}^{3}(\frac{1}{2}) ≪ K^{2} log^{3} K.  (2.12)

Although (2.12) is not stated explicitly in [13], it follows when one integrates the last formula on p. 130 (with G = T^{3/4}, say) from K to 2K with the help of the estimate for the sum in (2.12) in short intervals, obtained recently by the author in [9]. An asymptotic formula for the sum in (2.12) has been obtained recently by the author in [10]. This is

∑_{κ_{j}≤K} α^{j}H3 j(^{\frac{1}{2}}) = K^{2}P^{3}(log K)+R(K), R(K) = O(K^{5/4} log^{25/2} K), (2.13)

where P_{3}(x) is a cubic polynomial in x with leading coefficient equal to 4/(3π^{2}).

The proof of (1.5) consists of three steps: the first is to show that (2.2) can be simplified to give (1.7). Then we show that |ζ(\frac{1}{2} + it)|^{4} can be replaced by I(t, Δ) (with suitable Δ) and permissible error. The last step is to use the spectral decomposition (2.10) and obtain (1.5). The bound in (1.5) is actually the limit of the method, set by the condition V^{1/2} log^{−A} V ≤ Δ ≤ V^{3/4} in (2.10). Namely we wish Δ to be as small as possible, so any further improvements of (1.5) will necessitate the widening of this range, or obtaining another type of the asymptotic formula for the integral in (2.10).

We proceed now with the proof. By using (2.6) and the mean theorem for Dirichlet polynomials (e.g., see [2, Chapter 5]), it is seen that the contribution of ∑_{n≤c_{0}t} in (2.5) to (2.2) is ≪ T^{3/2} log^{10} T (the exponent of the logarithm is not optimal, but it is unimportant). Then we simplify the first sum in (2.5) by Taylor’s formula (truncating it at n = \sqrt{T}) to obtain (1.7), as claimed.

Before we go to the second step, let (1 ≪ N ≪ \sqrt{T})

g_{N}(t) := \frac{1}{2} (\frac{2}{π} )^{3/4} ∑_{n≤N}(−1)^{n}d(n)n^{−5/4} sin(\sqrt{8πnt} − \frac{1}{4}π).  (2.14)

24 A. Ivić

By using trivial estimation and (2.8) we have

∫_{T}^{2T} ∑_{n≥N}(−1)^{n}d(n)n^{−5/4} sin(\sqrt{8πnt} − \frac{1}{4}π)|ζ(\frac{1}{2} + it)|^{4}dt

≪ T N^{−1/4} log N log^{4} T ≪ T (log T)^{7/2} log log T

(2.15)

by choosing

N = (log T)^{2}.  (2.16)

Hence, in view of (1.6) and (2.7), it remains to prove that

∫_{T}^{2T} g_{N}(t)|ζ(\frac{1}{2} + it)|^{4}dt ≪ T (log T)^{7/2} log log T.  (2.17)

Now we have, with I(T, Δ) given by (2.11),

H(T, Δ) := ∫_{T}^{2T} g_{N}(t)(|ζ(\frac{1}{2} + it)|^{4} − I(t, Δ))dt

= \frac{1}{Δ\sqrt{π}} ∫_{−∞}^{∞} e^{−(u/Δ)^{2}} ∫_{T}^{2T} g_{N}(t)(|ζ(\frac{1}{2} + it)|^{4} − |ζ(\frac{1}{2} + it + iu)|^{4})dt du

= \frac{1}{Δ\sqrt{π}} ∫_{−∞}^{∞} e^{−(u/Δ)^{2}} ∫_{T}^{2T} g_{N}(t) (∫_{u}^{0} \frac{∂}{∂x}|ζ(\frac{1}{2} + it + ix)|^{4}dx) dt du

= \frac{1}{Δ\sqrt{π}} ∫_{−∞}^{∞} e^{−(u/Δ)^{2}} ∫_{u}^{0} (∫_{T}^{2T} g_{N}(t)\frac{∂}{∂t}|ζ(\frac{1}{2} + it + ix)|^{4}dt) dx du

= \frac{1}{Δ\sqrt{π}} ∫_{−∞}^{∞} e^{−(u/Δ)^{2}} ∫_{−|u|}^{|u|} O(|ζ(\frac{1}{2} + iT + ix)|^{4})dx du

+\frac{1}{Δ\sqrt{π}} ∫_{−∞}^{∞} e^{−(u/Δ)^{2}} ∫_{−|u|}^{|u|} O(|ζ(\frac{1}{2} + 2iT + ix)|^{4})dx du

−\frac{1}{Δ\sqrt{π}} ∫_{−∞}^{∞} e^{−(u/Δ)^{2}} ∫_{u}^{0} ∫_{T}^{2T} g′_{N}(t)|ζ(\frac{1}{2} + it + ix)|^{4}dt dx du.

The integrals over u can be truncated at |u| = Δ log T with a negligible error. Since we have (see [4] and [12])

∫_{T −G}^{T +G} |ζ(\frac{1}{2} + it)|^{4}dt ≪ G log^{4} T + T G^{−1/2} log^{C} T (C > 0, T^{ε} ≤ G ≤ T), and

g′_{N}(t) ≪ T^{−1/2}N^{1/4} log N  (T ≤ t ≤ 2T),  (2.18)

it follows that

H(T, Δ) ≪ \frac{1}{Δ} ∫_{−Δ log T}^{Δ log T} e^{−(u/Δ)^{2}}du( ∫_{T −Δ log T}^{T +Δ log T}|ζ(\frac{1}{2} + ix)|^{4}dx

+ ∫_{2T −Δ log T}^{2T +Δ log T}|ζ(\frac{1}{2} + ix)|^{4}dx)  (2.19)

+Δ^{−1}T^{−1/2}N^{1/4} log N ∫_{−Δ log T}^{Δ log T} e^{−(u/Δ)^{2}}|u|du T log^{4} T
≪ Δ log^{5} T + T Δ^{−1/2} log^{C} T + T^{1/2}ΔN^{1/4} log N log^{4} T
≪ T log^{6−A} T ≪ T (log T)^{7/2} log log T

in view of (2.16), where we choose with A > 0 sufficiently large (this is the lower bound in the permissible range for which (2.10) holds)

Δ = T^{1/2} log^{−A} T.

We are now at the final step of the proof of Theorem 1. From (2.10) and (2.11) we obtain, on integrating by parts (again Q_{4}(x) = P_{4}(x) + P′_{4}(x)),

∫_{T}^{2T} g_{N}(t)I(t, Δ)dt = ∫_{T}^{2T} g_{N}(t)Q_{4}(log t)dt

+O (Δ log^{5} T ∫_{T}^{2T} |g′_{N}(t)|dt) + O(Δ log^{5} T) + O(T Δ^{−1/2} log^{4} T)

−π ∫_{T}^{2T} g′_{N}(t) (\sqrt{\frac{1}{2}t} ∑_{κ_{j}≤T Δ^{−1} log T} α_{j}H_{j}^{3}(\frac{1}{2})c_{j} cos(κ_{j} log \frac{κ_{j}}{4et})e^{−(κ_{j}Δ/(2t))^{2}} ) dt,

where we used (2.12). We have

∫_{T}^{2T} g_{N}(t)Q_{4}(log t)dt ≪ T^{1/2} log^{4} T

by the first derivative test, and by using (2.18) we obtain

∫_{T}^{2T} g_{N}(t)I(t, Δ)dt  (2.20)

≪ T^{1/2} log^{4} T + T^{1/2}ΔN^{1/4} log N log^{5} T + T Δ^{−1/2} log^{4} T + |K(T, Δ)|, where we have set

K(T, Δ) = ∑_{n≤N}(−1)^{n}d(n)n^{−3/4} ∑_{κ_{j}≤T Δ^{−1} log T} α_{j}H_{j}^{3}(\frac{1}{2})c_{j} ×

× ∫_{T}^{2T} cos(\sqrt{8πnt} − \frac{1}{4}) cos(κ_{j} log \frac{κ_{j}}{4et})e^{−(κ_{j}Δ/(2t))^{2}}dt.

We write the cosines as exponentials and note that the saddle point of the ensuing integral is at t_{0} = κ_{j}^{2}/(2πn) ∈ [T, 2T] for κ_{j} ≍ \sqrt{T n}. By the saddle point method (see e.g., [2]) the main contribution will be a multiple of (F (t) = \sqrt{8πnt} − κ_{j} log t)

∑_{n≤N}(−1)^{n}d(n)n^{−3/4} ∑_{κ_{j}≍\sqrt{T n}} α_{j}H_{j}^{3}(\frac{1}{2})c_{j}e^{iF (t_{0})}(F′′(t_{0}))^{−1/2}e^{−(κ_{j}Δ/(2t_{0}))^{2}}

≪ ∑_{n≤N} d(n)n^{−7/4} ∑_{κ_{j}≍\sqrt{T n}} α_{j}H_{j}^{3}(\frac{1}{2})

≪ T N^{1/4} log^{3} T log N ≪ T (log T)^{7/2} log log T,

(2.21)

where we used again (2.12). Thus from (2.20) and (2.21) we obtain the bound in (2.17), as asserted. This completes the proof of Theorem 1. In concluding, note that the inner sum in (2.21) was estimated trivially. However, one hopes that there is a lot of cancellation in such type of exponential sum with α_{j}H_{j}^{3}(\frac{1}{2}). Indeed, it was conjectured by the author in [8] that such a cancellation occurs, and it was heuristically justified why one does expect this fact. Also there is hope to use the explicit expression which stands for the function R(K) in the proof of the asympotic formula (2.13). The small improvement of the bound in (1.5) of Theorem 1 over the bound in (2.9), which is relatively not difficult to obtain, is precisely significant for this reason: it does show that cancellation in a sum with α_{j}H_{j}^{3}(\frac{1}{2}) does occur.

3. Proof of Theorem 2 and Theorem 3

We shall first deal with Theorem 3, which is needed for the proof of Theorem 2. The proof is based on the method used by the author in [3]. Suppose 1 ≪ H ≪ T. We note that, by the first derivative test,

∫_{T}^{T +H} g(t)dt ≪ \sqrt{T}  (3.1)

holds uniformly in H, and proceed as follows. Let

E = \frac{1}{8} (\frac{2}{π} )^{3/2} \frac{ζ^{4}(\frac{5}{2})}{ζ(5)}.

Then we have (EH comes from the terms m = n), by the first derivative test,

\begin{matrix}∫_{T}^{T +H} g^{2}(t)dt = EH + O(\sqrt{T}) \\ +O( ∑_{m,n=1;m≠n}^{∞} d(m)d(n)(mn)^{−5/4} |∫_{T}^{T +H} exp(i\sqrt{8πmt} ± i\sqrt{8πnt})dt|) \\ = EH + O(\sqrt{T} ∑_{m,n=1;m≠n}^{∞} d(m)d(n)(mn)^{−5/4}|\sqrt{m} − \sqrt{n}|^{−1}) + O(\sqrt{T}) \\ = EH + O(\sqrt{T} ∑_{r=1}^{∞} ∑_{n=1}^{∞}(n(n + r))^{ε−5/4}r^{−1}(n + r)^{1/2}) + O(\sqrt{T}) \\ = EH + O(\sqrt{T}). \end{matrix} (3.2)

Thus (3.2) implies that

∫_{T}^{T +H} g^{2}(t)dt ≫ H,  (3.3)

provided that H = D\sqrt{T} with a sufficiently large constant D > 0. We shall show now that there exists τ ∈ [T, T + H] such that g(τ) > 2η for some constant η > 0 (and also a point τ_{1} such that g(τ_{1}) < −2η). Suppose on the contrary that g(t) < ε if g(t) > 0 for any given ε > 0 (the case when g(t) > −ε if g(t) < 0 is treated analogously). Let (cf. (1.8))

g_{+}(t) = max(g(t), 0),  g_{−}(t) = min(g(t), 0).  (3.4)

Since g(t) is bounded we obtain, for some constants C_{1}, C_{2} > 0, on using (3.1),

∫_{T}^{T +H} g^{2}(t)dt = ∫_{T}^{T +H} g_{+}^{2}(t)dt + ∫_{T}^{T +H}(−g_{−}(t))^{2}dt
≤ Hε^{2} + C_{1} ∫_{T}^{T +H}(−g_{−}(t))dt
= Hε^{2} − C_{1} ∫_{T}^{T +H} g(t)dt + C_{1} ∫_{T}^{T +H} g_{+}(t)dt
≤ Hε^{2} + C_{2}\sqrt{T} + C_{1}εH.

However the above bound contradicts (3.3) if ε is small enough and D is large enough, since H = D\sqrt{T}. Hence there exists τ ∈ [T, T + H] such that g(τ) > 2η for some constant η > 0. Setting for brevity c = \frac{1}{2}(2/π)^{3/4}, using | sin x − sin y| ≤ |x − y| (x, y ∈ ℝ), we have (N ≥ 3, x ≫ 1)

|g(τ + x) − g(τ)|

= |c ∑_{n≤N}(−1)^{n}d(n)n^{−5/4} (sin(\sqrt{8πn(τ + x)} − π/4) − sin(\sqrt{8πnτ} − π/4)) |

+O ( ∑_{n>N} d(n)n^{−5/4})

≪ ∑_{n≤N} d(n)n^{−5/4}n^{1/2}T^{−1/2}|x| + ∑_{n>N} d(n)n^{−5/4}

≪ T^{−1/2}|x|N^{1/4} log N + N^{−1/4} log N < η

if |x| ≤ L(T) := CηT^{1/2}N^{−1/4} log N and N = [η^{−8}], provided that η is sufficiently small (which may be assumed), and C > 0 is a suitable absolute constant. One obtains then

|g(τ + x)| ≥ g(τ) − |g(τ + x) − g(τ)| > η, |x| ≤ L(T).

This means that every interval [T_{1}, T_{1} + D\sqrt{T}] (T ≤ T_{1} ≤ 2T, D sufficiently large) contains a subinterval of length ≫_{η} \sqrt{T} in which g(t) > η. Consequently we divide [T, 2T] into ≫ \sqrt{T} subintervals of the form [T + (j − 1)D\sqrt{T} , T + jD\sqrt{T}], (j = 1, 2, …), and we obtain that g(t) > η on a set A(T) satisfying μ(A(T)) ≫_{η} T, as claimed.

We pass now to the proof of Theorem 2. The upper bounds in (1.9) and (1.10) follow easily from the fact that g(t) is bounded and that one has

∫_{T}^{2T} |ζ(\frac{1}{2} + it)|^{2k}dt ≪ T (log T)^{k^{2}} (k = 1, 2).

For the lower bounds in (1.9) and (1.10) (they actually hold for any fixed k ∈ ℕ) we use the well-known bound (see e.g., [2] or [4])

∫_{T}^{T +C\sqrt{T}} |ζ(\frac{1}{2} + it)|^{2k}dt ≫ \sqrt{T} (log T)^{k^{2}}  (C > 0, k ∈ ℕ).  (3.5)

In each interval of length ≥ C\sqrt{T} in the proof of Theorem 2 (where we had g(t) > η) we use (3.5) and gather the resulting lower bounds to obtain the lower bound in (1.9). The proof of the lower bound in (1.10) is analogous.

REFERENCES

[1] J. L. H a f n e r and A. I v i ć, On the mean square of the Riemann zeta-function on the critical line, J. Number Theory 32(1989), 151-191.

[2] A. I v i ć, The Riemann zeta-function, John Wiley & Sons, New York, 1985.

[3] A. I v i ć, Large values of certain number-theoretic error terms, Acta Arith. 56(1990), 135-159.

[4] A. I v i ć, Mean values of the Riemann zeta-function, LN’s 82, Tata Institute of Fundamental Research, Bombay, 1991 (distr. by Springer Verlag, Berlin etc.).

[5] A. I v i ć, On the fourth moment of the Riemann zeta-function, Publications InstṀath. (Belgrade) 57(71) (1995), 101-110.

[6] A. I v i ć, The Mellin transform and the Riemann zeta-function, Proceedings of the Conference on Elementary and Analytic Number Theory (Vienna, July 18-20, 1996), Universität Wien & Universität für Bodenkultur, Eds. W.G. Nowak and J. Schoißengeier, Vienna 1996, pp. 112-127.

[7] A. I v i ć, On the error term for the fourth moment of the Riemann zeta-function, J. London Math. Soc., (2)60(1999), 21-32.

[8] A. I v i ć, On some conjectures and results for the Riemann zeta-function and Hecke series, Acta Arith. 109(2001), 115-145.

[9] A. I v i ć, On sums of Hecke series in short intervals, J. Théorie des Nombres Bordeaux 14(2001), 554-568.

[10] A. I v i ć, On the moments of Hecke series at central points, Functiones et Approximatio 30(2002), 49-82.

[11] A. I v i ć and Y. M o t o h a s h i, The mean square of the error term for the fourth moment of the zeta-function, Proc. London Math. Soc. (3)f̄66(1994), 309-329.

[12] A. I v i ć and Y. M o t o h a s h i, The fourth moment of the Riemann zeta-function, J. Number Theory 51(1995), 16-45.

[13] Y. M o t o h a s h i, Spectral theory of the Riemann zeta-function, Cambridge University Press, Cambridge, 1997.

Katedra Matematike RGF-a Universiteta u Beogradu -Dušina 7 11000 Beograd Serbia and Montenegro

e-mail: aivic@rgf.bg.ac.yu, aivic@matf.bg.ac.yu