Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 213–238

ON HECKE L-FUNCTIONS ASSOCIATED WITH

CUSP FORMS II: ON THE SIGN CHANGES OF S_{f}(T)

A. Sankaranarayanan

Tata Institute of Fundamental Research, School of Mathematics Mumbai-400 005, India; sank@math.tifr.res.in

In honour of Professor T. N. Shorey on his sixtieth birthday

Abstract. We study the number of sign changes of S_{f} (t) (related to Hecke L-functions attached to holomorphic cusp forms of even positive integral weight with respect to the full modular group) over shorter intervals.

1. Introduction

Let

S(t) = π^{−1} arg ζ(\frac{1}{2} + it),

where the argument is obtained by continuous variation along the straight lines joining 2, 2 + it and \frac{1}{2} + it, starting with the value zero. When t is equal to the imaginary part of any zero of ζ(s), we put

S(t) = lim_{ε→0} \frac{1}{2}\{S(t + ε) + S(t − ε)\}.

As for Atle Selberg’s comment on a deep result of Littlewood on S(t), A. Ghosh established that (see Theorem 1 of [5] and also the paper of Selberg [16]) S(t) changes its sign at least

T (log T) exp(−A(δ)(log log T)(log log log T)^{−(1/2)+δ})

times in the interval (T, 2T). Here δ is any arbitrarily small positive constant, and A(δ) > 0 depending only on δ. In fact, he proved this result over shorter intervals.

Let f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz} be a holomorphic cusp form of even integral weight k > 0 with respect to the full modular group Γ = SL(2, Z). We define the associated Hecke L-function

(1.1)  L_{f}(s) = ∑_{n=1}^{∞} a_{n}n^{−s}

2000 Mathematics Subject Classification: Primary 11Mxx; Secondary 11M06, 11M41.

for Re s > (k + 1)/2. Throughout this paper, we assume that f(z) is a Hecke eigenform with a_{1} = 1. It is known (see [7]) that L_{f}(s) admits analytic continuation to C as an entire function and it satisfies the functional equation

(1.2)  (2π)^{−s}Γ(s)L_{f}(s) = (−1)^{k/2}(2π)^{−(k−s)}Γ(k − s)L_{f}(k − s).

L_{f}(s) has an Euler-product representation (for Re s > (k + 1)/2)

(1.3)  L_{f}(s) = ∏_{p}(1 − a_{p}p^{−s} + p^{k−1}p^{−2s})^{−1}.

The non-trivial zeros of L_{f}(s) lie within the critical strip (k − 1)/2 < Re s < (k +1)/2. These zeros are located symmetrically to the real axis and they are also symmetrical about the line Re s = k/2. The Riemann hypothesis in this situation asserts that all the non-trivial zeros are on the critical line Re s = k/2. From Deligne’s proof of Ramanujan–Petersson’s conjecture (see [1] and [2]), we have the bound for the coefficients

(1.4)  |a_{n}| ≤ d(n)n^{(k−1)/2}.

Several interesting deep results about the Hecke L-functions have been established lately. As a sample, a certain average growth of these L-functions in the weight aspect on the critical line has been investigated in the papers of Peter Sarnak (see [15]) and of Matti Jutila and Yoichi Motohashi (see [9]).

Let N_{f}(T) denote the number of zeros β + iγ of L_{f}(s) for which 0 < γ < T. If T is equal to the ordinate of any zero, then we define

(1.5)  N_{f}(T) := lim_{ε→0} \frac{1}{2}\{N_{f}(T + ε) + N_{f}(T − ε)\}.

Now, one can show that (following Theorem 9.3 of [18])

(1.6)  N_{f}(T) = \frac{T}{π} log \frac{T}{π} − \frac{T}{π} + 1 + S_{f}(T) + O(\frac{1}{T} ),

where

(1.7)  S_{f}(t) = \frac{1}{π} arg L_{f}(\frac{k}{2} + it).

The argument is obtained by a continuous variation along the straight lines joining the points \frac{1}{2}k+1, \frac{1}{2}k+1+it and \frac{1}{2}k+it, starting with the value \frac{1}{2}(k − 1). Hence the variation of S_{f}(t) is closely connected with the distribution of the imaginary parts of the zeros of L_{f}(s).

We now define, for σ ≥ k/2, T ≥ 1 and H ≤ T,

(1.8) N_{f}(σ, T, T + H) = #{β + iγ : L_{f}(β + iγ) = 0, β ≥ σ, T ≤ γ ≤ T + H}.

In [14], we proved the following two theorems:

On the sign changes of S_{f}(T) 215

Theorem A. For t ≥ 2, 2 ≤ x ≤ t^{2} , we have
S_{f}(t) = −\frac{1}{π} ∑_{n<x^{3}} \frac{Λ_{x,f}(n) sin(t log n)}{n^{σ_{x,t}} log n} + O((σ_{x,t} − k/2)| ∑_{n<x^{3}} \frac{Λ_{x,f}(n)}{n^{σ_{x,t}+it}} |)
+ O((σ_{x,t} − k/2) log t),

where

σ_{x,t} = k/2 + 2 max(β − k/2, 2/ log x),

ϱ = β + iγ running over those zeros for which

|t − γ| ≤ x^{3|β−k/2|}(log x)^{−1},

and Λ_{x,f}(n) is as in (2.6).

As corollaries we obtained (by choosing x = \sqrt{log t} )

S_{f}(t) = O(log t)

unconditionally, and assuming the Riemann hypothesis for L_{f}(s), we got

S_{f}(t) = O(\frac{log t}{log log t}).

Theorem A′. Let B be any fixed small positive constant. Let

B′ = \frac{19}{20} + \frac{13.505}{5}B

and B′ < α ≤ 1. Then for T^{α} ≤ H ≤ T, we have

N_{f}(σ, T, T + H) ≪ H(\frac{H}{T^{B′}} )^{−(B/(1−B′))(σ−k/2)} log T

uniformly for k/2 ≤ σ ≤ (k + 1)/2.

As an application to the above Theorems A and A′, the object of this paper is now to prove

Main theorem. Let B′ be the constant as in Theorem A′. Let B′ < α ≤ 1. If (T + 1)^{α} ≤ H ≤ T and δ > 0 is an arbitrarily small real number, there is an A = A(α, δ) > 0 and a T_{0} = T_{0}(α, δ) > 0 such that when T > T_{0} , S_{f}(t) changes its sign at least

H(log T) exp(−A(log log T)(log log log T)^{−(1/2)+δ})

times in the interval (T, T + H).

Remark 1. This main theorem is an analogous result of the theorem in the case of S(t) related to the ordinary Riemann zeta-function, which was established by A. Ghosh (see Theorem 1 of [5]). In the case of S(t), B′ can be replaced by \frac{1}{2} (or even by a better positive constant).

Remark 2. If we assume the Riemann hypothesis for L_{f}(s), then the main theorem is true with 0 < α ≤ 1.

The proof requires asymptotic formulae for integrals of the type

∫_{T}^{T +H} |S_{f}(t)|^{2l} dt

and

∫_{T}^{T +H} |S_{1,f}(t + h) − S_{1,f}(t)|^{2l} dt,

where

S_{1,f}(t) := ∫_{0}^{t} S_{f}(u) du

with the error terms uniform in integers l ≥ 1 and h > 0 with a suitable value of h. It should be mentioned that the asymptotic formulae for higher moments of S(t) over shorter intervals have been extensively studied earlier in [3], [4], [5] and [6].

In fact, first we establish the following theorems from which the main theorem follows. The constants B and B′ occurring in the sequel are as in Theorem A′, which we do not mention hereafter.

Theorem 1. Let B′ < α ≤ 1. If T^{α} ≤ H ≤ T, then there is an absolute positive constant A_{1} = A_{1}(α) such that for any integer l satisfying

1 ≤ l ≪ (log log T)^{1/3},

we have

∫_{T}^{T +H} |S_{f}(t)|^{2l} dt = \frac{(2l)!}{l!}(\frac{1}{2π} )^{2l}H(log log T)^{l}+O(A_{1}^{l}l^{l−(1/2)}H(log log T)^{l−(1/2)}),

where the implied constants depend at most on α.

Theorem 2. Let B′ < α ≤ 1. If (T + h)^{α} ≤ H ≤ T, then there is an absolute positive constant A_{2} = A_{2}(α) such that for any integer l, with

1 ≤ l ≪ (log log T)^{1/3},

and any h satisfying

(log T)^{1/2} < h^{−1} < \frac{1}{10l} log T,

we have

∫_{T}^{T +H} |S_{1,f}(t + h) − S_{1,f}(t)|^{2l} dt = \frac{(2l)!}{l!}(\frac{h}{2π} )^{2l}H(log h^{−1})^{l}
+ O(A_{2}^{l}l^{l−(1/2)}Hh^{2l}(log log T)^{l−(1/2)}).

Remark 3. Theorems 1 and 2 are analogous results of Theorems 2 and 3 of [5]. However, here the range of l as well as the error terms have been improved. In fact, Theorems 2 and 3 of [5] hold with this range of l as well as with this error term, which can be easily noticed from our arguments.

As a consequence of Theorems 1 and 2, we obtain

Theorem 3. Let B′ < α ≤ 1. If T^{α} ≤ H ≤ T, then for any given δ > 0, we have

∫_{T}^{T +H} |S_{f}(t)| dt = \frac{2}{\sqrt{π}} \frac{H}{2π} (log log T)^{(1/2)}
+ O_{δ}(H((log log T)(log log log T)^{−(1/2)+δ})^{(1/2)}),

where the implied constants depend on α and δ.

Theorem 4. Let B′ < α ≤ 1. If (T + h)^{α} ≤ H ≤ T, then for any given δ > 0 and any h satisfying

(log T)^{1/2} < h^{−1} < ε_{1} \frac{log T}{log log T} ,

for some suitable constant ε_{1} = ε_{1}(α) > 0, we have

∫_{T}^{T +H} |S_{1,f}(t + h) − S_{1,f}(t)| dt = \frac{2}{\sqrt{π}} \frac{Hh}{2π}(log h^{−1})^{1/2}

+ O(Hh((log log T)(log log log T)^{−(1/2)+δ})^{1/2}),

where the implied constants depend on α and δ.

Remark 4. We prove Theorems 1 and 2 in detail adapting the approach of [5] to our situation. However, we need an asymptotic estimate for the quantity ∑_{p≤x} a_{p}^{2} log p/p^{k−1} which is proved in Section 4 using Shimura’s split of the Rankin–Selberg L-function into the ordinary Riemann zeta-function and the symmetric square L-function associated to a Hecke eigenform f for the full modular group. Apart from this, Theorem A′ plays a crucial role (on the whole) particularly in proving the main theorem over shorter intervals.

Acknowledgement. The author wishes to thank the anonymous referee for the careful reading of the manuscript and for valuable comments.

2. Notation and preliminaries

Throughout the paper, the implied constants A are effective absolute positive constants and they need not be the same at each occurrence. When k is even, it is known that a_{n} s are real. In fact, they are totally real algebraic numbers. Hence a_{p} is real from (1.1) and (1.3). By Deligne’s estimate, we also have |a_{p}| ≤ 2p^{(k−1)/2} . We define a real number A′_{p} such that a_{p} = 2A′_{p}p^{(k−1)/2} , and hence, |A′_{p}| ≤ 1. Let α′_{p} and \overline{α′_{p}} be the roots of the equation x^{2} − 2A′_{p}x + 1 = 0 and we note that |α′_{p}| = 1. Therefore, from the Euler product of L_{f}(s), we can write

(2.1)  L_{f}(s) = ∏_{p}(1 − α_{p}p^{−s})^{−1}(1 − \overline{α_{p}}p^{−s})^{−1}

with |α_{p}| = p^{(k−1)/2} and a_{p} = α_{p} + \overline{α_{p}}. Taking the logarithm and differentiating both sides of (2.1) with respect to s, we find that

(2.2)  −\frac{L′_{f}(s)}{L_{f}(s)} = ∑_{m≥1,p}(α_{p}^{m} + \overline{α_{p}}^{m})p^{−ms}(log p).

Now we define

(2.3)  Λ_{f}(n) = (α_{p}^{m} + \overline{α_{p}}^{m})(log p) if n = p^{m}; 0 otherwise.

Hence we obtain

(2.4)  −\frac{L′_{f}(s)}{L_{f}(s)} = ∑_{n=2}^{∞} Λ_{f}(n)n^{−s} (in Re s > (k + 1)/2).

Note that

(2.5)  Λ_{f}(n) ≤ 2(log n)n^{(k−1)/2}.

For x > 1, we define

(2.6) Λ_{x,f}(n) = \{ \begin{matrix}Λ_{f}(n), & if 1 ≤ n ≤ x, \\ Λ_{f}(n)\frac{\{(log(\frac{x^{3}}{n} ))^{2} − 2(log(\frac{x^{2}}{n} ))^{2}\}}{2(log x)^{2}}, & if x ≤ n ≤ x^{2}, \\ Λ_{f}(n)\frac{(log(\frac{x^{3}}{n} ))^{2}}{2(log x)^{2}} & for x^{2} ≤ n ≤ x^{3}. \end{matrix}

3. Some lemmas

Lemma 3.1. Let τ be a real positive number and suppose that δ(n) are complex numbers satisfying

|δ(n)| ≤ C

for some fixed constant C > 0. Then, for any integer l ≥ 1, we have

\begin{aligned}S_{1} & := ∑_{p_{1},···,p_{l}<y, ; {q_{1},···,q_{l}<y, \atop p_{1}···p_{l}=q_{1}···q_{l}}} \frac{δ(p_{1}) ⋯ δ(p_{l})δ(q_{1}) ⋯ δ(q_{l})}{(p_{1} ⋯ p_{l}q_{1} ⋯ q_{l})^{τ}} \\ & = l!(∑_{p<y} \frac{δ^{2}(p)}{p^{2τ}})^{l} + O(C^{2l}l!(∑_{p<y} p^{−2τ} )^{l−2}(∑_{p<y} p^{−4τ} )). \end{aligned}

Proof. See, for example, Lemma 1 of [5].

For x ≥ 2, t > 0, we define the number σ_{x,t} by

σ_{x,t} = k/2 + 2 max(β − k/2, 2/ log x),

where ϱ = β + iγ runs over all zeros of L_{f}(s) for which

|t − γ| ≤ x^{3|β−k/2|}(log x)^{−1}.

Lemma 3.2. Suppose that T^{α} ≤ H ≤ T, where B′ < α ≤ 1 and x ≥ 2, 1 ≤ ξ ≤ x^{8l}, x^{3}ξ^{2} ≤ (H/T^{B′})^{1/4} . Then, for 0 ≤ ν ≤ 8l, we have

I_{1} := ∫_{T}^{T +H}(σ_{x,t} − \frac{k}{2} )^{ν}ξ^{σ_{x,t}−(k/2)} dt ≪ A^{l} \frac{H}{(log x)^{ν}}

+ A^{l}(H log T ((ν)!\frac{log T}{log x} (\frac{4}{log(H/T^{B′})})^{ν+1}

+ (ν)!\frac{1}{log x}(\frac{4}{log(H/T^{B′})})^{ν})).

Proof. The proof follows using Theorem A′ at the appropriate place of the proof of Lemma 12 of [16].

Lemma 3.3. Let H > 1, l ≥ 1 and 1 < y ≤ H^{1/l}. Suppose that β_{p} are complex numbers satisfying

(3.3.1)  |β_{p}| < B_{1} \frac{log p}{log y} for p < y.

Then, we have

(3.3.2)  ∫_{0}^{H} |∑_{p<y} β_{p}p^{−(1/2)−it}|^{2l} dt ≪ (AB_{1}^{2}l)^{l}H,

and if |β_{p}| < B_{1} , then we have

(3.3.3)  ∫_{0}^{H} |∑_{p<y} β_{p}p^{−1−2it}|^{2l} dt ≪ (AB_{1}^{2}l)^{l}H.

Proof. See, for example, Lemma 3 of [5].

Remark. It should be mentioned here that a general mean-value theorem for the Dirichlet polynomial with a better error term is also available, for which we refer to [10].

Lemma 3.4. Let B′ < α ≤ 1, T^{α} ≤ H ≤ T and x = T^{(α−B′)/(60l)} . Then, for T ≤ t ≤ T + H, we have

S_{f}(t) + \frac{1}{π} ∑_{p<x^{3}} \frac{(α_{p} + \overline{α_{p}}) sin(t log p)}{p^{k/2}}

= O(| ∑_{p<x^{3}} \frac{Λ_{f}(p) − Λ_{x,f}(p)}{p^{k/2} log p}p^{−it}|)

+ O(| ∑_{p<x^{3/2}} \frac{Λ_{x,f}(p^{2})}{p^{k} log p}p^{−2it}|) + O((σ_{x,t} − \frac{k}{2} ) log T )

+ O((σ_{x,t} − \frac{k}{2} )x^{(σ_{x,t}−(k/2))} ∫_{k/2}^{∞} x^{(k/2)−σ}| ∑_{p<x^{3}} \frac{Λ_{x,f}(p) log(xp)}{p^{σ+it}} | dσ).

Proof. From Theorem A (stated in the introduction), we obtain

(3.4.1)

S_{f}(t) = −\frac{1}{π} ∑_{p<x^{3}} \frac{Λ_{x,f}(p) sin(t log p)}{p^{σ_{x,t}} log p} − \frac{1}{π} ∑_{p^{2}<x^{3}} \frac{Λ_{x,f}(p^{2}) sin(t log p^{2})}{p^{2σ_{x,t}}(log p^{2})}

+ O((σ_{x,t} − \frac{k}{2} )| ∑_{p<x^{3}} \frac{Λ_{x,f}(p)}{p^{σ_{x,t}+it}} |)

+ O((σ_{x,t} − \frac{k}{2} )| ∑_{p^{2}<x^{3}} \frac{Λ_{x,f}(p^{2})}{p^{2σ_{x,t}+2it}} |)

+ O(| ∑_{{p^{r} <x^{3} \atop r>2}} \frac{Λ_{x,f}(p^{r}) sin(t log p^{r})}{p^{rσ_{x,t}}(log p^{r})}|)

+ O((σ_{x,t} − \frac{k}{2} )| ∑_{{p^{r} <x^{3} \atop r>2}} \frac{Λ_{x,f}(p^{r})}{p^{rσ_{x,t}+rit}} |) + O((σ_{x,t} − \frac{k}{2} ) log t).

On the sign changes of S_{f}(T) 221

Note that σ_{x,t} ≥ \frac{1}{2}k and

|Λ_{x,f}(n)| ≤ |Λ_{f}(n)| ≤ 2(log n)n^{(k−1)/2}.

Now, it is easy to see that

(3.4.2)  ∑_{{p^{r} <x^{3} \atop r>2}} \frac{Λ_{x,f}(p^{r}) sin(t log p^{r})}{p^{rσ_{x,t}}(log p^{r})} = O(1) = O((σ_{x,t} − \frac{k}{2} ) log T ),

(3.4.3)  \begin{aligned}(σ_{x,t} − \frac{k}{2} )| ∑_{{p^{r} <x^{3} \atop r>2}} \frac{Λ_{x,f}(p^{r})}{p^{rσ_{x,t}+rit}} | & = O((σ_{x,t} − \frac{k}{2} )) \\ & = O((σ_{x,t} − \frac{k}{2} ) log T ) \end{aligned}

and

(3.4.4)  \begin{aligned}(σ_{x,t} − \frac{k}{2} )| ∑_{p^{2}<x^{3}} \frac{Λ_{x,f}(p^{2})}{p^{2σ_{x,t}+2it}} | & = O((σ_{x,t} − \frac{k}{2} ) log x) \\ & = O((σ_{x,t} − \frac{k}{2} ) log T ). \end{aligned}

Now, we write the first four terms on the right-hand side of (3.4.1) in the following manner, namely,

(3.4.5)

S_{f}(t) + \frac{1}{π} ∑_{p<x^{3}} \frac{(α_{p} + \overline{α_{p}}) sin(t log p)}{p^{k/2}}

= O(| ∑_{p<x^{3}} \frac{Λ_{f}(p) − Λ_{x,f}(p)}{p^{k/2} log p}p^{−it}|)

+ O(| ∑_{p<x^{3}} \frac{Λ_{x,f}(p)}{p^{k/2} log p}(1 − p^{(k/2)−σ_{x,t}})p^{−it}|)

+ O((σ_{x,t} − \frac{k}{2} )| ∑_{p<x^{3}} \frac{Λ_{x,f}(p)}{p^{σ_{x,t}+it}} |)

+ O(| ∑_{p<x^{3/2}} \frac{Λ_{x,f}(p^{2})}{p^{k} log p}p^{−2it}|)

+ O(| ∑_{p<x^{3/2}} \frac{Λ_{x,f}(p^{2})}{p^{k} log p}(1 − p^{k−2σ_{x,t}})p^{−2it}|)

+ O((σ_{x,t} − \frac{k}{2} ) log T ).

We note that

(3.4.6)

Q_{1} := | ∑_{p<x^{3/2}} \frac{Λ_{x,f}(p^{2})}{p^{k} log p}(1 − p^{k−2σ_{x,t}})p^{−2it}|

< ∑_{p<x^{3/2}} \frac{2(log p)p^{k−1}}{p^{k} log p} (1 − p^{k−2σ_{x,t}})

< ∑_{p<x^{3/2}} \frac{4(σ_{x,t} − \frac{1}{2}k) log p}{p} = O((σ_{x,t} − \frac{1}{2}k) log T ),

since

σ_{x,t} ≥ \frac{k}{2} + \frac{4}{log x}

and 1 − e^{−x} < x. Further, we have

(3.4.7)  \begin{aligned}Q_{2} & := | ∑_{p<x^{3}} \frac{Λ_{x,f}(p)}{p^{k/2} log p}(1 − p^{(k/2)−σ_{x,t}})p^{−it}| \\ & = |∫_{k/2}^{σ_{x,t}} ∑_{p<x^{3}} \frac{Λ_{x,f}(p)}{p^{σ′+it}} dσ′| ≤ ∫_{k/2}^{σ_{x,t}} | ∑_{p<x^{3}} \frac{Λ_{x,f}(p)}{p^{σ′+it}} | dσ′. \end{aligned}

If \frac{1}{2}k ≤ σ′ ≤ σ_{x,t}, then

(3.4.8)

| ∑_{p<x^{3}} \frac{Λ_{x,f}(p)}{p^{σ′+it}} | = |x^{σ′−(k/2)} ∫_{σ′}^{∞} x^{(k/2)−σ} ∑_{p<x^{3}} \frac{Λ_{x,f}(p)(log xp)}{p^{σ+it}} dσ|

≤ x^{σ_{x,t}−(k/2)} ∫_{k/2}^{∞} x^{(k/2)−σ}| ∑_{p<x^{3}} \frac{Λ_{x,f}(p)(log xp)}{p^{σ+it}} | dσ,

and therefore, from (3.4.7) and (3.4.8), we get

(3.4.9) Q_{2} ≤ (σ_{x,t} − \frac{k}{2} )x^{σ_{x,t}−(k/2)} ∫_{k/2}^{∞} x^{(k/2)−σ}| ∑_{p<x^{3}} \frac{Λ_{x,f}(p)(log xp)}{p^{σ+it}} | dσ.

Now, the lemma follows from (3.4.5), (3.4.6) and (3.4.9).

Lemma 3.5. Let B′ < α ≤ 1 and suppose that T^{α} ≤ H ≤ T. Put x = T^{(α−B′)/(60l)} . Then, for l ≪ log T, we have

∫_{T}^{T +H} |S_{f}(t) + \frac{1}{π} ∑_{p<x^{3}} \frac{(α_{p} + \overline{α_{p}}) sin(t log p)}{p^{k/2}} |^{2l} dt ≪ A^{l}l^{2l}H.

On the sign changes of S_{f}(T) 223

Proof. Let

(3.5.1)  ∑_{1}(t) := ∑_{p<x^{3}} \frac{(α_{p} + \overline{α_{p}}) sin(t log p)}{p^{k/2}},
(3.5.2)  E_{1}(t) := ∑_{p<x^{3}} \frac{Λ_{f}(p) − Λ_{x,f}(p)}{p^{k/2} log p}p^{−it},
(3.5.3)  E_{2}(t) := ∑_{p<x^{3/2}} \frac{Λ_{x,f}(p^{2})}{p^{k} log p}p^{−2it},
(3.5.4)  E_{3}(t) := (σ_{x,t} − \frac{1}{2}k) log T,

and

(3.5.5) E_{4}(t) := (σ_{x,t} − \frac{1}{2}k)x^{(σ_{x,t}−(k/2))} ∫_{k/2}^{∞} x^{(k/2)−σ}| ∑_{p<x^{3}} \frac{Λ_{x,f}(p) log(xp)}{p^{σ+it}} | dσ.

Now, clearly from Lemma 3.4, we have

(3.5.6) |S_{f}(t) + π^{−1} ∑_{1}(t)|^{2l} ≪ A^{l}(|E_{1}(t)|^{2l} + |E_{2}(t)|^{2l} + |E_{3}(t)|^{2l} + |E_{4}(t)|^{2l}).

If we take

β_{p} = \frac{Λ_{f}(p) − Λ_{x,f}(p)}{p^{(k−1)/2} log p},

then from the definition of Λ_{f}(n) and Λ_{x,f}(n), we easily find that

β_{p} = 0 for 2 ≤ p ≤ x,
|β_{p}| ≤ 2(\frac{log p}{log x} − 1)^{2} ≤ 2\frac{log p}{log x} for x ≤ p ≤ x^{2},

and

|β_{p}| ≤ 6\frac{log p}{log x} for x^{2} ≤ p ≤ x^{3}.

Therefore,

|β_{p}| ≤ B_{1} \frac{log p}{log x} for p ≤ x^{3}

with some absolute positive constant B_{1} . Similarly, if we take

β′_{p} = \frac{Λ_{x,f}(p^{2})}{p^{k−1} log p},

then from the definition of Λ_{x,f}(n), we find that

Λ_{x,f}(p^{2}) ≤ 9p^{k−1}(log p),

and so we get |β′_{p}| < B_{2} with some absolute positive constant B_{2} . Therefore, from Lemma 3.3, ((3.3.2), (3.3.3), respectively), we obtain

(3.5.7)  ∫_{T}^{T +H} |E_{1}(t)|^{2l} dt ≪ (Al)^{l}H

and

(3.5.8)  ∫_{T}^{T +H} |E_{2}(t)|^{2l} dt ≪ (Al)^{l}H.

Note that we have fixed x = T^{(α−B′)/(60l)} . From Lemma 3.2, with ξ = 1 and ν = 2l, we get

(3.5.9)  ∫_{T}^{T +H} |E_{3}(t)|^{2l} dt ≪ A^{l}(l(2l)! + l^{2l})H ≪ A^{l}l(2l)^{2l−1}H ≪ A^{l}l^{2l}H,

since,

(3.5.10A)  (2l)! ≤ (2l)^{2l−1} for l ≥ 1,

(3.5.10B)

S_{2} := H log T ((ν)!\frac{log T}{log x} (\frac{4}{log(H/T^{B′})})^{ν+1} + (ν)!\frac{1}{log x}(\frac{4}{log(H/T^{B′})})^{ν})

≪ ν!\frac{H}{(log x)(log T)^{ν−1}}

and

(3.5.10C)  \frac{H}{(log x)^{ν}} ≪ A^{l}l^{2l}H.

Now, we notice that

(3.5.11)  ∫_{T}^{T +H} |E_{4}(t)|^{2l} dt ≤ Q_{3}Q_{4}

where

Q_{3} := (∫_{T}^{T +H}(σ_{x,t} − \frac{1}{2}k)^{4l}x^{4l(σ_{x,t}−(k/2))} dt)^{1/2}

On the sign changes of S_{f}(T) 225

and

Q_{4} := (∫_{T}^{T +H}(∫_{k/2}^{∞} x^{(k/2)−σ}| ∑_{p<x^{3}} \frac{Λ_{x,f}(p)(log xp)}{p^{σ+it}} | dσ)^{4l} dt)^{1/2}.

From Lemma 3.2, (with ξ = x^{4l}, ν = 4l), we obtain

(3.5.12)  Q_{3} ≪ (A^{l}(l^{4l} + l(4l)!)H(log T)^{−4l})^{1/2} ≪ A^{l}l^{2l}H^{1/2}(log T)^{−2l}.

By Hölder’s inequality, we get

(3.5.13)

Q_{4}^{2} ≤ ∫_{T}^{T +H}(∫_{k/2}^{∞} x^{(k/2)−σ} dσ)^{4l−1}

× (∫_{k/2}^{∞} x^{(k/2)−σ}| ∑_{p<x^{3}} \frac{Λ_{x,f}(p)(log xp)}{p^{σ+it}} |^{4l} dσ) dt

≤ (log x)^{1−4l}(∫_{k/2}^{∞} x^{(k/2)−σ}

× (∫_{T}^{T +H} | ∑_{p<x^{3}} \frac{Λ_{x,f}(p)(log xp)}{p^{σ+it}} |^{4l} dt) dσ).

By taking

β_{p} = \frac{Λ_{x,f}(p)(log xp)}{p^{(k−1)/2}(log x)^{2}} ,

we observe that |β_{p}| ≤ 10log p/log x. Now, by (3.3.2), we obtain

(3.5.14)  ∫_{T}^{T +H} | ∑_{p<x^{3}} \frac{Λ_{x,f}(p)(log xp)}{p^{σ+it}} |^{4l} dt ≪ (AB_{1}^{2}l)^{2l}H(log x)^{8l}.

Therefore, we get from (3.5.13) and (3.5.14)

(3.5.15)  Q_{4}^{2} ≪ (AB_{1}^{2}l)^{2l}H(log x)^{4l}.

From (3.5.11), (3.5.12) and (3.5.15), with our choice of x, we get

(3.5.16)  \begin{aligned}∫_{T}^{T +H} |E_{4}(t)|^{2l} dt & ≪ A^{l}l^{2l}H^{1/2}(log T)^{−2l}(AB_{1}^{2}l)^{l}H^{1/2}(log x)^{2l} \\ & ≪ A^{l}l^{l}H. \end{aligned}

This proves the lemma.

226 A. Sankaranarayanan

Lemma 3.6. Let B′ < α ≤ 1 and T^{α} ≤ H ≤ T. Then, if l ≥ 1 is an integer and

x^{3} = T^{(α−B′)/(20l)} ≤ z ≤ H^{1/l},

we have

(3.6.1)  Q_{5} := ∫_{T}^{T +H} |S_{f}(t) + \frac{1}{π} ∑_{p<z} \frac{(α_{p} + \overline{α_{p}}) sin(t log p)}{p^{k/2}} |^{2l} dt ≪ A^{l}l^{2l}H.

Proof. We clearly have

(3.6.2)  \begin{aligned}Q_{5} & ≪ 4^{l} ∫_{T}^{T +H} |S_{f}(t) + \frac{1}{π} ∑_{p<x^{3}} \frac{(α_{p} + \overline{α_{p}}) sin(t log p)}{p^{k/2}} |^{2l} dt \\ & + 4^{l} ∫_{T}^{T +H} | ∑_{x^{3}≤p<z} p^{−(1/2)−it}|^{2l} dt. \end{aligned}

From Lemma 3.5, we observe that

(3.6.3)  ∫_{T}^{T +H} |S_{f}(t) + \frac{1}{π} ∑_{p<x^{3}} \frac{(α_{p} + \overline{α_{p}}) sin(t log p)}{p^{k/2}} |^{2l} dt ≪ A^{l}l^{2l}H.

From (3.3.2) (with B_{1} = O(1)), we have

(3.6.4)  ∫_{T}^{T +H} | ∑_{x^{3}≤p<z} p^{−(1/2)−it}|^{2l} dt ≪ A^{l}l^{l}H.

In the notation of Lemma 3.3,

β_{p} = 1 = \frac{log p}{log z} \frac{log z}{log p} ≪ \frac{log p}{log z}

so that (3.3.1) is satisfied with z in place of y. This proves the lemma.

4. Prime number theorem related to the Dirichlet series ∑_{n=1}^{∞} a_{n}^{2}/n^{s} We know that

(4.1)  L_{f}(s) = ∏_{p}(1 − \frac{α_{p}}{p^{s}} )^{−1}(1 − \frac{\overline{α_{p}}}{p^{s}} )^{−1} = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

is an entire function, |α_{p}| = p^{(k−1)/2} , α_{p}\overline{α_{p}} = p^{k−1} and a_{p} = α_{p} + \overline{α_{p}}. Now, let

(4.2)  L_{f^{2}}(s) := ∑_{n=1}^{∞} \frac{a_{n}^{2}}{n^{s}}

and

(4.3)  L_{f⊗f}(s) = ∏_{p}(1 − \frac{α_{p}^{2}}{p^{s}} )^{−1}(1 − \frac{α_{p}\overline{α_{p}}}{p^{s}} )^{−1}(1 − \frac{\overline{α_{p}}^{2}}{p^{s}} )^{−1},

where the symbol ⊗ in (4.3) denotes the Rankin–Selberg convolution. The important relation between (4.2) and (4.3) is given by (see [12], [11], [17] and [13])

(4.4)  ζ(s − k + 1)L_{f⊗f}(s) = ζ(2s − 2k + 2)L_{f^{2}}(s),

where ζ(s) is the ordinary Riemann zeta-function. It has been proved by Rankin (see [12]) that L_{f^{2}}(s) has a simple pole at s = k with residue kα (α is a certain constant). Therefore, the series −(L′_{f^{2}}(k − 1 + s))/(L_{f^{2}}(k − 1 + s)) has a simple pole at s = 1 with residue 1.

We define (4.5)

Λ^{∗}(n) = \{ \begin{matrix}\frac{(α_{p}^{2m} + \overline{α_{p}}^{2m} + (α_{p}\overline{α_{p}})^{m} + (−1)^{m+1}(α_{p}\overline{α_{p}})^{m}) log p}{p^{m(k−1)}}, & if n = p^{m}, \\ 0, & otherwise. \end{matrix}

We have the usual von Mangoldt’s function, namely,

(4.6)  Λ(n) = \{ \begin{matrix}log p, & if n = p^{m}, \\ 0, & otherwise. \end{matrix}

We also define Ψ_{f^{2}}^{∗}(x) and Ψ_{f^{2}}(x) by

(4.7)  Ψ_{f^{2}}^{∗}(x) = ∑_{n≤x} Λ^{∗}(n)(x − n)

and

(4.8)  Ψ_{f^{2}}^{∗}(x) = ∫_{0}^{x} Ψ_{f^{2}}(u) du = ∫_{1}^{x} Ψ_{f^{2}}(u) du.

It is obvious that

(4.9)  Ψ_{f^{2}}(x) = ∑_{n≤x} Λ^{∗}(n).

The aim of this section is to prove:

228 A. Sankaranarayanan Theorem 4.1. For x ≥ x_{0} , we have

Ψ_{f^{2}}(x) = x + O(xe^{−C\sqrt{log x}} ).

To prove this theorem, we need the following lemmas.

Lemma 4.1. There exists a positive constant C (> 0) such that

L_{f^{2}}(k − 1 + s) ≠ 0 in σ > 1 − \frac{C}{log(|t| + 2)}.

Proof. See, for example, [8].

Lemma 4.2. Suppose that L_{f^{2}}(s) has no zeros in the domain

σ > 1 − η(|t|),

where η(t), for t ≥ 0, a decreasing function, has a continuous derivative η′(t) and satisfies

(i) 0 < η(t) < \frac{1}{2},
(ii) η′(t) → 0 as t → ∞,
(iii) \frac{1}{η(t)} = O(log t) as t → ∞.

Let α′_{1} be a fixed number satisfying 0 < α′_{1} < 1. Then,

−\frac{L′_{f^{2}}(s)}{L_{f^{2}}(s)} = O(log^{2}(|t|))

uniformly in the region σ ≥ 1 − α′_{1}η(|t|) as t → ±∞.

Proof. Since we have an Euler-product representation for L_{f^{2}}(s) from (4.3) and (4.4), the proof of this lemma follows in a similar fashion to that of Theorem 20 of [8].

Lemma 4.3. Under the conditions of Lemma 4.2, we have

Ψ_{f^{2}}^{∗}(x) = \frac{1}{2}x^{2} + O(x^{2}e^{−α′_{1}ω(x)})

as x → ∞, where ω(x) is the minimum of η(t) log x + log t for t ≥ 1.

Proof. First of all, we note that (for C > 1) (4.3.1)

Ψ_{f^{2}}^{∗}(x) = \frac{1}{2πi} ∫_{C−i∞}^{C+i∞} \frac{x^{s+1}}{s(s + 1)}(−\frac{L′_{f^{2}}(k − 1 + s)}{L_{f^{2}}(k − 1 + s)}) ds

= \frac{1}{2πi} ∫_{C−i∞}^{C+i∞} \frac{x^{s+1}}{s(s + 1)}(−\frac{L′_{f^{2}}(k − 1 + s)}{L_{f^{2}}(k − 1 + s)} − \frac{ζ′(2s)}{ζ(2s)} + \frac{ζ′(2s)}{ζ(2s)}) ds

= \frac{1}{2πi} ∫_{C−i∞}^{C+i∞} \frac{x^{s+1}}{s(s + 1)}(−\frac{L′_{f⊗f}(k − 1 + s)}{L_{f⊗f}(k − 1 + s)} − \frac{ζ′(s)}{ζ(s)} + \frac{ζ′(2s)}{ζ(2s)}) ds

= \frac{1}{2πi} ∫_{C−i∞}^{C+i∞} \frac{x^{s+1}}{s(s + 1)}(−\frac{L′_{f⊗f}(k − 1 + s)}{L_{f⊗f}(k − 1 + s)} − \frac{ζ′(s)}{ζ(s)}) ds + O(x^{7/4}),

since

\begin{aligned}\frac{1}{2πi} ∫_{C−i∞}^{C+i∞} \frac{x^{s+1}}{s(s + 1)}(−\frac{ζ′(2s)}{ζ(2s)}) ds & = \frac{1}{2πi} ∫_{C−iT}^{C+iT} \frac{x^{s+1}}{s(s + 1)}(−\frac{ζ′(2s)}{ζ(2s)}) ds \\ & + O(\frac{x^{C+1}}{T} ). \end{aligned}

Now, by moving the line of integration to σ = \frac{3}{4} , we see that the horizontal portions contribute an error which is in the absolute value at most O(x^{C+1}/T), and the vertical portion contributes at most O(x^{7/4}). We can choose C = 1 + ε (ε is a small positive constant) and T = x^{1/2} . From (4.3.1), we get (4.3.2)

\frac{Ψ_{f^{2}}^{∗}(x)}{x^{2}} = \frac{1}{2πi} ∫_{C−i∞}^{C+i∞} \frac{x^{s−1}}{s(s + 1)}(−\frac{L′_{f⊗f}(k − 1 + s)}{L_{f⊗f}(k − 1 + s)} − \frac{ζ′(s)}{ζ(s)}) ds + O(x^{−1/4}).

Now, we move the line of integration of the integral appearing on the right-hand side of (4.3.2) to σ = 1 − α′_{1}η(|t|). Therefore, this lemma follows when applying Lemmas 4.1 and 4.2.

Now, from Lemma 4.3, Theorem 4.1 follows by standard arguments (see, for example, [8]).

5. Proof of Theorem 1

We fix z = T^{α/(5l)} . Notice that α_{p} + \overline{α_{p}} = a_{p}. Let us write

(5.1)  Δ_{z}(t) := Δ(t) := S_{f}(t) + \frac{1}{π} ∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}}.

Then, from the binomial theorem, we have

(5.2)

(S_{f}(t))^{2l} = (\frac{1}{π} ∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}})^{2l}

+ ∑_{j=1}^{2l} \binom{2l}{j}Δ^{j}(t)(−\frac{1}{π} ∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}})^{2l−j}

= Q_{6} + Q_{7}, say.

We observe that

Q_{7} ≪ 4^{l} l |Δ(t)|(|Δ(t)|^{2l−1} + |∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}} |^{2l−1}).

Therefore, we obtain (using Hölder’s inequality)

(5.3)

Q_{8} := ∫_{T}^{T +H} |S_{f}(t)|^{2l} dt − \frac{1}{π^{2l}} ∫_{T}^{T +H} |∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}} |^{2l} dt

≪ A^{l} ∫_{T}^{T +H} |Δ(t)|^{2l} dt + A^{l} ∫_{T}^{T +H} |Δ(t)| |∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}} |^{2l−1} dt

≪ A^{l} ∫_{T}^{T +H} |Δ(t)|^{2l} dt

+ A^{l}(∫_{T}^{T +H} |Δ(t)|^{2l} dt)^{1/2l}(∫_{T}^{T +H} |∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}} |^{2l} dt)^{1−(1/2l)}.

Let η_{1} := η_{1}(t) := ∑_{p<z} a_{p}p^{−(k/2)−it}, and hence,

(5.4) ∑_{p<z} a_{p}p^{−k/2} sin(t log p) = \frac{i}{2}(η_{1} − \overline{η_{1}}).

Therefore, from the binomial expansion, we obtain

(5.5)  \begin{aligned}Q_{9} & := ∫_{T}^{T +H} |∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}} |^{2l} dt \\ & = (\frac{1}{2})^{2l} ∑_{j=0}^{2l}(−1)^{j}\binom{2l}{j} ∫_{T}^{T +H} η_{1}^{j}\overline{η_{1}}^{(2l−j)} dt \\ & = 2^{−2l} \frac{(2l)!}{(l!)^{2}} ∫_{T}^{T +H} |η_{1}(t)|^{2l} dt \\ & + O(4^{−l} ∑_{{j=0,1,...,2l \atop j≠l}} \binom{2l}{j}|∫_{T}^{T +H} η_{1}^{j}\overline{η_{1}}^{(2l−j)} dt|). \end{aligned}

We note that the integral in the error term of (5.5) is

(5.6)  ≪ ∑_{{p_{1},···,p_{j} <z \atop q_{1},...,q_{(2l−j)}<z}} \frac{a_{p_{1}} ⋯ a_{p_{j}}a_{q_{1}} ⋯ a_{q_{(2l−j)}}}{(p_{1} ⋯ p_{j}q_{1} ⋯ q_{(2l−j)})^{k/2}} |log(\frac{p_{1} ⋯ p_{j}}{q_{1} ⋯ q_{(2l−j)}} )|^{−1}.

We note that |a_{p}| ≤ 2p^{(k−1)/2} and z = T^{α/(5l)} . Since

(5.7)  min(\frac{1}{a}, \frac{1}{b}) ≤ |log(\frac{a}{b})|

for any two distinct positive integers a and b, from (5.6) and (5.7) (for j ≠ l), we get,

(5.8)  ∫_{T}^{T +H} η_{1}^{j}\overline{η_{1}}^{(2l−j)} dt ≪ z^{2l}(∑_{p<z} |a_{p}|p^{−k/2})^{2l} ≪ A^{l}z^{3l} ≪ A^{l}H.

Therefore, the error term in (5.5) is

(5.9)  ≪ A^{l}H.

Now,

(5.10)  \begin{aligned}I_{2} & := ∫_{T}^{T +H} |η_{1}(t)|^{2l} dt \\ & = H ∑_{p_{1},...,p_{l}<z ; {q_{1},...,q_{l}<z \atop p_{1}···p_{l}=q_{1}···q_{l}}} \frac{a_{p_{1}} ⋯ a_{p_{l}}a_{q_{1}} ⋯ a_{q_{l}}}{(p_{1} ⋯ p_{l}q_{1} ⋯ q_{l})^{k/2}} \\ & + O( ∑_{p_{1},...,p_{l}<z ; {q_{1},...,q_{l}<z \atop p_{1}···p_{l}≠q_{1}···q_{l}}} \frac{a_{p_{1}} ⋯ a_{p_{l}}a_{q_{1}} ⋯ a_{q_{l}}}{(p_{1} ⋯ p_{l}q_{1} ⋯ q_{l})^{k/2}} |log(\frac{p_{1} ⋯ p_{l}}{q_{1} ⋯ q_{l}} )|^{−1}). \end{aligned}

Arguments similar to (5.6) yield the error term in (5.10) as

(5.11)  ≪ A^{l}H.

Since |a_{p}| ≤ 2p^{(k−1)/2} , we have |δ(p)| := |a_{p}/p^{(k−1)/2}| ≤ 2. Therefore, choosing C = 2 and τ = \frac{1}{2} in Lemma 3.1, we obtain the first term on the right-hand side of (5.10) as

(5.12)  = Hl!(∑_{p<z} \frac{a_{p}^{2}}{p^{k}} )^{l} + O(H2^{2l}l!(∑_{p<z} p^{−1})^{l−2}(∑_{p<z} p^{−2})).

232 A. Sankaranarayanan We note that (from Theorem 4.1),

(5.13) Ψ_{f^{2}}(x) = ∑_{n≤x} Λ^{∗}(n) = ∑_{p≤x} \frac{a_{p}^{2} log p}{p^{k−1}} + O(x^{1/2} log x) = x + O(xe^{−C\sqrt{log x}} ),

and hence, using Abel’s identity, we obtain

(5.14) ∑_{p≤z} \frac{a_{p}^{2}}{p^{k}} = log log z + O(1) = log log T − log(5l) + O(1).

Hence, from (5.10), (5.11), (5.12) and (5.14), we get

(5.15) ∫_{T}^{T +H} |η_{1}(t)|^{2l} dt = l!H(log log T)^{l} + O(A^{l}l!(log l)H(log log T)^{l−1}).

Therefore, from (5.5), (5.9) and (5.15), we find that

(5.16)

∫_{T}^{T +H} |∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}} |^{2l} dt = \frac{(2l)!}{l!}4^{−l}H(log log T)^{l}
+ O(A^{l}l!(log l)H(log log T)^{l−1})
≪ A^{l}l!H(log log T)^{l},

since 1 ≤ l ≪ (log log T)^{1/3} . Note that we have used

\frac{(2l)!}{(l!)^{2}} = \binom{2l}{l} ≤ 2^{2l}.

From Lemma 3.6 and (5.16), we see that the right-hand side of (5.3) is

(5.17)  ≪ (Al)^{2l}H + A^{l}lH^{1/2l}(A^{l}l^{l−1}H(log log T)^{l})^{1−(1/2l)},

since (for l ≥ 1) we have

(5.18)  l! ≤ l^{l−1}.

Therefore, the right-hand side of (5.17) becomes the total error, which is

(5.19)  ≪ (Al)^{2l}H + A^{l}l^{l−(1/2)}H(log log T)^{l−(1/2)}.

Note that

l^{2l} ≪ l^{l−(1/2)}(log log T)^{l−(1/2)} provided l ≪ (log log T)^{(l−(1/2))/(l+(1/2))}, and

min_{l≥1}(\frac{l − \frac{1}{2}}{l + \frac{1}{2}} ) = min_{l≥1}(1 − \frac{1}{l + \frac{1}{2}} ) = \frac{1}{3}.

Hence, Theorem 1 holds with this error term

O(A^{l}l^{l−(1/2)}H(log log T)^{l−(1/2)}),

provided 1 ≤ l ≪ (log log T)^{1/3} . This proves Theorem 1.

On the sign changes of S_{f}(T) 233

6. Proof of Theorem 2

First, we write

Δ_{z}(t) := Δ(t) := S_{f}(t) + π^{−1} ∑_{p<z} \frac{a_{p} sin(t log p)}{p^{k/2}} := S_{f}(t) + π^{−1} ∑_{2}(t).

Then,

S_{1,f}(t + h) − S_{1,f}(t) = ∫_{t}^{t+h} S_{f}(u) du = −π^{−1} ∫_{t}^{t+h} ∑_{2}(u) du + ∫_{t}^{t+h} Δ(u) du.

Therefore,

(6.1)  \begin{aligned}|∫_{t}^{t+h} S_{f}(u) du|^{2l} & = \frac{1}{π^{2l}} |∫_{t}^{t+h} ∑_{2}(u) du|^{2l} \\ & + O(A^{l}|∫_{t}^{t+h} Δ(u) du|^{2l}) \\ & + O(A^{l}|∫_{t}^{t+h} Δ(u) du||∫_{t}^{t+h} ∑_{2}(u) du|^{2l−1}) \end{aligned}

exactly as in (5.3). We notice that

|∫_{t}^{t+h} Δ(u) du|^{2l} ≤ h^{2l−1} ∫_{t}^{t+h} |Δ(u)|^{2l} du,

and hence, by Hölder’s inequality, we get

(6.2)  \begin{aligned}Q_{10} & := ∫_{T}^{T +H} |∫_{t}^{t+h} S_{f}(u) du|^{2l} dt \\ & = \frac{1}{π^{2l}} ∫_{T}^{T +H} |∫_{t}^{t+h} ∑_{2}(u) du|^{2l} dt \\ & + O(A^{l}h^{2l−1} ∫_{T}^{T +H} ∫_{t}^{t+h} |Δ(u)|^{2l} du) \\ & + O(A^{l}(h^{2l−1} ∫_{T}^{T +H} ∫_{t}^{t+h} |Δ(u)|^{2l} du dt)^{1/2l} \\ & × (∫_{T}^{T +H} |∫_{t}^{t+h} ∑_{2}(u) du|^{2l} dt)^{1−(1/2l)}). \end{aligned}

We notice that

(6.3)  ∫_{T}^{T +H} ∫_{t}^{t+h} |Δ(u)|^{2l} du dt = ∫_{0}^{h} du ∫_{T +u}^{T +u+H} |Δ(t)|^{2l} dt,

and hence, by Lemma 3.6, with (T + h)^{α} ≤ H ≤ T, B′ < α ≤ 1 and

(T + h)^{(α−B′)/(20l)} ≤ z ≤ H^{1/l},

we have

(6.4)  ∫_{T}^{T +H} |Δ(t)|^{2l} dt ≪ (Al)^{2l}H.

With these restrictions, we have (6.5)

Q_{10} := ∫_{T}^{T +H} |∫_{t}^{t+h} S_{f}(u) du|^{2l} dt

= \frac{1}{π^{2l}} ∫_{T}^{T +H} |∫_{t}^{t+h} ∑_{2}(u) du|^{2l} dt

+ O((Al)^{2l}h^{2l}H + A^{l}lH^{1/2l}h(∫_{T}^{T +H} |∫_{t}^{t+h} ∑_{2}(u) du|^{2l} dt)^{1−(1/2l)}).

Now, the main term on the right-hand side of (6.5) (apart from the constant π^{−2l}) is

(6.6)  ∫_{T}^{T +H} |∑_{p<z} \frac{a_{p}(cos((t + h) log p) − cos(t log p))}{p^{k/2} log p}|^{2l} dt.

We put

(6.7)  η_{2} = η_{2}(t) = ∑_{p<z} a_{p}p^{−(k/2)−it}(log p)^{−1}(p^{−ih} − 1),

so that

(6.8)  ∑_{p<z} \frac{a_{p}(cos((t + h) log p) − cos(t log p))}{p^{k/2} log p} = \frac{η_{2} + \overline{η_{2}}}{2} .

The integral in (6.6) becomes equal to

(6.9) 2^{−2l} \frac{(2l)!}{(l!)^{2}} ∫_{T}^{T +H} |η_{2}(t)|^{2l} dt + O(4^{−l} ∑_{{j=0,1,...,2l \atop j≠l}} \binom{2l}{j}|∫_{T}^{T +H} η_{2}^{j}\overline{η_{2}}^{(2l−j)} dt|).

Now, (for j ≠ l)

(6.10)

Q_{11} := ∫_{T}^{T +H} η_{2}^{j}\overline{η_{2}}^{(2l−j)} dt

≪ ∑_{{p_{1},...,p_{j} <z \atop q_{1},...,q_{(2l−j)}<z}} \frac{a_{p_{1}} ⋯ a_{p_{j}}a_{q_{1}} ⋯ a_{q_{(2l−j)}}}{(p_{1} ⋯ p_{j}q_{1} ⋯ q_{(2l−j)})^{k/2}}

× ∏_{m=1}^{j} \frac{|p_{m}^{ih} − 1|}{(log p_{m})} × ∏_{n=1}^{2l−j} \frac{|q_{n}^{ih} − 1|}{(log q_{n})} × |log(\frac{p_{1} ⋯ p_{j}}{q_{1} ⋯ q_{(2l−j)}} )|^{−1}

≪ A^{2l}z^{2l}h^{2l}(∑_{p<z} p^{−1/2})^{2l},

since |a_{p}| ≤ 2p^{(k−1)/2} and

|p^{ih} − 1| = 2|sin(\frac{h log p}{2})| ≤ h log p.

Hence, the error term in (6.9) is

(6.11)  ≪ A^{l}h^{2l}H,

by taking z = T^{α/(5l)} .

Now, we have

(6.12)  \begin{aligned}Q_{12} & := ∫_{T}^{T +H} |η_{2}(t)|^{2l} dt \\ & = H ∑_{p_{1},...,p_{l}<z ; {q_{1},...,q_{l}<z \atop p_{1}···p_{l}=q_{1}···q_{l}}} \frac{a_{p_{1}} ⋯ a_{p_{l}}a_{q_{1}} ⋯ a_{q_{l}}}{(p_{1} ⋯ p_{l}q_{1} ⋯ q_{l})^{k/2}} \\ & × ∏_{j=1}^{l} \frac{(p_{j}^{ih} − 1)(q_{j}^{−ih} − 1)}{(log p_{j})(log q_{j})} + O(A^{l}h^{2l}H), \end{aligned}

in the exact way as we obtained (5.10) and (5.11). Now, by Lemma 3.1, with τ = \frac{1}{2} ,

δ(p_{j}) = \{ \begin{matrix}\frac{a_{p_{j}}(p_{j}^{ih} − 1)}{p_{j}^{(k−1)/2}(log p_{j})} & for 1 ≤ j ≤ l, \\ \frac{a_{p_{j}}(p_{j}^{−ih} − 1)}{p_{j}^{(k−1)/2}(log p_{j})} & for l + 1 ≤ j ≤ 2l, \end{matrix}

and C = 2h, the main term in (6.12) becomes equal to

(6.13)

Q_{13} := l!(∑_{p<z} \frac{a_{p}^{2}|p^{ih} − 1|^{2}}{p^{k}(log p)^{2}} )^{l}H + O(2^{2l}h^{2l}l!(∑_{p<z} p^{−1})^{l−2}H)

= l!H(4 ∑_{p<z} \frac{a_{p}^{2}}{p^{k}} (\frac{sin(\frac{1}{2}h log p)}{log p})^{2})^{l}

+ O(A^{l}l^{l}h^{2l}H(log log T)^{l−2}).

Now, assuming that 1 < h^{−1} < log T /(10l) < log z, we write the sum on the right-hand side of (6.13) as

(6.14)  ( ∑_{p<e^{1/h}} + ∑_{e^{1/h}≤p<z})\frac{a_{p}^{2}}{p^{k}} (\frac{sin(\frac{1}{2}h log p)}{log p})^{2}.

The first sum in (6.14) is

(6.15)  \frac{h^{2}}{4} ∑_{p<e^{1/h}} \frac{a_{p}^{2}}{p^{k}} + O(h^{2} ∑_{p<e^{1/h}} \frac{a_{p}^{2}}{p^{k}} h^{2}(log p)^{2}) = \frac{h^{2}}{4} log h^{−1} + O(h^{2}),

since

(6.16)  ∑_{p≤x} \frac{a_{p}^{2}}{p^{k}} = log log x + O(1)

as in (5.14), and in the error term, we have used a_{p}^{2} ≤ 4p^{k−1} and

(6.17)  ∑_{p≤x} \frac{(log p)^{2}}{p} = O((log x)^{2}).

The second sum in (6.14) is

(6.18)  ≪ ∑_{e^{1/h}≤p<z} \frac{log p}{p(log p)^{3}} ≪ ∫_{1/h}^{∞} v^{−3} dv ≪ h^{2}.

Hence, we obtain from (6.14), (6.15) and (6.18)

(6.19)  ∑_{p<z} \frac{a_{p}^{2}}{p^{k}} (\frac{sin(\frac{1}{2}h log p)}{log p})^{2} = \frac{h^{2}}{4} log h^{−1} + O(h^{2}).

On the sign changes of S_{f} (T) 237

Therefore, from (6.12), (6.13) and (6.19), we get

(6.20) ∫_{T}^{T +H} |η_{2}(t)|^{2l} dt = l!Hh^{2l}(log h^{−1} + O(1))^{l} + O(A^{l}l^{l}h^{2l}H(log log T)^{l−2}).

Substituting (6.11) and (6.20) in (6.5) and using the inequality l! ≤ l^{l−1} for l ≥ 1, we arrive at

(6.21)

Q_{10} := ∫_{T}^{T +H} |∫_{t}^{t+h} S_{f}(u) du|^{2l} dt

= \frac{(2l)!}{l!}(\frac{h}{2π} )^{2l}H(log h^{−1})^{l} + O(A^{l}l^{2l}h^{2l}H)

+ O(A^{l}Hh^{2l}l^{l−(1/2)}(log log T)^{l−(1/2)})

from which Theorem 2 follows.

7. Completion of the proof of the main theorem

The proofs of Theorems 3 and 4 are verbatim the same as in [5] (see Section 5 and 6 of [5]). Hence, the proof of the main theorem follows from the arguments similar to those in Section 7 of [5].

References

[1] Deligne, P.: Formes modulaires et représentations ℓ-adiques. - Sém. Bourbaki 355, Vol. 1968/69, Springer-Verlag, 1971.

[2] Deligne, P.: La conjecture de Weil, I. - Inst. Hautes Études Sci. Publ. Math. 43, 1974, 273–307.

[3] Fujii, A.: On the distribution of the zeros of the Riemann Zeta function in short intervals. - Bull. Amer. Math. Soc. 81, 1975, 139–142.

[4] Fujii, A.: On the difference between r consecutive ordinates of the Riemann Zeta function. - Proc. Japan Acad. 51, 1975, 741–743.

[5] Ghosh, A.: On the Riemann zeta-function - sign changes of S(T). - In: Recent Progress in Analytic Number Theory 1, Academic Press, 1981, edited by H. Halberstam and C. Hooley, 25–46.

[6] Ghosh, A.: On the Riemann Zeta function—Mean value theorems and the distribution of |S(T)|. - J. Number Theory 17, 1983, 93–102.

[7] Hecke, E.: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung. - Math. Ann. 112, 1936, 664–669.

[8] Ingham, A. E.: The distribution of prime numbers. - Stechert–Hafner Service Agency, New York–London, edited by G. H. Hardy and E. Cunningham, 1964.

[9] Jutila, M., and Y. Motohashi: Uniform bound for Hecke L-functions. - Acta Math. (to appear).

[10] Montgomery, H. L., and R. C. Vaughan: Hilbert’s inequality. - J. London Math. Soc. 8, 1974, 73–82.

238 A. Sankaranarayanan

[11] Perelli, A.: On the prime number theorem for the coefficients of certain modular forms. - In: Elementary and Analytic Theory of Numbers, Banach Center Publ. 17, PWN-Polish Scientific Publishers, Warsaw, 1985, 405–410.

[12] Rankin, R. A.: Contributions to the theory of Ramanujan’s function τ(n) and similar arithmetic functions. - Proc. Cambridge Philos. Soc. 35, 1939, 357–372.

[13] Sankaranarayanan, A.: Fundamental properties of symmetric square L-functions, I. - Illinois J. Math. 46, 2002, 23–43.

[14] Sankaranarayanan, A.: On Hecke L-functions associated with cusp forms. - Acta Arith. 108, 2003, 217–234.

[15] Sarnak, P.: Estimates for Rankin–Selberg L-functions and quantum unique ergodicity. - J. Functional Analysis 184, 2001, 419–453.

[16] Selberg, A.: Contributions to the theory of the Riemann zeta-function. - In: Collected Papers 1, Springer-Verlag, 1989, 214–280.

[17] Shimura, G.: On the holomorphy of certain Dirichlet series. - Proc. London Math. Soc. 31, 1975, 79–98.

[18] Titchmarsh, E. C.: The Theory of the Riemann Zeta-Function. - 2nd edition, revised by D. R. Heath-Brown, Oxford Publications, 1986.

Received 3 January 2005