PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 92(106) (2012), 97–108 DOI: 10.2298/PIM1206097I

ON A RELATION BETWEEN SUMS OF ARITHMETICAL FUNCTIONS AND DIRICHLET SERIES

Hideaki Ishikawa and Yuichi Kamiya

Communicated by Aleksandar Ivić

Abstract. We introduce a concept called good oscillation. A function is called good oscillation, if its m-tuple integrals are bounded by functions having mild orders.

We prove that if the error terms coming from summatory functions of arithmetical functions are good oscillation, then the Dirichlet series associated with those arithmetical functions can be continued analytically over the whole plane.

We also study a sort of converse assertion that if the Dirichlet series are continued analytically over the whole plane and satisfy a certain additional assumption, then the error terms coming from the summatory functions of Dirichlet coefficients are good oscillation.

1. Introduction

Let s = σ + it be a complex variable, where σ and t are real. Let ζ(s) be the Riemann zeta-function which is defined for σ > 1 by the absolutely convergent Dirichlet series

(1.1)  ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} .

ζ(s) can be continued analytically over the whole s-plane beyond the line σ = 1, and its only singularity is a simple pole at s = 1.

Many proofs of the analytic continuation of ζ(s) are known (see, for example, Titchmarsh [6]). Among them, we recall the proof based on the Euler–Maclaurin summation formula. Applying the partial summation formula to (1.1), one has

ζ(s) = s ∫_{1}^{∞} \frac{∑ _{n⩽x} 1}{x^{s+1}} dx = s ∫_{1}^{∞} \frac{[x]}{x^{s+1}} dx, σ > 1,

2010 Mathematics Subject Classification: Primary 11M06. Key words and phrases: arithmetical function, Dirichlet series, analytic continuation.

where [x] is the largest integer not exceeding x. The right-hand side of the above equation is rewritten as

(1.2)  ζ(s) = \frac{1}{s − 1} + \frac{1}{2} + s ∫_{1}^{∞} \frac{[x] − x + \frac{1}{2}}{x^{s+1}} dx,

and (1.2) gives the analytic continuation of ζ(s) for σ > 0 except for s = 1 because of the boundedness of [x] − x + \frac{1}{2}. Let B_{m}(x), m ∈ N, be the Bernoulli polynomials which are inductively defined from B_{1}(x) = x − \frac{1}{2} by keeping the properties \frac{d}{dx} \frac{B_{m+1} (x)}{(m+1)!} = \frac{B_{m} (x)}{m!} and ∫_{0}^{1} B_{m}(x)dx = 0, e.g., B_{1}(x) = x − \frac{1}{2}, B_{2}(x) = x^{2} − x + \frac{1}{6}, and so on (see, for example, Apostol [1, p. 226] ). Then, applying the integration by parts to (1.2), one has

(1.3)  ζ(s) = \frac{1}{s − 1} + \frac{1}{2} + ∑_{m=1}^{M−1} (s)_{m} \frac{B_{m+1}}{(m + 1)!} − \frac{(s)_{M}}{M!} ∫_{1}^{∞} \frac{B_{M} (x − [x])}{x^{s+M}} dx,

where B_{m} = B_{m}(0) is the Bernoulli number and (s)_{m} is the function defined by (s)_{m} = s(s + 1) ⋯ (s + m − 1). Since B_{M} (x − [x]), M ∈ N, are periodic functions, B_{M} (x − [x]) are all bounded. Hence the integral on the right-hand side of (1.3) is analytic on the half-plane σ > −M + 1, and, by taking M to ∞, ζ(s) is continued analytically over the whole s-plane except for s = 1.

Now let us consider the general Dirichlet series F (s) = ∑ _{n=1}^{∞} \frac{a(n)}{n^{s}} , where a(n), n ∈ N, are complex numbers. F (s) will be continued analytically over the whole s-plane except for finite poles by the same argument as above, if the following assumption is satisfied:

Consider the summatory function of the Dirichlet coefficients, ∑ _{n⩽x} a(n). Let J (x) be a certain function concerning with location of poles. Define the function E_{0}(x) by E_{0}(x) = ∑ _{n⩽x} a(n) − J (x). Then there exists a sequence of functions {E_{m}(x)}_{m=1}^{∞} such that

\{ \begin{matrix}\frac{d}{dx} E_{1}(x) = E_{0}(x), & x ∈ (0, ∞) − N, \\ \frac{d}{dx} E_{m+1} (x) = E_{m}(x), & x ∈ (0, ∞), m ⩾ 1, \end{matrix}

and E_{m}(x) are all bounded.

To prove the analytic continuation of F (s) by the same argument as above, the assumption for the boundedness of E_{m}(x) may be relaxed. We introduce the following definition.

Definition 1.1. Let g_{0}(x) : (0, ∞) → C be a function which is continuous on (0, ∞) − N, bounded on every finite open interval (0, c), and bounded by O(x^{α_{0}} ) as x → ∞, where α_{0} is a nonnegative constant. Let C_{m}, m ∈ N, be arbitrary constants, and g_{m}(x; C_{m}), m ∈ N, be the functions defined by

(1.4)  g_{1}(x; C_{1}) = ∫_{0}^{x} g_{0}(v) dv + C_{1}, x ∈ (0, ∞),

and

(1.5)  g_{m}(x; C_{m}) = ∫_{0}^{x} g_{m−1} (v; C_{m−1} ) dv + C_{m}, x ∈ (0, ∞), m ⩾ 2.

Then the function g_{0}(x) is called good oscillation, if there exists a nonnegative sequence {α_{m}}_{m=1}^{∞} such that g_{m}(x; C_{m}) = O(x^{α_{m}} ) as x → ∞ and m+1−α_{m} → ∞ as m → ∞.

By this definition, g_{1}(x; C_{1}) is differentiable on (0, ∞) − N and g_{m}(x; C_{m}), m ⩾ 2, are differentiable on (0, ∞), which satisfy

\{ \begin{matrix}\frac{d}{dx} g_{1}(x; C_{1}) = g_{0}(x), x ∈ (0, ∞) − N, \\ \frac{d}{dx} g_{m+1} (x; C_{m+1} ) = g_{m}(x; C_{m}), x ∈ (0, ∞), m ⩾ 1. \end{matrix}

We notice that the sequence {C_{m}}_{m=1}^{∞} in Definition 1.1 is uniquely determined. We will verify this in Section 2.

The purpose of this paper is to give a condition, for which the analytic continuation of the Dirichlet seires is valid, from a point of view of the concept of good oscillation.

Theorem 1.1. Let a(n), n ∈ N, be complex numbers. Assume the following condition (X):

(X) There exist constants l ∈ N ∪ {0}, J_{h}, and J such that the function g_{0}(x) defined by

g_{0}(x) = ∑_{n⩽x} a(n) − ( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J)

is good oscillation, where the empty sum ∑ _{h=0}^{l−1} in the case l = 0 is defined to be 0.

Then the following assertion (Y) holds:

(Y) There exists a constant σ_{1} ⩾ 1 such that the Dirichlet series

(1.6)  F (s) = ∑_{n=1}^{∞} \frac{a(n)}{n^{s}}

is absolutely convergent for σ > σ_{1}. Moreover, F (s) can be continued analytically over the whole s-plane beyond the line σ = σ_{1}, and its only singularity is a pole of the order l at s = 1.

The following result states that a sort of converse assertion holds under additional assumptions.

Theorem 1.2. Assume the condition (Y) in Theorem 1.1 and the following conditions (A1), (A2):

(A1) For any nonnegative integer m, there exists a nonnegative constant c_{m}

such that F (−m − \frac{1}{2} + it) = O((1 + |t|)^{c_{m}} ). (A2) If {c_{m}}_{m=0}^{∞} is the sequence above, then lim_{m→∞} c_{m}/m^{2} = 0

Then the assertion (X) in Theorem 1.1 holds.

The functional equation of ζ(s) and the Phragmén–Lindelöf convexity principle give the well-known estimate ζ(σ+it) = O((1+|t|)^{\frac{1}{2} −σ} ), where the implied constant is uniform for s in the vertical strip −M ⩽ σ ⩽ δ < 0, and hence c_{m} can be chosen as m+1 and lim_{m→∞} \frac{c_{m}}{m^{1+ε}} = 0 holds for every arbitrary small ε > 0. The property lim_{m→∞} \frac{c_{m}}{m^{1+ε}} = 0 holds for other L-functions, e.g., Dirichlet L-functions, cusp form L-functions, and the power moments of those L-functions, because those functions have functional equations.

Since the Bernoulli polynomials B_{M} (x − [x]), where M ∈ N, are all bounded, −B_{1}(x − [x]) = [x] − x + \frac{1}{2} is good oscillation. Moreover, B_{1}(x − [x]) is expressed as

(1.7)  B_{1}(x − [x]) = − ∑_{n=1}^{∞} \frac{sin2nπx}{nπ} , x ∉ Z.

Using (1.2) and (1.7), one can prove the functional equation of ζ(s) [6, pp. 13–15].

This example shows that more properties about g_{0}(x) than good oscillation are required to study analytic continuations with functional equations. On the other hand, the multiple L-series L_{j} (s), discussed in the last section, can be continued analytically over the whole s-plane, but any functional equation has not been obtained. This example may suggest that the concept of good oscillation is appropriate to study analytic continuations off functional equations.

2. A note on the concept of good oscillation

As is mentioned in Introduction, the sequence {C_{m}}_{m=1}^{∞} in Definition 1.1 is uniquely determined. Here we prove this.

Let {C_{m}}_{m=1}^{∞} and {D_{m}}_{m=1}^{∞} be the sequences of constants, and g_{m}(x; C_{m}) and g_{m}(x; D_{m}) be the functions defined by (1.4) and (1.5) with the same g_{0}(x). If g_{0}(x) is good oscillation, then there exists a nonnegative sequence {α_{m}}_{m=1}^{∞} (resp. {β_{m}}_{m=1}^{∞} ) such that g_{m}(x; C_{m}) = O(x^{α_{m}} ) (resp. g_{m}(x; D_{m}) = O(x^{β_{m}} )) as x → ∞ and m + 1 − α_{m} → ∞ (resp. m + 1 − β_{m} → ∞) as m → ∞. Assume C_{m} ≠ D_{m} for some m, and let K be the least m such that C_{m} ≠ D_{m}. Then, by (1.4) and (1.5),

g_{K} (x; D_{K}) = g_{K} (x; C_{K}) + D_{K} − C_{K} , D_{K} − C_{K} ≠ 0,
g_{K+1} (x; D_{K+1} ) = (D_{K} − C_{K})x + g_{K+1} (x; C_{K+1} ) + D_{K+1} − C_{K+1}
= (D_{K} − C_{K})x + g_{K+1} (x; C_{K+1} ) + O(1),

and, for m ⩾ K + 1,

g_{m}(x; D_{m}) = (D_{K} − C_{K})\frac{x^{m−K}}{(m − K)!} + g_{m}(x; C_{m}) + O(x^{m−K−1} )

= (D_{K} − C_{K})\frac{x^{m−K}}{(m − K)!} + O(x^{α_{m}} ) + O(x^{m−K−1} ).

From the assumption m + 1 − α_{m} → ∞ it follows that α_{m} ⩽ m − K − 1 for all large m. Hence we have

g_{m}(x; D_{m}) = (D_{K} − C_{K})\frac{x^{m−K}}{(m − K)!} + O(x^{m−K−1} ).

This shows β_{m} ⩾ m − K, which contradicts m + 1 − β_{m} → ∞ as m → ∞.

3. Proof of Theorem 1.1

Let N_{1} and N_{2} be positive integers, f(x) be a C^{M} function defined on the closed interval [N_{1}, N_{2}], and a(n), n ∈ N, be complex numbers. Let g_{0}(x) : (0, ∞) → C be a function defined by

g_{0}(x) = ∑_{n⩽x} a(n) − ( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J) ,

where l ∈ N ∪ {0}, and J_{h}, J are constants. For this g_{0}(x), let g_{m}(x; C_{m}), m ∈ N, be the functions defined by (1.4) and (1.5). Then, by the integration by parts in the sense of Stieltjes,

∑_{N_{1} <n⩽N_{2}} f(n)a(n) = ∫_{N_{1}}^{N_{2}} f(x) d( ∑_{n⩽x} a(n))

= ∫_{N_{1}}^{N_{2}} f(x)\frac{d}{dx}( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J) dx + ∫_{N_{1}}^{N_{2}} f(x) d(g_{0}(x))

= ∫_{N_{1}}^{N_{2}} f(x)\frac{d}{dx}( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J) dx

+ [f(x) g_{0}(x)]_{N_{1}}^{N_{2}} − ∫_{N_{1}}^{N_{2}} f′(x)g_{0}(x) dx.

Repeating the integration by parts, we have

(3.1) ∑_{N_{1} <n⩽N_{2}} f(n) a(n) = ∫_{N_{1}}^{N_{2}} f(x)\frac{d}{dx}( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J) dx

+ [f(x) g_{0}(x)]_{N_{1}}^{N_{2}} + ∑_{m=1}^{M−1} (−1)^{m}[f^{(m)} (x) g_{m}(x; C_{m})]_{N_{1}}^{N_{2}}

+ (−1)^{M} ∫_{N_{1}}^{N_{2}} f^{(M)} (x) g_{M−1} (x; C_{M−1} )dx.

From now on, we put f(x) = x^{−s} and N_{1} = N in (3.1), and we abbreviate g_{m}(x; C_{m}) to g_{m}(x). Then

∑_{N<n⩽N_{2}}  \frac{a(n)}{n^{s}} = ∫_{N}^{N_{2}} \frac{1}{x^{s}} \frac{d}{dx}( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J) dx

+ ∑_{m=0}^{M−1} [(s)_{m} \frac{g_{m}(x)}{x^{s+m}}]_{N}^{N_{2}} + (s)_{M} ∫_{N}^{N_{2}} \frac{g_{M−1} (x)}{x^{s+M}} dx.

Now let us assume the condition (X) of Theorem 1.1. Then there exists a nonnegative sequence {α_{m}}_{m=0}^{∞} such that g_{m}(x) = O(x^{α_{m}} ) as x → ∞ and m+1−α_{m} → ∞ as m → ∞. Hence, for s with σ > max_{0⩽m⩽M−1} {1, α_{m}}, we can take N_{2} to be ∞, and have

(3.2)  F (s) − ∑_{n=1}^{N} \frac{a(n)}{n^{s}} = ∫_{N}^{∞} \frac{1}{x^{s}} \frac{d}{dx}( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J) dx

− ∑_{m=0}^{M−1} (s)_{m} \frac{g_{m}(N)}{N^{s+m}} + (s)_{M} ∫_{N}^{∞} \frac{g_{M−1} (x)}{x^{s+M}} dx.

Let W_{r} be a number defined by

W_{h} = \{ \begin{matrix}J_{h} + (h + 1)J_{h+1} , & if 0 ⩽ h ⩽ l − 2 , \\ J_{h}, & if h = l − 1 . \end{matrix}

Applying

\frac{d}{dx}( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J) = ∑_{h=0}^{l−1} W_{h}(logx)^{h}

and

∫_{N}^{∞} \frac{(logx)^{h}}{x^{s}} dx = ∑_{j=0}^{h} \binom{h}{j} j!(−1)^{j+1} \frac{N^{1−s} (logN)^{h−j}}{(1 − s)^{j+1}}

to (3.2), we have, for σ > max_{0⩽m⩽M−1} {1, α_{m}}, that

(3.3)  F (s) = ∑_{n=1}^{N} \frac{a(n)}{n^{s}} + ∑_{h=0}^{l−1} W_{h}h! ∑_{j=0}^{h} \frac{(−1)^{j+1}}{(h − j)!} \frac{N^{1−s} (logN)^{h−j}}{(1 − s)^{j+1}}

− ∑_{m=0}^{M−1} (s)_{m} \frac{g_{m}(N)}{N^{s+m}} + (s)_{M} ∫_{N}^{∞} \frac{g_{M−1} (x)}{x^{s+M}} dx.

The integral on the right-hand side of (3.3) is analytic on the half-plane σ > 1−(M −α_{M−1} ), and hence (3.3) is valid for σ > 1−(M −α_{M−1} ). Since M −α_{M−1} tends to ∞ as M → ∞, F (s) can be continued analytically over the whole s-plane except for s = 1. This completes the proof.

Remark 3.1. In the condition (X) of Theorem 1.1, we assume existence of constants l ∈ N ∪ {0}, J_{h}, and J such that the function g_{0}(x) defined by

g_{0}(x) = ∑_{n⩽x} a(n) − ( x ∑_{h=0}^{l−1} J_{h}(logx)^{h} + J)

is good oscillation. These constants l ∈ N∪{0}, J_{h}, and J are uniquely determined, i.e., if there exist l̃∈ N ∪ {0}, J̃_{h}, and J̃ such that the function g̃_{0}(x) defined by

g̃_{0}(x) = ∑_{n⩽x} a(n) − ( x ∑_{h=0}^{_{l}−1} J̃_{h}(logx)^{h} + J̃)

is good oscillation, then l = l̃, J_{h} = J̃_{h}, and J = J̃. This assertion is equivalent to that if

G_{0}(x) = x ∑_{h=0}^{L} D_{h}(logx)^{h} + D

is good oscillation, then D_{h} = 0 for all 0 ⩽ h ⩽ L and D = 0. Here we prove this.

Assume that there exists an h such that D_{h} ≠ 0, and let H = max{h | D_{h} ≠ 0}. Then

G_{0}(x) = D_{H} x(logx)^{H} + O(x(logx)^{H−1} ).

For G_{0}(x), let G_{m}(x), m ∈ N, be the functions defined by (1.4) and (1.5). Then

G_{m}(x) = D_{H} \frac{x^{m+1}}{(m + 1)!} (logx)^{H} + O(x^{m+1} (logx)^{H−1} ).

This shows α_{m} ⩾ m + 1, which contradicts that G_{0}(x) is good oscillation. Thus D_{h} = 0 for all 0 ⩽ h ⩽ L and G_{0}(x) = D. D = 0 is similarly proved as above.

Remark 3.2. In Definition 1.1 we required m+1−α_{m} → ∞ as m → ∞. Even if we replace this with lim sup_{m→∞} (m + 1 − α_{m}) = ∞, Theorem 1.1 still holds, and Theorem 1.2 holds with the relaxed condition (A2)’ lim inf_{m→∞} c_{m}/m^{2} = 0. By this replacement, however, the properties described in Section 2 and Remark 3.1 are lost.

4. Proof of Theorem 1.2

Firstly, we only assume the condition (Y).

Let A_{m}(x) : (0, ∞) → C, m ∈ N ∪ {0}, be the function defined by the Riesz sum of the Dirichlet coefficients a(n) in (1.6):

A_{m}(x) = \{ \begin{matrix}\frac{1}{m!} ∑ _{n⩽x} a(n)(x − n)^{m}, & if x ⩾ 1 and m ⩾ 1, \\ ∑ _{n⩽x} a(n) − ã(x), & if x ⩾ 1 and m = 0, \\ 0, & if 0 < x < 1, \end{matrix}

where ã(x) is the function defined by ã(x) = \frac{a(x)}{2} if x is an integer, and ã(x) = 0 otherwise. Let SR_{m}(x) : (0, ∞) → C, m ∈ N ∪ {0}, be the function defined by

(4.1)  SR_{m}(x) = ∑_{j=−1}^{m} Res_{w=−j} F (w)\frac{x^{w+m}}{w(w + 1) …(w + m)} .

Let E_{m}(x) : (0, ∞) → C, m ∈ N ∪ {0}, be the function defined by

E_{m}(x) = A_{m}(x) − SR_{m}(x). (4.2)

Then it is easily verified that

\{ \begin{matrix}\frac{d}{dx} E_{1}(x) = E_{0}(x), x ∈ (0, ∞) − N, \\ \frac{d}{dx} E_{m+1} (x) = E_{m}(x), x ∈ (0, ∞), m ⩾ 1. \end{matrix}

The function E_{m}(x) was used in [5] for another purpose.

The following lemma can be proved similarly to Davenport [2, p.105, Lemma], and we omit the proof.

Lemma 4.1. Let c > 0 and y > 0. It follows that, for m ∈ N,

I_{m}(y) = \frac{1}{2πi} ∫_{c−i∞}^{c+i∞} \frac{y^{w}}{w(w + 1) ⋯ (w + m)} dw = \{ \begin{matrix}0, & 0 < y ⩽ 1 \\ \frac{1}{m!} (1 − y^{−1} )^{m}, & y > 1, \end{matrix}

and

I_{0}(y) = lim_{T →∞} \frac{1}{2πi} ∫_{c−iT}^{c+iT} \frac{y^{w}}{w} dw = \{ \begin{matrix}0, & 0 < y < 1 \\ \frac{1}{2}, & y = 1 \\ 1, & y > 1. \end{matrix}

For m ∈ N ∪ {0}, let

R_{m}(y, T) = I_{m}(y) − \frac{1}{2πi} ∫_{c−iT}^{c+iT} \frac{y^{w}}{w(w + 1) ⋯ (w + m)} dw.

Then, for T > 2m,

(4.3)  |R_{m}(y, T)| < \{ \begin{matrix}y^{c} min\{ \frac{2^{m}}{T^{m}} , \frac{1}{T^{m+1} | log y|} \}, & y ≠ 1 \\ \frac{1}{T^{m}} , & y = 1. \end{matrix}

Now we assume (Y), (A1), and the new condition (B): there exists a nonnegative λ such that a(n) = O(n^{λ+ε} ), where ε > 0 is chosen arbitrarily small. Then we have the following.

Lemma 4.2. Assume the conditions (Y) in Theorem 1.1, (A1) in Theorem 1.2, and (B) introduced above. Let x > 2 and T > 2. Choose c in R_{m}(y, T) introduced above as c = σ_{1} + ε, where σ_{1} and ε are the same ones as in (Y) and (B), respectively. Then

| ∑_{n=1}^{∞} a(n)R_{m}(x/n, T)| ≪ \frac{x^{σ_{1} +ε}}{T^{m+1}} + \frac{x^{1+λ+ε} logx}{T^{m+1}} + \frac{x^{λ+ε}}{T^{m}} .

Proof. From (4.3) it follows that

(4.4) |∑_{n=1}^{∞} a(n)R_{m}(x/n, T)|

< ∑_{\begin{matrix}n=1 \\ n≠x \end{matrix}}^{∞} |a(n)|( \frac{x}{n})^{σ_{1} +ε} min\{ \frac{2^{m}}{T^{m}}, \frac{1}{T^{m+1} | log(x/n)|}\} + \frac{|a(x)|}{T^{m}},

where the second term on the right-hand side only appears in the case that x is an integer, and this is O(x^{λ+ε} /T^{m}) by (B).

Divide the sum on the right-hand side of (4.4) into three sums ∑ _{1}, ∑ _{2}, and ∑ _{3} accordingly as the range of n, n ⩽ \frac{3}{4}x or n ⩾ \frac{5}{4}x, \frac{3}{4}x < n < x, and x < n < \frac{5}{4}x, respectively. By (Y) and (B), ∑ _{1} is O(x^{σ_{1} +ε} /T^{m+1} ).

To estimate ∑ _{2}, let x_{1} be the largest integer not greater than x. The contribution of the term n = x_{1} is O(x^{λ+ε} /T^{m}) by (B). For the other terms we put n = x_{1} − ν, 0 < ν < \frac{1}{4}x, to get log(x/n) ⩾ ν/x_{1} ≫ ν/x. Then, by (B),

∑ 2 ≪ \frac{x}{T^{m+1}} ∑_{0<ν< \frac{1}{4} x} \frac{|a(x_{1} − ν)|}{ν} + \frac{x^{λ+ε}}{T^{m}} ≪ \frac{x^{1+λ+ε} logx}{T^{m+1}} + \frac{x^{λ+ε}}{T^{m}} .

To estimate ∑ _{3}, let x_{2} be the least integer greater than x. The contribution of the term n = x_{2} is O(x^{λ+ε} /T^{m}) by (B). For the other terms we put n = x_{2} + ν, 0 < ν < \frac{1}{4}x, to get − log(x/n) ⩾ ν/(2x_{2} + ν) ≫ ν/x. Then, by (B),

∑ 3 ≪ \frac{x}{T^{m+1}} ∑_{0<ν< \frac{1}{4} x} \frac{|a(x_{2} + ν)|}{ν} + \frac{x^{λ+ε}}{T^{m}} ≪ \frac{x^{1+λ+ε} logx}{T^{m+1}} + \frac{x^{λ+ε}}{T^{m}} .

Thus we obtain the desired estimate. □

Lemma 4.3. Assume the conditions (Y) in Theorem 1.1, (A1) in Theorem 1.2, and (B) introduced above. Let x > 2, T > 2, and c_{m} be the same one as in (A1). Then

E_{m}(x) ≪ \frac{x^{m+σ_{1} +ε}}{T^{m+1}} + \frac{x^{m+1+λ+ε} logx}{T^{m+1}} + \frac{x^{m+λ+ε}}{T^{m}} + x^{− \frac{1}{2}} max\{T^{c_{m} −m} , logT \}

+ \frac{x^{m}}{T^{m+1}} exp_{T} ( \frac{c_{m}(σ_{1} + ε)}{m + \frac{1}{2} + σ_{1} + ε} ) ∫_{−m− \frac{1}{2}}^{σ_{1} +ε} ( x exp_{T} ( − \frac{c_{m}}{m + \frac{1}{2} + σ_{1} + ε} ) )^{σ}dσ.

Proof. By the residue theorem the function SR_{m}(x) defined by (4.1) can be expressed as

(4.5)  SR_{m}(x) = \frac{1}{2πi} ∫_{C}F (w)\frac{x^{w+m}}{w(w + 1) ⋯ (w + m)} dw,

where C is the boundary of the rectangle with vertices σ_{1} + ε − iT, σ_{1} + ε + iT, −m − \frac{1}{2} + iT, and −m − \frac{1}{2} − iT in positive orientation. C consists of four sides C_{1}, C_{2}, C_{3}, C_{4} taken in this order starting with the right side. For ∫_{C_{1}} …dw, we have by (1.6) and Lemma 4.1 that

\frac{1}{2πi} ∫_{C_{1}} F (w)\frac{x^{w+m}}{w(w + 1) ⋯ (w + m)} dw = A_{m}(x) − x^{m} ∑_{n=1}^{∞} a(n) R_{m}(x/n, T). (4.6)

By (4.5), (4.6), and Lemma 4.2, we have

(4.7)  E_{m}(x) = −\frac{1}{2πi} ∫_{C_{2} +C_{3} +C_{4}} F (w)\frac{x^{w+m}}{w(w + 1) ⋯ (w + m)} dw

+ O( \frac{x^{m+σ_{1} +ε}}{T^{m+1}} + \frac{x^{m+1+λ+ε} logx}{T^{m+1}} + \frac{x^{m+λ+ε}}{T^{m}} ) .

For ∫_{C_{3}} …dw, we have by (A1) that
(4.8)  ∫_{C_{3}} F (w)\frac{x^{w+m}}{w(w + 1) ⋯ (w + m)} dw ≪ x^{− \frac{1}{2}} ∫_{−T}^{T} \frac{(1 + |y|)^{c_{m}}}{(1 + |y|)^{m+1}} dy

≪ x^{− \frac{1}{2}} × \{ \begin{matrix}T^{c_{m} −m} /(c_{m} − m ), & if c_{m} > m , \\ logT, & if c_{m} ⩽ m \end{matrix}

≪ x^{− \frac{1}{2}} max\{T^{c_{m} −m} , logT \}.

The treatments of ∫_{C_{2}} …dw and ∫_{C_{4}} …dw are similar, and we show the former case only. By the conditions (Y) and (A1), and by the Phragmén–Lindelöf convexity principle,

(4.9)  F (σ + it) = O( exp_{|t|} ( \frac{−c_{m}(σ − σ_{1} − ε)}{m + \frac{1}{2} + σ_{1} + ε} ) )

holds uniformly for s with −m − \frac{1}{2} ⩽ σ ⩽ σ_{1} + ε and |t| ⩾ 2. Using (4.9), we have

(4.10) ∫_{C_{2}} F (w)\frac{x^{w+m}}{w(w + 1) ⋯ (w + m)} dw

≪ \frac{x^{m}}{T^{m+1}} exp_{T} ( \frac{c_{m}(σ_{1} + ε)}{m + \frac{1}{2} + σ_{1} + ε} ) ∫_{−m− \frac{1}{2}}^{σ_{1} +ε} ( x exp_{T} ( − \frac{c_{m}}{m + \frac{1}{2} + σ_{1} + ε} ) )^{σ}dσ.

Substituting (4.8) and (4.10) into (4.7), we obtain the desired estimate. □

In Lemma 4.3 we have assumed the condition (B): a(n) = O(n^{λ+ε} ). This λ can be chosen as λ = σ_{1} under the condition (Y). In fact, since ∑ _{n=1}^{∞} \frac{|a(n)|}{n^{σ_{1} +ε}} converges, the terms corresponding to all large n are uniformly bounded by some constant. Thus we can put λ = σ_{1} in Lemma 4.3, and have the following estimate under the conditions (Y) and (A1):

(4.11) E_{m}(x) ≪ \frac{x^{m+1+σ_{1} +ε} logx}{T^{m+1}} + \frac{x^{m+σ_{1} +ε}}{T^{m}} + x^{− \frac{1}{2}} max\{T^{c_{m} −m} , logT \}

+ \frac{x^{m}}{T^{m+1}} exp_{T} ( \frac{c_{m}(σ_{1} + ε)}{m + \frac{1}{2} + σ_{1} + ε} ) ∫_{−m− \frac{1}{2}}^{σ_{1} +ε} ( x exp_{T} ( − \frac{c_{m}}{m + \frac{1}{2} + σ_{1} + ε} ) )^{σ}dσ.

Put T = x^{\frac{m+σ_{1} + \frac{1}{2}}{cm +1}} in (4.11). Then

x exp_{T} ( − \frac{c_{m}}{m + \frac{1}{2} + σ_{1} + ε} ) > x^{\frac{1}{cm +1}} > 1,

and hence the fourth term on the right-hand side of (4.11) is estimated as

≪ \frac{x^{m+σ_{1} +ε}}{T^{m+1}} .

The third term on the right-hand side of (4.11) is estimated as

≪ x^{− \frac{1}{2}} (logT + T^{c_{m} −m} ) ≪ 1 + \frac{x^{− \frac{1}{2}} T^{c_{m} +1}}{T^{m+1}} ≪ 1 + \frac{x^{m+σ_{1}}}{T^{m+1}} .

Hence, with T = x^{\frac{m+σ_{1} + \frac{1}{2}}{cm +1}} ,

(4.12)  E_{m}(x) ≪ \frac{x^{m+1+σ_{1} +ε} logx}{T^{m+1}} + \frac{x^{m+σ_{1} +ε}}{T^{m}} + 1
≪ x^{m+1+σ_{1} +2ε− \frac{m+σ_{1} + \frac{1}{2}}{cm +1} (m+1)} + x^{m+σ_{1} +ε− \frac{m+σ_{1} + \frac{1}{2}}{cm +1} m} + 1
≪ x^{m+1+σ_{1} − \frac{m+σ_{1} + \frac{1}{2}}{cm +1} m} + 1.

By (4.12), we have E_{m}(x) = O(x^{α_{m}} ) with

m + 1 − α_{m} = −σ_{1} + \frac{m + σ_{1} + \frac{1}{2}}{c_{m} + 1} m or m + 1,

and, under the condition (A2), the right-hand side tends to ∞ as m → ∞.

Now we define the function g_{0}(x) by g_{0}(x) = E_{0}(x) + ã(x). Then g_{0}(x) is good oscillation under the conditions (Y), (A1), and (A2), and has the expression

g_{0}(x) = ∑_{n⩽x} a(n) − ∑_{j=−1}^{0} Res_{w=−j} F (w)\frac{x^{w}}{w}

= ∑_{n⩽x} a(n) − ( x ∑_{h=0}^{l−1} \frac{(− logx)^{h}}{h!} ∑_{r=h}^{l−1} C_{−(r+1)} (−1)^{r} + F (0)) ,

where the constants C_{−(r+1)} come from the Laurant expansion of F (s) at s = 1:

F (s) = \frac{C_{−l}}{(s − 1)^{l}} + ⋯+ \frac{C_{−1}}{s − 1} + O(1).

This completes the proof of Theorem 1.2.

5. An application of Theorem 1.2

Let j be a positive integer. Let s_{i}, i = 1, 2, …, j, be complex variables, and χ_{i}, i = 1, 2, …, j, be Dirichlet characters of the same conductor q ⩾ 2. The multiple L-series is defined by

L_{j} (s_{1}, …, s_{j} | χ_{1}, …, χ_{j}) = ∑_{\begin{matrix}0<n_{1} <···<n_{j} \\ n_{1} ,...,n_{j} ∈N \end{matrix}} \frac{χ_{1}(n_{1})}{n_{1}^{s_{1}}} \frac{χ_{2}(n_{2})}{n_{2}^{s_{2}}} … \frac{χ_{j} (n_{j})}{n_{j}^{s_{j}}} .

If Re(s_{i}) ⩾ 1, i = 1, 2, …, j − 1, and Re(s_{j}) > 1, then the series are absolutely convergent.

When all s_{i} are equal to s, we abbreviate the multiple L-series to L_{j} (s). Then

L_{j} (s) = ∑_{n=1}^{∞} \frac{h_{j} (n)}{n^{s}}

with

(5.1) h_{j} (n) = ∑_{\begin{matrix}m_{1} m_{2} ...m_{j−1} |n \\ m_{1} <···<m_{j−1} < \frac{n}{m_{1} ... m_{j−1}} \end{matrix}}  χ_{1}(m_{1}) …χ_{j−1} (m_{j−1} )χ_{j} ( \frac{n}{m_{1}m_{2} …m_{j−1}} ) .

The first named author studied the analytic continuation of L_{j} (s) in [3]. As an application of the analytic continuation, properties of ∑ _{n⩽x} h_{j} (n) were studied in [4].

The function L_{j} (s) can be continued meromorphically over the whole s-plane. Moreover, if j = 2, χ_{1}χ_{2}(−1) = 1, and χ_{1}, χ_{2} are primitive characters, then L_{2}(s) is continued to an entire function. Hence L_{2}(s) satisfies the condition (Y) in Theorem 1.2 with l = 0. The proof of the analytic continuation of L_{2}(s) is based on the Euler–Maclaurin summation formula and much technique. In fact, the following expression of L_{2}(s) can be obtained (see [3] for details): for σ > 1 − \frac{M}{2} ,

L_{2}(s) = \frac{1}{q^{2s}} ∑_{a_{1} =1}^{q} ∑_{a_{2} =1}^{q} χ_{1}(a_{1})χ_{2}(a_{2})

× ( \frac{1}{s − 1} ζ(2s − 1, a_{1}/q) + ∑_{m=0}^{M−1} \frac{B̃_{m+1} (\frac{a_{1} −a_{2}}{q} )}{(m + 1)!}(s)_{m}ζ(2s + m, a_{1}/q)

− ∑_{m_{1} =0}^{∞} \frac{1}{(m_{1} + a_{1}/q)^{s}} \frac{(s)_{M}}{M!} ∫_{m_{1} + \frac{a_{1} −a_{2}}{q}}^{∞} \frac{B̃_{M} (x)}{(x + a_{2}/q)^{s+M}} dx) ,

where ζ(s, a) is the Hurwitz zeta-function and B̃_{M} (x) = B_{M} (x − [x]). Using this expression, we have for any nonnegative integer m that

L_{2}(−m − \frac{1}{2} + it) ≪ (1 + |t|)^{4+2m} .

Hence L_{2}(s), where χ_{1}χ_{2}(−1) = 1 and χ_{1}, χ_{2} are primitive characters, satisfies the conditions (Y), (A1), and (A2) in Theorem 1.2, and consequently, there exists a constant J such that the function ∑ _{n⩽x} h_{2}(n) − J is good oscillation. It seems difficult to give a proof of this directly from (5.1).

References

1. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, UTM, 1976.

2. H. Davenport, Multiplicative Number Theory, third ed., Springer-Verlag, 2000.

3. H. Ishikawa, On analytic properties of a multiple L-function; in: S. Saitoh et al.(eds.), Analytic Extension Formulas and their Applications, Kluwer, Dordrecht, 2001, 105–122.

4. H. Ishikawa, A multiple character sum and a multiple L-function, Arc. Math. 79 (2002), 439–448.

5. H. Ishikawa and Y. Kamiya, Spectral sets of certain functions associated with Dirichlet series, J. Math. Anal. Appl. 347 (2008), 204–223.

6. E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., Clarendon Press, Oxford, 1986.

Faculty of Education, Nagasaki University (Received 18 03 2010) 1-14 Bunnkyoumachi, Nagasaki city, Nagasaki, 852-8521, Japan (Revised 24 10 2011)

    isikawah@nagasaki-u.ac.jp

Department of Modern Economics, Faculty of Economics

Daito Bunka University

560 Iwadono, Higashi-Matsuyama, Saitama, 355-8501, Japan

    ykamiya@ic.daito.ac.jp