Balanced Derivatives, Identities, and Bounds for Trigonometric and Bessel Series
Bruce C. Berndt, Martino Fassina, Sun Kim, Alexandru Zaharescu
aa r X i v : . [ m a t h . N T ] F e b BALANCED DERIVATIVES, IDENTITIES, AND BOUNDS FORTRIGONOMETRIC AND BESSEL SERIES
BRUCE C. BERNDT, MARTINO FASSINA, SUN KIM, AND ALEXANDRU ZAHARESCUA
BSTRACT . Motivated by two identities published with Ramanujan’s lost notebook and con-nected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in anearlier paper, three of the present authors derived representations for certain sums of productsof trigonometric functions as double series of Bessel functions. These series are generalizedin the present paper by introducing the novel notion of balanced derivatives, leading to fur-ther theorems. As we will see below, the regions of convergence in the unbalanced caseare entirely different than those in the balanced case. From this viewpoint, it is remarkablethat Ramanujan had the intuition to formulate entries that are, in our new terminology, “bal-anced”. If x denotes the number of products of the trigonometric functions appearing in oursums, in addition to proving the identities mentioned above, theorems and conjectures forupper and lower bounds for the sums as x → ∞ are established.
1. I
NTRODUCTION AND MAIN RESULTS
In a series of papers [5], [6], [7], [8], [11] written by three of the present authors andJ. Li, they examined two formulas of Ramanujan in an unpublished fragment found in [17,p. 335]. The two formulas are connected with the famous
Gauss circle problem and theequally famous
Dirichlet divisor problem . Each of the two formulas has three distinct inter-pretations. Ramanujan’s formulas and the methods developed to prove them have generatedfurther research, in particular, in [6] and [8]. In this paper, we continue our study by ex-amining “balanced” derivatives of the series representations and making applications to thetrigonometric sums studied in [8].In order to state Ramanujan’s formulas, the
Gauss circle problem , and the
Dirichlet divisorproblem , it is necessary to first define the relevant Bessel functions appearing in Ramanujan’sidentities. Let J ν ( z ) denote the ordinary Bessel function of order ν . Define I ν ( z ) := − Y ν ( z ) − π K ν ( z ) , (1.1)where Y ν ( z ) denotes the Bessel function of imaginary argument of order ν given by Y ν ( z ) := J ν ( z ) cos( νπ ) − J − ν ( z )sin( νπ ) , | z | < ∞ , (1.2)and K ν ( z ) denotes the modified Bessel function of order ν defined by K ν ( z ) := π e πiν/ J − ν ( iz ) − e − πiν/ J ν ( iz )sin( νπ ) , − π < arg z < π. (1.3) Mathematics Subject Classification.
Primary 11L03. Secondary 33C10; 11L07.
Key words and phrases.
Balanced derivatives; Bessel functions; Dirichlet divisor problem; Ramanujan’slost notebook; Trigonometric series. If ν is an integer n , it is understood that we define the functions by taking the limits as ν → n in (1.2) and (1.3).We now describe the Gauss circle problem and the
Dirichlet divisor problem . Detaileddiscussions and references for these two famous problems can be found in [9].Let r ( n ) denote the number of ways in which the positive integer n can be expressed as asum of two squares, where different orders and different signs of the summands are regardedas distinct representations of n as a sum of two squares. For example, ± + ( ± =( ± + ( ± , and so r (5) = 8 . Let R ( x ) := X ≤ n ≤ x ′ r ( n ) , where r (0) = 1 and the prime ′ indicates that, if n = x , only r ( x ) is counted. Then R ( x ) =: πx + P ( x ) = πx + ı X n =1 r ( n ) (cid:16) xn (cid:17) / J (2 π √ nx ) , (1.4)where the representation for P ( x ) on the far right side is due to Hardy and Ramanujan [14,p. 265]. Finding the precise order of magnitude of P ( x ) , as x → ∞ , is known as the Gausscircle problem .Throughout this paper, for arithmetic functions f and g , let X nm ≤ x ′ f ( n ) g ( m ) := (P nm ≤ x f ( n ) g ( m ) , if x is not an integer, P nm ≤ x f ( n ) g ( m ) − P nm = x f ( n ) g ( m ) , if x is an integer.In [7], three of the authors proved the following enigmatic identity of Ramanujan from hislost notebook [17]. Entry 1.1. [17, p. 335] If < θ < and x > , then X n ≤ x ′ h xn i sin(2 πnθ ) = πx (cid:18) − θ (cid:19) −
14 cot( πθ )+ 12 √ x ı X m =1 ∞ X n =0 J (cid:16) π p m ( n + θ ) x (cid:17)p m ( n + θ ) − J (cid:16) π p m ( n + 1 − θ ) x (cid:17)p m ( n + 1 − θ ) , (1.5) where, as customary, [ x ] denotes the greatest integer less than or equal to x . The identity (1.5) can be seen as a 2-variable analogue of (1.4). If we set θ = , by anelementary theorem on r ( n ) [15, p. 313], the left sides of (1.4) and (1.5) are identical.Let d ( n ) denote the number of positive divisors of the positive integer n , and let D ( x ) := X n ≤ x ′ d ( n ) . RIGONOMETRIC AND BESSEL SERIES 3
Then D ( x ) = x (log x + (2 γ − x ) (1.6) = x (log x + (2 γ − ı X n =1 d ( n ) (cid:16) xn (cid:17) / I (4 π √ nx ) , (1.7)where x > , γ denotes Euler’s constant, the identity (1.6) is due to Dirichlet [13] and definesthe “error term” ∆( x ) , I ( x ) is defined in (1.1), and the series representation for ∆( x ) on theright side of (1.7) is due to Vorono¨ı [18]. Finding the optimal bound for ∆( x ) as x → ∞ isthe Dirichlet divisor problem . Vorono¨ı [18] proved that ∆( x ) = O ( x / log x ) . (1.8)The upper bound for ∆( x ) given in (1.8) is not the best currently known. See [9] for a list ofupper bounds that have been obtained for ∆( x ) . Furthermore [14], [16], [12, p. 130], ∆( x ) = Ω ± ( x / ) , (1.9)as x → ∞ . We say that f ( x ) = Ω + ( x θ ) if there exists a sequence { x n } → ∞ such that f ( x n ) ≤ C ( x n ) θ fails to hold for every positive constant C . Similarly, f ( x ) = Ω − ( x θ ) if there exists asequence { x ′ n } → ∞ such that f ( x ′ n ) ≥ − C ( x ′ n ) θ fails to hold for every positive constant C .A proof of the following second enigmatic identity of Ramanujan from [17, p. 335] hasbeen given by J. Li and two of the present authors [10]. When θ = 0 , the left-hand side of(1.10) below reduces to the left-hand side of (1.7). Entry 1.2. [17, p. 335]
For x > and < θ < , X n ≤ x ′ h xn i cos(2 πnθ ) = 14 − x log(2 sin( πθ )) (1.10) + 12 √ x ı X m =1 ı X n =0 I (cid:16) π p m ( n + θ ) x (cid:17)p m ( n + θ ) + I (cid:16) π p m ( n + 1 − θ ) x (cid:17)p m ( n + 1 − θ ) . Remark 1.3.
Examining (1.7) and (1.10) , we observe that “big O” conjectures and theoremsabout the error term ∆( x ) , which traditionally and frequently involve the series of Besselfunctions on the right-hand side of (1.7) , pertain to the double series of Bessel functions onthe right-hand side of (1.10) , although, as of the present, (1.10) has not been employed inderiving “big O” theorems. In this paper we prove new identities in the spirit of Ramanujan, where on the left sides aresums of products of two trigonometric functions, while on the right sides are double seriesof Bessel functions. Our formulas involve two parameters σ, θ in the interval (0 , and stemfrom identities that three of the present authors proved in [8]. The novelty in the currentpaper consists in the possibility of taking termwise derivatives with respect to σ and θ . Sincewe are only allowed (by reasons of convergence) to take the same number of derivatives BRUCE C. BERNDT, MARTINO FASSINA, SUN KIM, AND ALEXANDRU ZAHARESCU in σ and in θ , we say that the identities thus obtained are “balanced”. As an interestingapplication, we consider two identities that were independently proved in [8], and we showthat one of them can be obtained as the first balanced derivative of the other (see Section 5).In the second portion of the paper, our goal is to derive “big O” and Ω theorems for sumsof two trigonometric functions. Unfortunately, in many cases, for lower bounds we are onlyable to make conjectures. We also extend our study of sine sums X mn ≤ x ′ mn sin(2 πna/p ) sin(2 πmb/q ) to sums of k sines, k ≥ .2. I DENTITIES FOR T RIGONOMETRIC S UMS IN T ERMS OF B ESSEL F UNCTIONS
Here is our first main result.
Theorem 2.1.
Let σ, θ be in the interval (0 , , and let x > . Then for every non-negativeinteger k , ∂ k ∂σ k ∂θ k ( X mn ≤ x ′ cos(2 πmσ ) sin(2 πnθ ) + cot( πθ )4 ) = √ x ∞ X m =0 ∞ X n =0 ∂ k ∂σ k ∂θ k ( J (4 π p ( m + σ )( n + θ ) x ) p ( m + σ )( n + θ ) + J (4 π p ( m + 1 − σ )( n + θ ) x ) p ( m + 1 − σ )( n + θ ) − J (4 π p ( m + σ )( n + 1 − θ ) x ) p ( m + σ )( n + 1 − θ ) − J (4 π p ( m + 1 − σ )( n + 1 − θ ) x ) p ( m + 1 − σ )( n + 1 − θ ) ) . (2.1)The proof of Theorem 2.1 relies on a detailed study of the convergence of a double seriesmore general than the one appearing on the right-hand side of (2.1).Let σ, θ, x be as in Theorem 2.1. Let α, β be non-negative integers and s, w complexnumbers. Consider the double series G α,β ( x, σ, θ, s, w ) := ∞ X m =0 ∞ X n =0 ∂ α + β ∂σ α ∂θ β ( J (4 π p ( m + σ )( n + θ ) x )( m + σ ) s ( n + θ ) w + J (4 π p ( m + 1 − σ )( n + θ ) x )( m + 1 − σ ) s ( n + θ ) w − J (4 π p ( m + σ )( n + 1 − θ ) x )( m + σ ) s ( n + 1 − θ ) w − J (4 π p ( m + 1 − σ )( n + 1 − θ ) x )( m + 1 − σ ) s ( n + 1 − θ ) w ) . (2.2)We determine values of α, β, s, w for which the double series G α,β ( x, σ, θ, s, w ) converges. Theorem 2.2.
Let G α,β ( x, σ, θ, s, w ) be defined as above. Assume that ( s ) + 2 α − β > , w ) + 2 β − α > . RIGONOMETRIC AND BESSEL SERIES 5
Moreover, if x is an integer, assume that Re( s ) + Re( w ) > , while if x is not an inte-ger, assume that Re( s ) + Re( w ) > . Then the double series G α,β ( x, σ, θ, s, w ) convergesuniformly with respect to σ and θ in any compact subset of (0 , . Consider the interesting case s = = w , which corresponds to the setting of Theorem2.1. To meet the conditions of Theorem 2.2 ensuring the convergence of (2.2), the onlypossibility is to take α = β = k (“balanced” situation). Remark 2.3.
Theorem 2.2 also shows, for every choice of non-negative integers α, β , thatthere exists an unbounded region D α,β of C such that for every ( s, w ) ∈ D α,β , the corre-sponding series (2.2) converges. With similar methods as the those used to prove Theorem 2.1, we establish other “bal-anced” identities similar to (2.1). In these new identities the left-hand side contains onlycosines or only sines, respectively.First recall, for each integer ν , that I ν was defined in (1.1). Let then T ( x ) := Z ∞ J ( u ) J (cid:16) xu (cid:17) du. (2.3)(In [8, p. 71], the definition (2.3) is misprinted; replace J ( x ) by J ( xu ) there.) Theorem 2.4.
Let σ, θ be in the interval (0 , , and let x > . Then for every non-negativeinteger k , ∂ k ∂σ k ∂θ k ( X mn ≤ x ′ cos(2 πmσ ) cos(2 πnθ ) − ) = √ x ∞ X m =0 ∞ X n =0 ∂ k ∂σ k ∂θ k ( I (4 π p ( m + σ )( n + θ ) x ) p ( m + σ )( n + θ ) + I (4 π p ( m + 1 − σ )( n + θ ) x ) p ( m + 1 − σ )( n + θ )+ I (4 π p ( m + σ )( n + 1 − θ ) x ) p ( m + σ )( n + 1 − θ ) + I (4 π p ( m + 1 − σ )( n + 1 − θ ) x ) p ( m + 1 − σ )( n + 1 − θ ) ) . (2.4) Theorem 2.5.
Let σ, θ be in the interval (0 , , and let x > . Then for every non-negativeinteger k , ∂ k ∂σ k ∂θ k ( X mn ≤ x ′ mn sin(2 πmσ ) sin(2 πnθ ) ) = x √ x ∞ X m =0 ∞ X n =0 ∂ k ∂σ k ∂θ k ( T (4 π ( m + σ )( n + θ ) x ) p ( m + σ )( n + θ ) − T (4 π ( m + 1 − σ )( n + θ ) x ) p ( m + 1 − σ )( n + θ ) − T (4 π ( m + σ )( n + 1 − θ ) x ) p ( m + σ )( n + 1 − θ ) + T (4 π ( m + 1 − σ )( n + 1 − θ ) x ) p ( m + 1 − σ )( n + 1 − θ ) ) . (2.5) BRUCE C. BERNDT, MARTINO FASSINA, SUN KIM, AND ALEXANDRU ZAHARESCU
3. C
ONVERGENCE OF “ ALMOST BALANCED ” DOUBLE SERIES
This section is devoted to the proof of Theorem 2.2. We start with a simple calculation.
Lemma 3.1.
The following identity holds: ∂ α + β ∂σ α ∂θ β J (4 π p ( m + σ )( n + θ ) x )( m + σ ) s ( n + θ ) w = P ν ∈ A c ν J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) s + α − β ( n + θ ) w + β − α + · · · . (3.1) Here A is a finite subset of Z , the constants c ν are non-negative, and the dots · · · denote asum of terms that have the form r ν J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) γ ( n + θ ) δ , ν ∈ Z , r ν ∈ R , (3.2) where ( Re( γ ) ≥ Re( s ) + α − β , Re( δ ) ≥ Re( w ) + β − α , (3.3) with at least one of the two inequalities in (3.3) being strict.Proof. Recall from [19, p. 17] the following identity, which holds for every integer ν : J ν − ( z ) − J ν +1 ( z ) = 2 J ′ ν ( z ) . (3.4)We argue by induction on the total number of derivatives k = α + β . For k = 0 there isnothing to prove. Assume now that the statement holds for some k ≥ . We will prove thatit holds for k + 1 . Write k + 1 = α + β , and assume without loss of generality that α ≥ .By the inductive hypothesis, ∂ k +1 ∂σ α ∂θ β J (4 π p ( m + σ )( n + θ ) x )( m + σ ) s ( n + θ ) w = ∂∂σ " P ν ∈ A c ν J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) s + α − − β ( n + θ ) w + β − α − + · · · . (3.5)The dots · · · in (3.5) stand for terms of the form (3.2), with the exponents γ and δ satisfying ( Re( γ ) ≥ Re( s ) + α − − β , Re( δ ) ≥ Re( w ) + β − α − , (3.6)where at least one of the two inequalities is strict. By the chain rule and (3.4), ∂∂σ " J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) s + α − − β ( n + θ ) w + β − α − = π √ x ( J ν − − J ν +1 )(4 π p ( m + σ )( n + θ ) x )( m + σ ) s + α − β ( n + θ ) w + β − α + (cid:16) β − α − − s (cid:17) J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) s + α +12 − β ( n + θ ) w + β − α − . (3.7)Similarly, for the “error terms” in (3.5), ∂∂σ " J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) γ ( n + θ ) δ = π √ x ( J ν − − J ν +1 )(4 π p ( m + σ )( n + θ ) x )( m + σ ) γ + ( n + θ ) δ − − γ J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) γ +1 ( n + θ ) δ . (3.8) RIGONOMETRIC AND BESSEL SERIES 7
The second term on the right side of (3.7) and both terms on the right side of (3.8) are of theform r ν J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) γ ( n + θ ) δ , ν ∈ Z , r ν ∈ R , where by (3.6) the complex numbers γ and δ satisfy ( Re( γ ) ≥ Re( s ) + α − β , Re( δ ) ≥ Re( w ) + β − α , with at least one of the two inequalities being strict. Substituting the identities (3.7) and (3.8)into (3.5), we obtain (3.1), thus completing the proof. (cid:3) The same analysis can be repeated for the other summands appearing in (2.2), separatingin each case the “main term” from the “error terms.” We then obtain identities analogous to(3.1). For instance, ∂ α + β ∂σ α ∂θ β J (4 π p ( m + 1 − σ )( n + θ ) x )( m + 1 − σ ) s ( n + θ ) w = ( − α P ν ∈ A c ν J ν (4 π p ( m + 1 − σ )( n + θ ) x )( m + 1 − σ ) s + α − β ( n + θ ) s + β − α + · · · . (3.9)Note that the c ν are the same exact constants that appear in (3.1).We shift our attention to the double series G ν ( x, σ, θ, s, w ) := ∞ X m =0 ∞ X n =0 J ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) s ( n + θ ) w ± J ν (4 π p ( m + 1 − σ )( n + θ ) x )( m + 1 − σ ) s ( n + θ ) w ± J ν (4 π p ( m + σ )( n + 1 − θ ) x )( m + σ ) s ( n + 1 − θ ) w ± J ν (4 π p ( m + 1 − σ )( n + 1 − θ ) x )( m + 1 − σ ) s ( n + 1 − θ ) w ! , (3.10)where x, σ, θ, s, w are as before, and ν is an integer. The designation ± indicates eitherchoice of sign, so that G ν ( x, σ, θ, s, w ) actually represents eight different double series. Weneed to consider these different combinations of signs because of the powers of − appearingas factors in (3.9) and the analogous formulas for the other terms of (2.2). We will see belowthat the choice of signs does not affect the convergence of the series.Theorem 2.2 follows by combining Lemma 3.1 with the following result on the conver-gence of (3.10). Theorem 3.2.
Let G ν ( x, σ, θ, s, w ) be defined as above. Assume that s ) > , and that w ) > . Moreover, if x is an integer, assume Re( s ) + Re( w ) > , while if x is notan integer, assume Re( s ) + Re( w ) > . Then the double series G ν ( x, σ, θ, s, w ) convergesuniformly with respect to σ and θ in any compact subset of (0 , . The remaining part of this section is devoted to proving Theorem 3.2.
BRUCE C. BERNDT, MARTINO FASSINA, SUN KIM, AND ALEXANDRU ZAHARESCU
We start by recalling the following fact. Let z be a complex number, and j a non-negativeinteger. Recall the notation for the binomial coefficient (cid:18) zj (cid:19) = z ( z − z − · · · ( z − j + 1) j ! . Then, for every complex number ζ , with | ζ | < , we have the binomial theorem (1 + ζ ) z = ∞ X j =0 (cid:18) zj (cid:19) ζ j . Using (3), we can easily show that, for every θ ∈ (0 , , n + θ ) z = 1 n z + O (cid:18) n z +1 (cid:19) . (3.11)This simple formula will be used several times in the following discussion.Now recall the following asymptotic formulas, which hold for any positive integer ν [19,p. 199]: J ν ( z ) = ν (cid:18) πz (cid:19) cos (cid:18) z − νπ − π (cid:19) + O (cid:18) z (cid:19) , (3.12) J − ν ( z ) = ν (cid:18) πz (cid:19) cos (cid:18) z + 12 νπ − π (cid:19) + O (cid:18) z (cid:19) . (3.13)Let ν be a fixed non-zero integer, and let β = − νπ − π . By (3.12), (3.13), and (3.11), tostudy the convergence of G ν ( x, σ, θ, s, w ) , it is sufficient to investigate the convergence of S := ∞ X m =0 ∞ X n =0 cos( a p ( m + σ )( n + θ ) + β )( m + σ ) s + ( n + θ ) w + ± cos( a p ( m + σ )( n + 1 − θ ) + β )( m + σ ) s + ( n + 1 − θ ) w + ± cos( a p ( m + 1 − σ )( n + θ ) + β )( m + 1 − σ ) s + ( n + θ ) w + ± cos( a p ( m + 1 − σ )( n + 1 − θ ) + β )( m + 1 − σ ) s + ( n + 1 − θ ) w + ! . Here for convenience we have set a = 4 π √ x .3.1. Large values of n . We examine the double sum S for large values of n . We followthe arguments in [10, p. 576–577]. Let M and N be integers with M < N . By the Euler-Maclaurin summation formula [1, p. 619], N X n = M +1 cos( a p ( m + σ )( n + θ ) + β )( n + θ ) w + = Z N + θM + θ cos( a p ( m + σ ) t + β ) t w + dt + Z N + θM + θ { t − θ } ddt (cid:18) cos( a p ( m + σ ) t + β ) t w + (cid:19) dt, RIGONOMETRIC AND BESSEL SERIES 9 where { t − θ } denotes the fractional part of t − θ . Note that ddt (cid:18) cos( a p ( m + σ ) t + β ) t w + (cid:19) = 14 t w + (cid:18) − a p ( m + σ ) t sin( a p ( m + σ ) t + β ) − (4 w + 1) cos( a p ( m + σ ) t + β ) (cid:19) = O (cid:18) a √ m + σt Re w + (cid:19) . Hence, Z N + θM + θ { t − θ } ddt (cid:18) cos( a p ( m + σ ) t + β ) t w + (cid:19) dt = O a p ( m + σ )( M + θ ) Re w − ! . (3.14)Now let u = a p ( m + σ ) t . Then t = u a ( m + σ ) and dt = ua ( m + σ ) du . Thus, Z N + θM + θ cos( a p ( m + σ ) t + β ) t w + dt = 2( a ( m + σ )) w − Z a √ ( m + σ )( N + θ ) a √ ( m + σ )( M + θ ) cos( u + β ) u w − du. (3.15)Let c, e be real constants. An integration by parts shows that Z BA c sin u + e cos uu w − du = − c cos u + e sin uu w − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) AB + (cid:18) w − (cid:19) Z BA − c cos u + e sin uu w + du = O w A w − + 1 B w − ! , (3.16)as A, B → ∞ . Recall the identity cos( u + β ) = cos u cos β − sin u sin β . By (3.15) and(3.16), Z N + θM + θ cos( a p ( m + σ ) t + β ) t w + dt = O a √ m + σ M + θ ) Re w − + 1( N + θ ) Re w − !! . Hence, by (3.14), as M → ∞ , ∞ X n = M cos( a p ( m + σ )( n + θ ) + β )( n + θ ) w + = lim N →∞ N X n = M cos( a p ( m + σ )( n + θ ) + β )( n + θ ) w + = O a √ m + σ ( M + θ ) Re w − ! . Analogous results hold when σ is replaced by − σ and when θ is replaced by − θ . Wecan thus let M = [ m / (Re w − ) ] and conclude that ∞ X n ≥ m / (Re w −
14 ) cos( a p ( m + σ )( n + θ ) + β )( m + σ ) s + ( n + θ ) w + ± cos( a p ( m + σ )( n + 1 − θ ) + β )( m + σ ) s + ( n + 1 − θ ) w + ± cos( a p ( m + 1 − σ )( n + θ ) + β )( m + 1 − σ ) s + ( n + θ ) w + ± cos( a p ( m + 1 − σ )( n + 1 − θ ) + β )( m + 1 − σ ) s + ( n + 1 − θ ) w + ! = O am Re s +3 / ! . Recall that Re s > . Hence, in our study of the uniform convergence of the sum S , we canreplace it with the sum S , defined by ∞ X m =0 X ≤ n ≤ m / (Re w −
14 ) cos( a p ( m + σ )( n + θ ) + β )( m + σ ) s + ( n + θ ) w + ± cos( a p ( m + σ )( n + 1 − θ ) + β )( m + σ ) s + ( n + 1 − θ ) w + ± cos( a p ( m + 1 − σ )( n + θ ) + β )( m + 1 − σ ) s + ( n + θ ) w + ± cos( a p ( m + 1 − σ )( n + 1 − θ ) + β )( m + 1 − σ ) s + ( n + 1 − θ ) w + ! . Small values of n . Let δ > be a small positive number to be specified later, andlet S be the same double series as S but with the sum on n performed over the interval ≤ n ≤ m − δ . We now prove convergence of S , using the following result from [7]. Lemma 3.3. [7, p. 31–33]
Consider the sum S ( α, β, µ, H , H ) = X H
Exchanging the order of summation in a term of S yields M X m = M X ≤ n
Assume that δ < / (Re w − ) − . Let S be the same doubleseries as S but with the sum on n performed over the interval m δ < n ≤ m / (Re w − ) .The techniques used above to prove the convergence of S also show the convergence of S .Indeed, Lemma 3.3, applied with α = a √ n + θ, µ = θ, H = m δ , and H = m / (Re w − ) yields M X m = M X m δ 14 ) cos( a p ( m + σ )( n + θ ) + β )( m + σ ) s + ( n + θ ) w + = O a,δ M X m = M m + σ ) s + m (1+ δ ) / (Re w − ) ! = O a,δ,s M (1+ δ ) / (Re w − )1 ! , (3.17)where the second equality follows from Re s > . Since Re w > , the upper bound (3.17)can be used to apply Cauchy’s criterion to the partial sums of S , thus showing convergence.Given our analysis of S and S , we can replace the series S in our study of convergence bythe sum S , defined by ∞ X m =0 X m − δ 4. B ALANCED IDENTITIES Having established the convergence result in Theorem 2.2, we are now ready to proveTheorem 2.1. Proof of Theorem 2.1. When k = 0 , Equation (2.1) becomes X mn ≤ x ′ cos(2 πmσ ) sin(2 πnθ ) + cot( πθ )4= √ x ∞ X m =0 ∞ X n =0 ( J (4 π p ( m + σ )( n + θ ) x ) p ( m + σ )( n + θ ) + J (4 π p ( m + 1 − σ )( n + θ ) x ) p ( m + 1 − σ )( n + θ ) − J (4 π p ( m + σ )( n + 1 − θ ) x ) p ( m + σ )( n + 1 − θ ) − J (4 π p ( m + 1 − σ )( n + 1 − θ ) x ) p ( m + 1 − σ )( n + 1 − θ ) ) . (4.1)Note that convergence of the double sum on the right-hand side of (4.1) is a consequence ofTheorem 2.2. To prove (4.1) it is therefore sufficient to compute the Fourier coefficients ofboth sides of the equation and show that they are equal.The general statement of Theorem 2.1 follows from the case k = 0 by taking derivativeson both sides of (4.1). Such derivatives can be brought inside the infinite sums because thedouble series is uniformly convergent by Theorem 2.2 applied in the special case s = w = .In [6, Theorems 4.1, 4.4], three of the present authors first proved (4.1) by showing thatthe Fourier sine series of both sides are identical. Secondly, they proved (4.1), but with theorder of summation reversed, by demonstrating that the Fourier cosine series of both sidesare identical. Thus, it was shown that one could reverse the order of summation by provingthat the two iterated sums converge to the same limit. In our analysis above, using uniformconvergence, we also demonstrated that the two iterated series converged to the same limit.But now appealing to the aforementioned two theorems in [6], we have completed the proofof (4.1), and consequently of Theorem 2.1. (cid:3) Theorem 2.4 and Theorem 2.5 can be proved similarly to Theorem 2.1. We describe belowthe necessary modifications to the arguments presented above. Proof of Theorems 2.4 and 2.5. In the case k = 0 the two theorems yield known identities(see [8, Theorem 2.1] and [8, Theorem 2.3], respectively). In [8] these identities were provedwith the iterated sums replaced by double sums where, for brevity, the products of the indices m and n tend to infinity. However, in each case, the same arguments developed there can beused to prove the identities with iterated sums, yielding (2.4) and (2.5) in the case k = 0 .Theorems 2.4 and 2.5 then follow from the case k = 0 by taking derivatives on both sides ofthe identities.To conclude the proofs of Theorems 2.4 and 2.5 we need theorems on the uniform con-vergence of the two double series appearing on the right sides of (2.4) and (2.5). To this end,we prove that Theorem 2.2 holds even when, in the definition (2.2) of G α,β ( x, σ, θ, s, w ) , theexpression inside the brackets is replaced with a double series involving the function I (re-spectively T ) corresponding to the one on the right-hand side of (2.4) (respectively (2.5)).We denote these two new series by G α,βI ( x, σ, θ, s, w ) and G α,βT ( x, σ, θ, s, w ) , respectively. In the case of G α,βI ( x, σ, θ, s, w ) the proof of Theorem 2.2 carries over with minor mod-ifications. One starts from an analogue of Lemma 3.1, which holds if one simply replacesin the statement every occurrence of J ν with I ν . In the proof, instead of Equation (3.4), oneuses the following similar recurrence relation [19, p. 79], which holds for every integer ν : I ν − ( z ) + I ν +1 ( z ) = 2 I ′ ν ( z ) . (4.2)Later in the proof, the asymptotic formulas (3.12) and (3.13), which hold for every positiveinteger ν , are replaced by I ν ( z ) = − ν (cid:18) πz (cid:19) sin (cid:18) z − νπ − π (cid:19) + O (cid:18) z (cid:19) , (4.3) I − ν ( z ) = ν (cid:18) πz (cid:19) sin (cid:18) z + 12 νπ − π (cid:19) + O (cid:18) z (cid:19) . (4.4)These asymptotic formulas arise from combining the definition (1.1) with the asymptoticexpressions for Y ν and K ν given in [19, p. 199] and [19, p. 202], respectively. One can nowfollow the proof of Theorem 2.2 with the appropriate minor changes due to the fact that thetrigonometric function cos has been replaced with sin . It can be readily checked that thischange does not affect the convergence of the corresponding double series.Let us now consider G α,βT ( x, σ, θ, s, w ) . In order to prove the convergence of this doubleseries we need the following analogue of Lemma 3.1. Lemma 4.1. The following identity holds: ∂ α + β ∂σ α ∂θ β T (4 π ( m + σ )( n + θ ) x )( m + σ ) s ( n + θ ) w = P ν ∈ A c ν Q ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) s + α − β ( n + θ ) w + β − α + · · · . Here A is a finite subset of Z , the c ν are constants, and the dots · · · denote a sum of termsthat have the form r ν Q ν (4 π p ( m + σ )( n + θ ) x )( m + σ ) γ ( n + θ ) δ , ν ∈ Z , r ν ∈ R , where ( Re( γ ) ≥ Re( s ) + α − β , Re( δ ) ≥ Re( w ) + β − α , with at least one of the two inequalities in (4.1) being strict. With Q ν we denote one of theBessel functions I ν , K ν , Y ν . The first step in the proof of this lemma is to recall a formula for T ( y ) as a linearcombination of Bessel functions. From [8, p. 90, Equation (5.9)], if y = 2 π p nx/pq , where p and q are primes, then T ( y ) = 12 y I (2 y ) − y I ′ (2 y ) + Y (2 y ) − π K (2 y ) . Lemma 4.1 can now be proved by induction, similarly to Lemma 3.1. One needs to use,instead of (3.4), the corresponding recurrence relations for the functions I ν , K ν , Y ν [19, p. 79formula (2); p. 66 formula (2)]. To finish the proof of the convergence of G α,βT ( x, σ, θ, s, w ) , RIGONOMETRIC AND BESSEL SERIES 15 one can now follow the same steps as in the proof of the convergence of (3.10), using theasymptotic expansions given in (4.3), (4.4), [19, p. 199], and [19, p. 202]. (cid:3) 5. A PPLICATION : A BALANCED IDENTITY OF ORDER k = 0 from Theorems 2.4 and 2.5. As indicated above, theywere proved in [8, pp. 70–71, Theorems 2.1, 2.3] under hypotheses that were stronger thannecessary. Theorem 5.1. Let I ( x ) be defined by (1.1) . If < θ, σ < and x > , then X nm ≤ x ′ cos(2 πnθ ) cos(2 πmσ ) (5.1) = 14 + √ x X n,m ≥ ( I (4 π p ( n + θ )( m + σ ) x ) p ( n + θ )( m + σ ) + I (4 π p ( n + 1 − θ )( m + σ ) x ) p ( n + 1 − θ )( m + σ )+ I (4 π p ( n + θ )( m + 1 − σ ) x ) p ( n + θ )( m + 1 − σ ) + I (4 π p ( n + 1 − θ )( m + 1 − σ ) x ) p ( n + 1 − θ )( m + 1 − σ ) ) . Theorem 5.2. If < θ, σ < and x > , then X nm ≤ x ′ nm sin(2 πnθ ) sin(2 πmσ ) (5.2) = x √ x X n,m ≥ ( T (cid:0) π ( n + θ )( m + σ ) x (cid:1)p ( n + θ )( m + σ ) − T (cid:0) π ( n + 1 − θ )( m + σ ) x (cid:1)p ( n + 1 − θ )( m + σ ) − T (cid:0) π ( n + θ )( m + 1 − σ ) x (cid:1)p ( n + θ )( m + 1 − σ ) + T (cid:0) π ( n + 1 − θ )( m + 1 − σ ) x (cid:1)p ( n + 1 − θ )( m + 1 − σ ) ) . where T / is defined in (2.3) . From our remarks above on the uniform convergence of the right-hand side of (2.4), dif-ferentiating the identity (2.4) for k = 1 yields an identity for the left-hand side of (2.5) for k = 0 . However, we only know the precise nature of the right-hand side of (2.5) because ofthe independent proof of (2.5) in [8], but under stronger hypotheses, as emphasized above.Our goal is to show directly that the first mixed partial derivative of (5.1) is equal to (5.2).The needed termwise differentiation is justified by Theorem 2.4.Let u = 4 π p ( n + θ )( m + σ ) x. (5.3)Then, ∂∂θ ( I ( u ) p ( n + θ )( m + σ ) ) = I ′ ( u )2 π √ x ( n + θ ) − I ( u )2( n + θ ) / ( m + σ ) / and ∂ ∂σ∂θ ( I ( u ) p ( n + θ )( m + σ ) ) = I ′′ ( u )2 π √ x ( n + θ ) 2 π s ( n + θ ) x ( m + σ ) − I ′ ( u )2( n + θ ) / ( m + σ ) / π s ( n + θ ) x ( m + σ ) + I ( u )4( n + θ ) / ( m + σ ) / = 4 π xI ′′ ( u ) p ( n + θ )( m + σ ) − π √ xI ′ ( u )( n + θ )( m + σ ) + I ( u )4( n + θ ) / ( m + σ ) / . (5.4)We now use [19, p. 66, formula (3); p. 79, formula (3)], respectively, uY ′ ( u ) + Y ( u ) = uY ( u ) and uK ′ ( u ) + K ( u ) = − uK ( u ) . (5.5)Also, [19, p. 66, formula (4); p. 79, formula (4)], respectively, Y ′ ( u ) = − Y ( u ) and K ′ ( u ) = − K ( u ) . (5.6)Thus, from (1.1), (5.5), and (5.6), I ′ ( u ) = − Y ′ ( u ) − π K ′ ( u )= 1 u Y ( u ) − Y ( u ) + 2 πu K ( u ) + 2 π K ( u ) , (5.7)and from (5.6) and (5.7), I ′′ ( u ) = 1 u Y ′ ( u ) − u Y ( u ) − Y ′ ( u ) + 2 πu K ′ ( u ) − πu K ( u ) + 2 π K ′ ( u )= 1 u (cid:18) − u Y ( u ) + Y ( u ) (cid:19) − u Y ( u ) + Y ( u ) + 2 πu (cid:18) − u K ( u ) − K ( u ) (cid:19) − πu K ( u ) − π K ( u )= 2 u (cid:18) − Y ( u ) − π K ( u ) (cid:19) + 1 u (cid:18) Y ( u ) − π K ( u ) (cid:19) + Y ( u ) − π K ( u ) . (5.8) RIGONOMETRIC AND BESSEL SERIES 17 Now return to (5.4) and substitute from (5.3), (1.1), (5.7), and (5.8) to deduce that ∂ ∂σ∂θ ( I ( u ) p ( n + θ )( m + σ ) ) = 16 π x / u (cid:26) − u Y ( u ) − πu K ( u ) + 1 u Y ( u ) − πu K ( u ) + Y ( u ) − π K ( u ) (cid:27) − π x / u (cid:26) u Y ( u ) − Y ( u ) + 2 πu K ( u ) + 2 π K ( u ) (cid:27) − π x / u (cid:26) Y ( u ) + 2 π K ( u ) (cid:27) = x / (cid:26) − π u Y ( u ) − π u K ( u ) + 16 π u Y ( u ) − π u K ( u )+ 32 π u Y ( u ) − π u K ( u ) (cid:27) , or π ∂ ∂σ∂θ ( I ( u ) p ( n + θ )( m + σ ) ) = x / (cid:26) − πu Y ( u ) − u K ( u ) + 4 πu Y ( u ) − u K ( u ) + 8 πu Y ( u ) − u K ( u ) (cid:27) . (5.9)We now turn to T / ( z ) . If we set u = 2 y in (4), we find that T / ( u ) = 2 u I ( u ) − u I ′ ( u ) + Y ( u ) − π K ( u ) . (5.10)Appealing to (5.7), we see from (5.10) that T / ( u ) = − u Y ( u ) − πu K ( u ) − u Y ( u ) + 2 u Y ( u ) − πu K ( u ) − πu K ( u )+ Y ( u ) − π K ( u )= − u Y ( u ) − πu K ( u ) + 2 u Y ( u ) − πu K ( u ) + Y ( u ) − π K ( u ) . (5.11)Recalling the definition (5.3), we can write (5.11) in the form T / ( u ) p ( n + θ )( m + σ ) = 4 π √ xT / ( u ) u = √ x (cid:18) − πu Y ( u ) − u K ( u ) + 8 πu Y ( u ) − u K ( u ) + 4 πu Y ( u ) − u K ( u ) (cid:19) . (5.12)Hence, from (5.9) and (5.12), we conclude that the first balanced derivative of the termsin the first series on the right side of (5.1), multiplied by / (4 π ) , are equal to the terms of the first series on the right-hand side of (5.2). If we successively set u =4 π p ( n + 1 − θ )( m + σ ) x,u =4 π p ( n + θ )( m + 1 − σ ) x,u =4 π p ( n + 1 − θ )( m + 1 − σ ) x, we can make analogous conclusions for the second, third, and fourth series terms on the right-hand sides of (5.1) and (5.2). In conclusion, we have shown that taking the first balancedderivative of (5.1) yields (5.2), as expected.6. G ENERAL T HEOREMS OF C HANDRASEKHARAN AND N ARASIMHAN We offer two general theorems of Chandrasekharan and Narasimhan [12], which we em-ploy in the sequel. First, we provide the general setting [12, p. 93–96]. Definition 6.1. Let a ( n ) and b ( n ) be two sequences of complex numbers, where not allterms are equal to 0 in either sequence. Let λ n and µ n be two sequences of positive numbers,strictly increasing to ∞ . Let δ > . Throughout, s = σ + it , where σ and t are both real. Let ∆( s ) := N Y n =1 Γ( α n s + β n ) , (6.1) where N ≥ , β n , ≤ n ≤ N , is any complex number, and α n > , ≤ n ≤ N . Assumethat A := N X n =1 α n ≥ . Let ϕ ( s ) := ∞ X n =1 a ( n ) λ sn and ψ ( s ) := ∞ X n =1 b ( n ) µ sn converge absolutely in some half-plane, and suppose they satisfy the functional equation ∆( s ) ϕ ( s ) = ∆( δ − s ) ψ ( δ − s ) . (6.2) Furthermore, assume that there exists in the s -plane a domain D , which is the exterior of acompact set S , in which there exists an analytic function χ with the properties lim | t |→∞ χ ( s ) = 0 , uniformly in every interval −∞ < σ ≤ σ ≤ σ < ∞ , and χ ( s ) = ∆( s ) ϕ ( s ) , σ > α,χ ( s ) = ∆( δ − s ) ψ ( δ − s ) , σ < β, where α and β are particular constants. For ρ ≥ , let A ρ ( x ) := 1Γ( ρ + 1) X λ n ≤ x a ( n )( x − λ n ) ρ , (6.3) RIGONOMETRIC AND BESSEL SERIES 19 where the prime ′ indicates that if x = λ n and ρ = 0 , the last term is to be multiplied by .Furthermore, let Q ρ ( x ) := 12 πi Z C Γ( s ) ϕ ( s )Γ( s + ρ + 1) x s + ρ ds, (6.4)where C ρ is a closed curve enclosing all of the singularities of the integrand to the right of σ = − ρ − − k , where k is chosen such that k > | δ/ − / (4 A ) | , and all of the singularities of ϕ ( s ) lie in σ > − k . In Sections 7–9, ρ = 0 , and so in these sections we write A ρ ( x ) = A ( x ) and Q ρ ( x ) = Q ( x ) . In Section 10, we consider the more general case when ρ > . Theorem 6.2. [12, p. 98, Theorem 3.1] Suppose that ϕ ( s ) and ψ ( s ) satisfy (6.2) . Supposethat { µ n } contains a subset { µ n k } such that no number µ / (2 A ) n is represented as a linearcombination of the numbers n µ / (2 A ) n k o with the coefficients ± , unless µ / (2 A ) n = µ / (2 A ) n r forsome r , in which case µ / (2 A ) n has no other representation. Suppose furthermore that ∞ X n =1 | Re b ( n k ) | µ ( Aδ + ρ +1 / / (2 A ) n k = + ∞ . (6.5) Set θ := Aδ + ρ (2 A − − A . (6.6) Then lim x →∞ Re { A ρ ( x ) − Q ρ ( x ) } x θ = + ∞ , (6.7) lim x →∞ Re { A ρ ( x ) − Q ρ ( x ) } x θ = −∞ . (6.8) If in assumption (6.5) , we replace Re b ( n k ) by Im b ( n k ) , then (6.7) and (6.8) remain valid. The conclusion (6.7) is equivalent to the following statement: There exists a sequence { x n } tending to ∞ , such that there does not exist any positive constant C such that, for all n , Re { A ρ ( x n ) − Q ρ ( x n ) } ≤ Cx θn . In such a situation, we write Re { A ρ ( x n ) − Q ρ ( x n ) } = Ω + ( x θ ) . Similar remarks hold for (6.8). Theorem 6.3. [12, p. 106, Theorem 4.1] Suppose that the functional equation ∆( s ) ϕ ( s ) = ∆( δ − s ) ψ ( δ − s ) is satisfied with δ > , and that ϕ ( s ) is an entire function. Then, if A ( x ) is defined by (6.3) and Q ( x ) is defined by (6.4) , as x → ∞ , A ( x ) − Q ( x ) = O (cid:0) x δ/ − / (4 A )+2 Auη (cid:1) + O X x<λ n ≤ x ′ | a ( n ) | ! , (6.9) for every η ≥ , where u := β − δ − A , (6.10) and β is chosen so that P ∞ n =1 | b ( n ) | µ − βn converges. Furthermore, x ′ = x + O ( x − η − / (2 A ) ) . (6.11)7. T HE F IRST Ω AND “B IG O” T HEOREMS AND C ONJECTURE Let χ and χ be primitive, non-principal even characters modulo p and q , respectively.Let τ ( χ ) and τ ( χ ) denote their corresponding Gauss sums. Lastly, we use the notation d χ ,χ ( n ) = X d | n χ ( d ) χ ( n/d ) . (7.1) Theorem 7.1. Assume that χ and χ are non-principal even characters modulo the primes p and q , respectively. Let D χ ,χ ( x ) := X n ≤ x d χ ,χ ( n ) . (7.2) Then lim x →∞ Re D χ ,χ ( x ) x / = + ∞ , (7.3) lim x →∞ Re D χ ,χ ( x ) x / = −∞ . (7.4) Both (7.3) and (7.4) remain valid if we replace Re by Im in each of them.Proof. Recall that [8, p. 74] (if s is replaced by s ) (cid:18) π pq (cid:19) − s/ Γ (cid:18) s (cid:19) L ( s, χ ) L ( s, χ )= τ ( χ ) τ ( χ ) √ pq (cid:18) π pq (cid:19) − (1 − s ) / Γ (cid:18) 12 (1 − s ) (cid:19) L (1 − s, χ ) L (1 − s, χ ) . (7.5)We now apply Theorem 6.2. The parameters from Definition 6.1 and Theorem 6.2 are: N = 2 , δ = 1 , A = 1 , θ = 14 , (7.6) a ( n ) = d χ ,χ ( n ) , b ( n ) = τ ( χ ) τ ( χ ) d χ ,χ ( n ) √ pq , λ n = µ n = πn √ pq . (7.7)From the functional equation (7.5), since the analytic continuations of L ( s, χ ) and L ( s, χ ) are entire functions, we see that L (0 , χ ) = L (0 , χ ) = 0 . It follows that Q ( x ) = 0 . Theorem7.1 now follows immediately provided that we can show that (6.5) holds, that is, there existsa subset n k , ≤ n k < ∞ , such that ∞ X n k =1 | Re { τ ( χ ) τ ( χ ) d χ ,χ ( n k ) }| n / k = + ∞ . (7.8)Recall that p and q are primes. First, suppose that τ ( χ ) τ ( χ ) is not purely imaginary.Choose n such that n ≡ mod p ) and n ≡ mod q ) . Thus, we consider the set S pq := RIGONOMETRIC AND BESSEL SERIES 21 { mpq, ≤ m < ∞} . By Dirichlet’s theorem on primes in arithmetic progressions, S pq contains an infinite number of primes { P k } . Note that d χ ,χ ( P m ) = 2 , m ≥ . Thus, we have located an infinite subset n k = P k , where the series terms in (7.8) are positiveand where the series diverges.Suppose now that τ ( χ ) τ ( χ ) is purely imaginary. Choose r such that χ ( r ) + χ ( r ) isnot real. The set S pq := { r + mpq, ≤ m < ∞} contains an infinite number of primes { Q k } by Dirichlet’s theorem. Hence, d χ ,χ ( Q k ) = χ ( r ) + χ ( r ) . is not real. Now proceed as in the previous case with n k = Q k . This completes the proof. (cid:3) We provide the motivation for studying d χ ,χ ( n ) . We see from [8, p. 78, Equation (3.11)]that CC ( x ) : = X nm ≤ x ′ cos (cid:16) πnap (cid:17) cos (cid:16) πmbq (cid:17) (7.9) = 1 φ ( p ) φ ( q ) (cid:16) X χ mod pχ = χ , even X χ mod qχ = χ , even χ ( a ) χ ( b ) τ ( χ ) τ ( χ ) X n ≤ x ′ d χ ,χ ( n ) (cid:17) − φ ( p ) X n ≤ x ′ h xn i cos (cid:16) πnbq (cid:17) − φ ( q ) X n ≤ x ′ h xn i cos (cid:16) πnap (cid:17) + pφ ( p ) X pn ≤ x ′ (cid:20) xpn (cid:21) cos (cid:16) πnbq (cid:17) + qφ ( q ) X qn ≤ x ′ (cid:20) xqn (cid:21) cos (cid:16) πnap (cid:17) − φ ( p ) φ ( q ) (cid:16)X n ≤ x ′ d ( n ) − q X n ≤ x/q ′ d ( n ) − p X n ≤ x/p ′ d ( n ) + pq X ≤ x/pq ′ d ( n ) (cid:17) , (7.10)where φ ( n ) denotes Euler’s φ -function. The sums X n ≤ x ′ d χ ,χ ( n ) in (7.10) were examined in Theorem 7.1. To examine the next four sums recall the identity[5, p. 2063] X n ≤ x ′ h xn i cos (cid:18) πnaq (cid:19) = X n ≤ x/q ′ d ( n ) + X d | qd> φ ( d ) X χ mod dχ even χ ( a ) τ ( χ ) X n ≤ dx/q ′ d χ ( n ) . (7.11)where φ ( d ) denotes Euler’s φ -function, and where d χ ( n ) = X d | n χ ( d ) . (7.12)Chandrasekharan and Narasimhan [12, p. 133] established an Ω theorem for D χ ( x ) := X n ≤ x ′ d χ ( n ) (7.13) analogous to Theorem 7.1. For the last four sums on the right-hand side of (7.10), an Ω theorem is found in (1.9). In summary, although we can apply an Ω theorem to each ofthe divisor sums appearing in (7.10), we cannot apply Theorem 6.2 and the aforementionedanalogue from [12] to a sum of these arithmetic sums for which we have Ω theorems. Thus,we formulate the following conjecture. Conjecture 7.2. If CC ( x ) is defined in (7.9) , then lim x →∞ CC ( x ) x / = + ∞ , lim x →∞ CC ( x ) x / = −∞ . Theorem 7.3. For each ǫ > , as x → ∞ , D χ ,χ ( x ) = O ( x / ǫ ) . Proof. To prove Theorem 7.3, we apply Theorem 6.3. For the parameters that are needed,we refer to (7.6) and (7.7). First, recall that δ = A = 1 . We need the series P ∞ n =1 | b ( n ) | n − β to converge, and so we take β = 1 + ǫ , for every ǫ > . (The parameter ǫ will not necessarilybe the same with each occurrence.) Thus, from (6.10), u := β − δ − A = 14 + ǫ. The first “Big O” power in (6.9) is then 14 + (cid:18) 12 + 2 ǫ (cid:19) η, where η ≥ and is yet to be determined. We also need to determine the order of X x<λ n ≤ x ′ | a ( n ) | , where a ( n ) is given in (7.7) and x ′ = x + O ( x / − η ) is defined in (6.11). Trivially, | a ( n ) | ≤ d ( n ) . By (1.8), an upper bound for this sum is O ( x / − η log x ) . In order to achieve the mosteffective upper bound, we find the solution to 14 + (cid:18) 12 + 2 ǫ (cid:19) η = 12 − η + ǫ ′ , (7.14)where ǫ, ǫ ′ are arbitrarily small positive numbers. Thus, our optimal choice is η = . Hence,by (7.14), our proof of Theorem 7.3 is complete. (cid:3) Theorem 7.4. For each ǫ > , as x → ∞ , CC ( x ) = O ( x / ǫ ) . Proof. Again, appearing in (7.10), there are three kinds of divisor sums. For the formersums, we have the bound in Theorem 7.3. For the “middle” four sums, we recall (7.11).Since | d χ ( n ) | ≤ d ( n ) , we can apply the bound given in (1.8) for each of these four “middle”sums. For the latter four sums, we can invoke (1.8). In summary, each of the sums in (7.10)has the bound expressed in Theorem 7.4, and so the proof is complete. (cid:3) RIGONOMETRIC AND BESSEL SERIES 23 8. T HE S ECOND Ω AND “B IG O” T HEOREMS AND C ONJECTURE Let χ and χ be non-principal, primitive even and odd characters modulo p and q , re-spectively. Let τ ( χ ) and τ ( χ ) denote the corresponding Gauss sums. Lastly, recall that d χ ,χ ( n ) is given by (7.1). Theorem 8.1. Assume that χ is a non-principal primitive even character modulo p andthat χ is a non-principal primitive odd character modulo q , where p and q are primes. Let D χ ,χ ( x ) be defined by (7.2) . Then lim x →∞ Re D χ ,χ ( x ) x / = + ∞ , (8.1) lim x →∞ Re D χ ,χ ( x ) x / = −∞ . (8.2) Both (8.1) and (8.2) remain valid if we replace Re by Im , respectively, in (8.1) and (8.2) .Proof. Recall from [8, p. 82] that (cid:16) π √ pq (cid:17) − s Γ( s ) L ( s, χ ) L ( s, χ )= − iτ ( χ ) τ ( χ ) √ pq (cid:16) π √ pq (cid:17) s − Γ(1 − s ) L (1 − s, χ ) L (1 − s, χ ) . We now apply Theorem 6.3. The parameters from Definition 6.1 and (6.6) are: N = 1 , δ = 1 , A = 1 , θ = 14 ,a ( n ) = d χ ,χ ( n ) , b ( n ) = − iτ ( χ ) τ ( χ ) d χ ,χ ( n ) √ pq , λ n = µ n = 2 πn √ pq . Since L (0 , χ ) = 0 , and both L ( s, χ ) and L ( s, χ ) are entire functions, then Q ( x ) = 0 .Theorem 8.1 now follows immediately provided that we can show that (6.5) holds. Theproof that (6.5) is valid is exactly the same as in the previous theorem. (cid:3) Our motivation for studying d χ ,χ ( n ) is similar to that for Theorem 7.1. From [8, p. 85,Equation (4.5)] CS ( x ) := X nm ≤ x ′ cos (cid:16) πnap (cid:17) sin (cid:16) πmbq (cid:17) = 1 iφ ( p ) φ ( q ) X χ mod pχ = χ , even X χ mod qχ odd χ ( a ) χ ( b ) τ ( χ ) τ ( χ ) X n ≤ x ′ d χ ,χ ( n ) − φ ( p ) X m ≤ x ′ h xm i sin (cid:16) πmbq (cid:17) + pφ ( p ) X m ≤ x ′ (cid:20) xpm (cid:21) sin (cid:16) πmbq (cid:17) . (8.3)As in our study of CC ( x ) , we observe that multiple sums of the form X n ≤ x ′ d χ ,χ ( n ) arise. For and Ω theorem for the first set of sums on the right-hand side of (8.3) appeal toTheorem 8.1. For the second set of sums, we recall the identity [5, p. 2068, Lemma 11] X n ≤ x ′ h xn i sin (cid:18) πnap (cid:19) = − i X d | qd> φ ( d ) X χ mod dχ odd χ ( a ) τ ( χ ) X n ≤ dx/q ′ d χ ( n ) . (8.4)Recall that the character sums on the far right-hand side of (8.4) were defined in (7.12) and(7.13). An Ω theorem analogous to Theorem 8.1 can also be established [12, p. 133]. Wecannot appeal directly to Theorem 8.1 and the aforementioned analogue in order to establishan Ω theorem for CS ( x ) . We therefore must content ourselves to making the followingconjecture. Conjecture 8.2. If CS ( x ) is defined on the far left-hand side of (8.3) , then lim x →∞ CS ( x ) x / = + ∞ , lim x →∞ CS ( x ) x / = −∞ . Theorem 8.3. For each ǫ > , as x → ∞ , D χ ,χ ( x ) = O ( x / ǫ ) . Proof. Because the values of A and δ are identical to those in the proof of Theorem 7.3, theproof of Theorem 8.3 is the same as that for Theorem 7.3. (cid:3) Theorem 8.4. For each ǫ > , as x → ∞ , CS ( x ) = O ( x / ǫ ) . Proof. Referring to (8.3), we see that each member of the first set of sums on the right-handside of (8.3) satisfies the bounds of Theorem 8.3. For the second set of sums, refer to (8.4).Since | d χ ( n ) | ≤ d ( n ) , each of the divisor sums on the right-hand side of (8.3) has an upperbound given by (1.8). Hence, the proof of Theorem 8.4 is complete. (cid:3) 9. T HE T HIRD Ω AND “B IG O” T HEOREMS AND C ONJECTURE The third general Ω theorem is similar to Theorems 7.1 and 8.1. Theorem 9.1. Assume that χ and χ are non-principal primitive odd characters modulo p and q , respectively. Define D ∗ χ ,χ ( x ) := X n ≤ x ′ nd χ ,χ ( n ) . (9.1) Then lim x →∞ Re D ∗ χ ,χ ( x ) x / = + ∞ , (9.2) lim x →∞ Re D ∗ χ ,χ ∗ ( x ) x / = −∞ . (9.3) Both (9.2) and (9.3) remain valid if we replace Re by Im in each of (9.2) and (9.3) . RIGONOMETRIC AND BESSEL SERIES 25 Proof. The relevant functional equation is [8, top of page 89] (with s replaced by s ), π − s ( pq ) − s/ Γ (cid:18) s (cid:19) L ( s − , χ ) L ( s − , χ )= − τ ( χ ) τ ( χ ) √ pq π − (3 − s ) ( pq ) − (3 − s ) / Γ (cid:18) 12 (3 − s ) (cid:19) L (2 − s, χ ) L (2 − s, χ ) . Note that L ( s − , χ ) L ( s − , χ ) = ∞ X n =1 χ ( n ) n s − ∞ X m =1 χ ( m ) m s − = ∞ X n =1 nd χ ,χ ( n ) n s . In the notation of Definition 6.1 and (6.6), N = 2 , δ = 3 , A = 1 , θ = 54 , (9.4) a ( n ) = nd χ ,χ ( n ) , b ( n ) = − τ ( χ ) τ ( χ ) nd χ ,χ ( n ) √ pq , λ n = µ n = πn √ pq . (In [8, p. 89] the factor n is missing from the definition of a ( n ) .) Since χ and χ are oddprimitive characters, L (2 s − , χ ) and L (2 s − , χ ) are both entire functions of s whichvanish at s = 0 . Hence, Q ( x ) = 0 . Theorem 9.1 now follows if we can show that (6.5) isvalid. This can be shown in the same way as given in the proof of Theorem 7.1. (cid:3) Let SS ( x ) := X mn ≤ x ′ mn sin(2 πna/p ) sin(2 πmb/q ) . (9.5)From [8, p. 91, Equation (5.13)], SS ( x ) = − φ ( p ) φ ( q ) X χ mod pχ odd X χ mod qχ odd χ ( a ) χ ( b ) τ ( χ ) τ ( χ ) D ∗ χ ,χ ( x ) , (9.6)where D ∗ χ ,χ ( x ) is defined in (9.1). Thus, since SS ( x ) is a linear combination of the sums D ∗ χ ,χ ( x ) , we cannot appeal directly to Theorem 6.2. Thus, we make the following conjec-ture. Conjecture 9.2. If SS ( x ) is defined by (9.5) , then lim x →∞ SS ( x ) x / = + ∞ , lim x →∞ SS ( x ) x / = −∞ . We now use Chandrasekharan and Narasimhan’s Theorem 6.3 to obtain an upper boundfor D ∗ χ ,χ ( x ) . From (9.4), A = 1 and δ = 3 . Observe that | a ( n ) | ≤ nd ( n ) . Furthermore, forsome constant C > , we also see that | b ( n ) | ≤ Cnd ( n ) . Thus, for each ǫ > , we shall take β = 2 + ǫ . Hence, from (6.10), u = 2 + ǫ − − 14 = 14 + ǫ. With a reference to (6.9), we need to calculate, for η ≥ , δ − A + 2 Auη = 32 − 14 + 2 (cid:18) 14 + ǫ (cid:19) η = 54 + (cid:18) 12 + 2 ǫ (cid:19) η. (9.7)For the second power in (6.9), we use partial summation and (1.8) to deduce that X x<µ n ≤ x + O ( x / − η ) nd ( n ) = O ( x / − η log x ) . (9.8)From (9.7) and (9.8), we seek the optimal power of x by solving 54 + (cid:18) 12 + 2 ǫ (cid:19) η = 32 − η + ǫ. Solving this simple equation, we see that η = + ǫ . Therefore, we have established thefollowing theorem. Theorem 9.3. As x → ∞ , for every ǫ > , D ∗ χ ,χ ( x ) = O ( x / ǫ ) , where D ∗ χ ,χ ( x ) is defined by (9.1) . Using (9.6) and Theorem 9.3, we can immediately deduce the following theorem. Theorem 9.4. As x → ∞ , for every ǫ > , SS ( x ) = O ( x / ǫ ) . We conclude this section with a special case. Let ap = bq = . Then sin(2 πn/ 4) = ( ( − ( n − / , n odd , , n even , and SS ( , , x ) := − X mn ≤ xm,n odd mn ( − ( m + n ) / = X (2 j +1)(2 k +1) ≤ x ( − j + k (2 j + 1)(2 k + 1) , where we set m = 2 j + 1 , n = 2 k + 1 . This is a rather interesting lattice point problem. Weare counting lattice points under the hyperbola ab ≤ x , but we require both coordinates to beodd, and we put a weight on them.We restate Conjecture 9.2 and Theorem 9.4 in this particular case ap = bq = . Conjecture 9.5. If SS ( x ) is defined by (9.5) , then lim x →∞ SS ( , , x ) x / = + ∞ , lim x →∞ SS ( , , x ) x / = −∞ . RIGONOMETRIC AND BESSEL SERIES 27 Theorem 9.6. As x → ∞ , for every ǫ > , SS ( , , x ) = O ( x / ǫ ) . 10. S UMS WITH A P RODUCT OF AN A RBITRARY N UMBER OF sin ’ S We begin this section with a definition. Let J ν ( x ) denote the ordinary Bessel function oforder ν . Define K ν ( x ; µ ; m ) := Z ∞ u ν − µ − m − J µ ( u m − ) du m − Z ∞ u ν − µ − m − J µ ( u m − ) du m − · · · · · · Z ∞ u ν − µ − J µ ( u ) J ν ( x/u u · · · u m − ) du , (10.1)provided that µ, ν > − / , so that the integral converges.In the sequel, we apply Theorems 2 and 4 of [2, pp. 351, 356], which, for convenience,we offer below. Theorem 10.1. Let ϕ ( s ) and ψ ( s ) satisfy the functional equation (6.1) with ∆( s ) = Γ k ( s ) .If k ≥ , suppose that δ > − . Assume that ρ > kσ ∗ a − kδ − , where σ ∗ a is the abscissa ofabsolute convergence of ψ ( s ) . If x > , then ρ + 1) X λ n ≤ x ′ a ( n )( x − λ n ) ρ = 2 ρ (1 − k ) ∞ X n =1 b ( n ) (cid:18) xµ n (cid:19) ( δ + ρ ) / K δ + ρ (2 k ( µ n x ) / ; δ − k ) + Q ρ ( x ) , where Q ρ ( x ) is defined in (6.4) . The following Theorem 10.2 is an extension of Theorem 10.1. In our application, thehypotheses are readily verified. Theorem 10.2. Suppose that for σ > σ a ∗ , sup ≤ h ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X r k ≤ µ n ≤ ( r + h ) k b ( n ) µ σ − / (2 k ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o (1) , as r → ∞ . Then (10.1) is valid for q > kσ ∗ a − kδ − and for those positive values of x such that the left-hand side of (10.1) is defined. The series on the right-hand side of (10.1) converges uniformly on any interval for x > where the left-hand side is continuous. Theconvergence is bounded on any interval < x ≤ x ≤ x when ρ = 0 . Let k be an arbitrary positive integer. Let χ , χ , . . . , χ k be odd, primitive, non-principal,characters modulo p , p , . . . , p k , respectively. Throughout this section, we assume that χ , χ , . . . , χ k and χ are odd, and so we normally make this assumption without comment.Let a , a , . . . , a k denote positive integers such that ( a j , p j ) = 1 , ≤ j ≤ k . Definition 10.3. Let n , n , . . . , n k denote positive integers. Define d χ ,χ ,...,χ k ( n ) := X n n ··· n k = n χ ( n ) χ ( n ) · · · χ k ( n k ) and D k ( x ) := X n ≤ x ′ nd χ ,χ ,...,χ k ( n ) , (10.2) where, as customary, the prime ′ on the summation sign indicates that if x is an integer, only of the term is counted. Note that if k = 2 , then (10.2) is identical to (9.1). Theorem 10.4. Recall the definition of K ρ +3 / ( x ; ; k ) defined by (10.1) , as well as thedefinitions and notation above. Then for ρ > ( k − / , X ′ n ≤ x nd χ ,χ ,...,χ k ( n )( x − n ) ρ = ( − i ) k ( p p · · · p k ) ρ − / τ ( χ ) τ ( χ ) · · · τ ( χ k ) π kρ × ∞ X n =1 nd χ ,χ ,...,χ k ( n ) (cid:16) xn (cid:17) ρ +3 / K ρ +3 / (cid:18) k π k nxp p · · · p k ; 12 ; k (cid:19) . The integral K ρ +3 / (cid:18) k π k nxp p · · · p k ; ; k (cid:19) can be represented by a Meijer G -function, but we do not provide it here. Proof. Recall that the Dirichlet L -function L ( s, χ ) of modulus q satisfies the functional equa-tion [8, p. 82] (cid:18) πq (cid:19) − (2 s +1) / Γ (cid:18) s + 12 (cid:19) L (2 s, χ ) = − iτ ( χ ) √ q (cid:18) πq (cid:19) − (1 − s ) Γ (1 − s ) L (1 − s, χ ) . (10.3)We replace s by s − , q by p j , and χ by χ j in (10.3), ≤ j ≤ k . Multiply the k functionalequations together to obtain π − ks ( p p · · · p k ) − s Γ k ( s ) L (2 s − , χ ) L (2 s − , χ ) · · · L (2 s − , χ k )= ( − i ) k τ ( χ ) τ ( χ ) · · · τ ( χ k ) √ p p · · · p k π − k (3 − s ) / ( p p · · · p k ) − (3 − s ) / Γ k (cid:18) − s (cid:19) × L (2 − s, χ ) L (2 − s, χ ) · · · L (2 − s, χ k ) . (10.4)Note that δ = and that L (2 s − , χ ) L (2 s − , χ ) · · · L (2 s − , χ k )= ∞ X n =1 χ ( n ) n s − ∞ X n =1 χ ( n ) n s − · · · ∞ X n k =1 χ k ( n ) n s − k = ∞ X n =1 nd χ ,χ ,...,χ k ( n ) n s , σ = Re s > . We apply Theorems 10.1 and 10.2. Observe that Q ρ ( x ) = 0 , because, for ≤ j ≤ k , L ( s, χ j ) is an entire function and L ( − , χ j ) = 0 , since χ j is odd. Also observe that a ( n ) = nd χ ,χ ,...,χ k ( n ) , b ( n ) = ( − i ) k τ ( χ ) τ ( χ ) · · · τ ( χ k ) √ p p · · · p k nd χ ,χ ,...,χ k ( n ) , RIGONOMETRIC AND BESSEL SERIES 29 and λ n = µ n = π k n p p · · · p k . In Theorem 10.1 the sum is over λ n ≤ x . Replace x by π k x p p · · · p k . Thus, the amended sum will be over n ≤ x and ( x − λ n ) ρ ⇒ π kρ ( p p · · · p k ) ρ ( x − n ) ρ . Also appearing in Theorem 10.1 is a quotient in the summands on the right side that will betransformed by the change in variable above, i.e., (cid:18) xµ n (cid:19) (3 / ρ ) / ⇒ (cid:16) xn (cid:17) / ρ . Lastly, on the right-hand side of (10.1), k ( µ n x ) / ⇒ k (cid:18) π k n p p · · · p k · π k x p p · · · p k (cid:19) / = 2 k π k nxp p · · · p k . With all of these substitutions, we deduce that π kρ ( p p · · · p k ) ρ X ′ n ≤ x nd χ ,χ ,...,χ k ( n )( x − n ) ρ = ( − i ) k τ ( χ ) τ ( χ ) · · · τ ( χ k ) √ p p · · · p k ∞ X n =1 nd χ ,χ ,...,χ k ( n ) (cid:16) xn (cid:17) ρ +3 / K ρ +3 / (cid:18) k π k nxp p · · · p k ; 12 ; k (cid:19) , where K ρ +3 / is defined in (10.1). The identity above is precisely (10.4), and so the proof iscomplete. (cid:3) We next prove an identity for the sum S ρ ( a , a , . . . a k ; p , p , . . . p k ; x ) := X ′ ≤ n n ··· n k ≤ x n n · · · n k sin(2 πn a /p ) sin(2 πn a /p ) · · · sin(2 πn k a k /p k )( x − n ) ρ , (10.5)where ( a j , p j ) = 1 , ≤ j ≤ k and n = n n · · · n k . Lemma 10.5. [8, p. 72] Let ( a, q ) = 1 and n ∈ Z . Suppose that χ is an odd, primitive,non-principal character of order q . Then sin(2 πna/q ) = 1 iφ ( q ) X χ mod qχ odd χ ( a ) τ ( χ ) χ ( n ) , where φ ( q ) denotes Euler’s φ -function. Lemma 10.6. [3, p. 3806, Equation (4.12)] For any primitive character χ modulo q , X χ mod qχ odd χ ( a ) χ ( b ) = ( ± φ ( q ) , if a ≡ ± b ( mod q ) and ( a, q ) = 1 , , otherwise . We let X ± n , ± n ,..., ± n k denote a sum over all k pairs n ≡ ± a ( mod p ) , n ≡ ± a ( mod p ) , n k ≡ ± a k ( mod p k ) . Theorem 10.7. In the notation above, for ρ > ( k − / , S ρ ( a , a , . . . , a k ; p , p , . . . , p k ; x )= x ρ +3 / ( p p · · · p k ) ρ +1 / k π kρ X ± n , ± n ,..., ± n k ( − sgn K ρ +3 / (cid:18) k π k n n · · · n k xp p · · · p k ; 12 ; k (cid:19) √ n n · · · n k , where sgn denotes the number of minus signs in a particular k -tuple, ± n , ± n , . . . , ± n k . Proof. By (10.5) and Lemma 10.5, if n = n n · · · n k , S ρ ( a , a , . . . , a k ; p , p , · · · , p k ; x ) = ( − i ) k φ ( p ) φ ( p ) · · · φ ( p k ) X ′ ≤ n n ··· n k ≤ x n n · · · n k ( x − n ) ρ × X χ mod p X χ mod p · · · X χ k mod p k χ ( a ) χ ( a ) · · · χ ( a k ) τ ( χ ) τ ( χ ) · · · τ ( χ k ) χ ( n ) χ ( n ) · · · χ ( n k )= ( − i ) k φ ( p ) φ ( p ) · · · φ ( p k ) × X χ mod p X χ mod p · · · X χ k mod p k χ ( a ) χ ( a ) · · · χ ( a k ) (10.6) × τ ( χ ) τ ( χ ) · · · τ ( χ k ) X ′ n ≤ x nd χ ,χ ,...,χ k ( n )( x − n ) ρ . On the other hand, by Lemma 10.6, x ρ +3 / ( p p · · · p k ) ρ +1 / k X ± n , ± n ,..., ± n k ( − sgn K ρ +3 / (cid:18) k π k n n · · · n k xp p · · · p k ; 12 ; k (cid:19) ( n n · · · n k ) ρ +1 / = x ρ +3 / ( p p · · · p k ) ρ +1 / φ ( p ) φ ( p ) · · · φ ( p k ) ∞ X n =1 ∞ X n =1 · · · ∞ X n k =1 K ρ +3 / (cid:18) k π k n n · · · , n k xp p · · · p k ; 12 ; k (cid:19) ( n n · · · n k ) ρ +1 / × X χ mod p X χ mod p · · · X χ k mod p k χ ( a ) χ ( a ) · · · χ k ( a k ) χ ( n ) χ ( n ) · · · χ k ( n k )= x ρ +3 / ( p p · · · p k ) ρ +1 / φ ( p ) φ ( p ) · · · φ ( p k ) X χ mod p X χ mod p · · · X χ k mod p k χ ( a ) χ ( a ) · · · χ k ( a k ) RIGONOMETRIC AND BESSEL SERIES 31 × ∞ X n =1 d χ ,χ ,...,χ k ( n ) K ρ +3 / (cid:18) k π k nxp p · · · p k ; 12 ; k (cid:19) n ρ +1 / = ( p p · · · p k ) ρ +1 / φ ( p ) φ ( p ) · · · φ ( p k ) X χ mod p X χ mod p · · · X χ k mod p k χ ( a ) χ ( a ) · · · χ k ( a k ) × ∞ X n =1 nd χ ,χ ,...,χ k ( n ) (cid:16) xn (cid:17) ρ +3 / K ρ +3 / (cid:18) k π k nxp p · · · p k ; 12 ; k (cid:19) = ( p p · · · p k ) ρ +1 / φ ( p ) φ ( p ) · · · φ ( p k ) X χ mod p X χ mod p · · · X χ k mod p k χ ( a ) χ ( a ) · · · χ k ( a k ) × π kρ ( − i ) k τ ( χ ) τ ( χ ) · · · τ ( χ k )( p p · · · p k ) ρ − / X ′ n ≤ x nd χ ,χ ,...,χ k ( n )( x − n ) ρ = ( − i ) k π kρ φ ( p ) φ ( p ) · · · φ ( p k ) X χ mod p X χ mod p · · · X χ k mod p k χ ( a ) χ ( a ) · · · χ k ( a k ) × τ ( χ ) τ ( χ ) · · · τ ( χ k ) X ′ n ≤ x nd χ ,χ ,...,χ k ( n )( x − n ) ρ , (10.7)where in the penultimate step we applied Theorem 10.4, and in the last step, used the factthat for odd χ [4, p. 45], τ ( χ j ) τ ( χ j ) = − p j , ≤ j ≤ k. If we now compare (10.6) with (10.7), we deduce Theorem 10.7. (cid:3) Next, we offer a generalization of Theorem 9.1. Theorem 10.8. Assume that χ , χ , . . . , χ k are non-principal primitive odd characters mod-ulo p , p , . . . , p k , respectively. Recall the definition (10.2) of D k ( x ) . Then lim x →∞ Re D k ( x ) x (3 k − / (2 k ) = + ∞ , (10.8) lim x →∞ Re D k ( x ) x (3 k − / (2 k ) = −∞ . (10.9) Both (10.8) and (10.9) remain valid if we replace Re by Im in each of (10.8) and (10.9) .Proof. Replacing s by s/ in (10.4), we see that A = k and δ = 3 . Thus, from the definition(6.6), θ = k · − · k = 3 k − k . With the use of Theorem 6.2, the remainder of the proof follows along the same lines as theproof of Theorem 9.1. (cid:3) Recall from (10.5) that S ( a , a , . . . a k ; p , p , . . . p k ; x ) := S ( a , a , . . . a k ; p , p , . . . p k ; x )= X ′ ≤ n n ··· n k ≤ x n n · · · n k sin(2 πn a /p ) sin(2 πn a /p ) · · · sin(2 πn k a k /p k ) . From (10.6) with ρ = 0 , we see that S ( a , a , . . . , a k ; p , p , . . . , p k ; x ) is a linear combina-tion of terms of the form D k ( x ) = X ′ n ≤ x nd χ ,χ ,...,χ k ( n ) . In analogy with Conjecture 9.2, a similar conjecture can be made for S ( a , a , . . . , a k ; p , p , . . . , p k ; x ) . Thus, we apply Theorem 10.8 to each of these sums to obtain the following conjecture. Conjecture 10.9. If S ( a , a , . . . a k ; p , p , . . . p k ; x ) is defined by (10.5) , then lim x →∞ S ( a , a , . . . , a k ; p , p , . . . , p k ; x ) x (3 k − / (2 k ) = + ∞ , lim x →∞ S ( a , a , . . . , a k ; p , p , . . . , p k ; x ) x (3 k − / (2 k ) = −∞ . Note that Conjecture 9.2 is the special case k = 2 of Conjecture 10.9.Similarly, we can use (10.6) and Theorem 6.3 to obtain an upper bound for the order of S ( a , a , . . . , a k ; p , p , . . . , p k ; x ) . Theorem 10.10. For every ǫ > , as x → ∞ , S ( a , a , . . . , a k ; p , p , . . . , p k ; x ) = O (cid:0) x k/ ( k +1)+ ǫ (cid:1) . (10.10) Proof. Note that β = 2 + ǫ for each ǫ > . Then, by (6.10), u = 2 + ǫ − − k = 12 + ǫ − k . Next, by (6.9), we need to calculate δ − A + 2 Auη = 32 − k + k (cid:18) 12 + ǫ − k (cid:19) η = 32 − k + k − η + kǫη, (10.11)where η is a non-negative number to be determined.Let d k ( n ) denote the number of ways n can be written as a product of k factors. Then, by(1.8) and induction on k , X n ≤ x ′ d k ( n ) = xP k − (log x ) + O ( x ( k − / ( k +1) log k − x ) , (10.12)where k ≥ and P r (log x ) is a polynomial of degree r in log x . (See also [12, p. 133,Equation (10.10)]). Next, use (10.12) and partial summation to deduce that | X x<µ n ≤ x + O ( x − /k − η ) nd χ ,χ ,...,χ k ( n ) | ≤ X x<µ n ≤ x + O ( x − /k − η ) nd k ( n ) = O ( x − /k − η log k − x ) . (10.13) RIGONOMETRIC AND BESSEL SERIES 33 Appealing to Theorem 6.3, we should find the optimal value of η by equating the powers in(10.11) and (10.13). Thus, we should solve − k − η = 32 − k + k − η. Hence, η = k − k ( k + 1) , and so the optimal power is, for every ǫ > , − k − k ( k + 1) + ǫ = 2 kk + 1 + ǫ. This completes the proof of (10.10). (cid:3) Note that if k = 2 , (10.10) reduces to Theorem 9.4. Note also that k + 1) k + 2 − kk + 1 = 2( k + 1)( k + 2) is the difference in the exponents of (10.10) for successive values of k . Thus, increasing thenumber of sin ’s by 1 in S ( a , a , . . . , a k ; p , p , . . . , p k ; x ) increases the upper bound for thepower in the error term by a “small” amount, i.e., O (1 /k ) .Suppose that a j p j = 14 , ≤ j ≤ k, so that the terms are equal to 0 if one or more of the n j , ≤ j ≤ k are even. For odd n j , let n j = 2 m j + 1 , ≤ j ≤ k. Then, X ′ ≤ n ,n ,...,n k ≤ x n n · · · n k sin(2 πn a /p ) sin(2 πn a /p ) · · · sin(2 πn k a k /p k )= X ′ ≤ m +1 , m +1 ,..., m k +1 ≤ x (2 m + 1)(2 m + 1) · · · (2 m k + 1)( − m + m + ··· + m k = X ′ ≤ m ,m ,...,m k ≤ x (2 m + 1)(2 m + 1) · · · (2 m k + 1)( − m + m + ··· + m k , after replacing x by x + 1 . Thus, Theorem 10.7 gives an identity for the weighted sumof products of positive odd lattice points m + 1 , m + 1 , . . . , m k + 1 in k -dimensionalspace weighted by ( − m + m + ··· + m k . However, instead of applying Theorem 10.7 directly,if we apply Equation (10.6) instead, we obtain a “Big O” bound for this sum weighted by ( − m + m + ··· + m k . Recall that we addressed the case k = 2 earlier.Finding a representation for a sum of k cos -functions appears to be enormously compli-cated. Similarly, finding a representation for a sum with a mixture of sin ’s and cos ’s appearsalso to be extremely complicated. 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