A note on the gaps between zeros of Epstein's zeta-functions on the critical line
aa r X i v : . [ m a t h . N T ] M a r A NOTE ON THE GAPS BETWEEN ZEROS OF EPSTEIN’SZETA-FUNCTIONS ON THE CRITICAL LINE
STEPHAN BAIER, SRINIVAS KOTYADA AND USHA KESHAV SANGALE
Abstract.
It is proved that Epstein’s zeta-function ζ Q ( s ), related to a positive definiteintegral binary quadratic form, has a zero 1 / iγ with T ≤ γ ≤ T + T / ε for sufficientlylarge positive numbers T . This is an improvement of the result by M. Jutila and K. Srinivas(Bull. London Math. Soc. 37 (2005) 45–53). To Professor Matti Jutila with deep regards Introduction
Let the quadratic form Q ( x, y ) = ax + bxy + cy be positive definite, have integercoefficients and let r Q ( n ) count the number of solutions of the equation Q ( x, y ) = n inintegers x and y . The Epstein zeta-function associated to Q is denoted by ζ Q ( s ) and isgiven by the series ζ Q ( s ) = X ( x,y ) ∈ Z − (0 , Q ( x, y ) s = ∞ X n =1 r Q ( n ) n s in the half-plane σ >
1, where (as usual) s = σ + it . Throughout the paper, we shall write∆ to denote the number ∆ := | ac − b | , the modulus of the discriminant of Q . ζ Q ( s ) hasmany analytical properties in common with the Riemann zeta-function, ζ ( s ). For example,it admits analytic continuation into the entire complex plane except for a simple pole at s = 1 with residue 2 π ∆ − / . It satisfies the following functional equation √ ∆2 π ! s Γ( s ) ζ Q ( s ) = √ ∆2 π ! − s Γ(1 − s ) ζ Q (1 − s ) . (1.1)The analogue of Hardy’s theorem for ζ ( s ) also holds true for ζ Q ( s ), i.e., ζ Q ( s ) admitsinfinitely many zeros on the critical line σ = 1 /
2. In fact, much more is true. In 1934,Potter and Tichmarsh [7] showed that every interval of the type [
T, T + T / ε ] contains azero 1 / iγ of ζ Q ( s ) for any fixed ε and for all sufficiently large T . Sankaranarayanan in1995 [9] showed that the same result holds true for intervals of the type [ T, T + cT / log T ].In 2005, Jutila and Srinivas [4] proved that the same is true for intervals of the type[ T, T + cT / ε ], thus surpassing the classical barrier of 1 / T . In this notewe improve this result further. More precisely, we prove the following Mathematics Subject Classification.
Theorem 1.
Let Q be a positive definite binary integral quadratic form. Then for any fixed ε > and T ≥ T ( ε, Q ) , there is a zero / iγ of the corresponding Epstein zeta-function ζ Q ( s ) with | γ − T |≤ T / ε . (1.2)In this paper we indicate those steps in [4] which enabled us to improve the result ofJutila and Srinivas mentioned earlier. Therefore, for technical details the readers are urgedto refer to [4] and [3]. However, for the sake of completeness, we shall discuss the mainideas contained in [4].The paper is organized as follows: In section 2 we describe the basic idea of the proof,section 3 contains basic results used in the proof of the main theorem and in section 4 weestimate a double exponential sum non-trivially, which leads to the improvement.2. Basic idea of the proof
Hardy and Littlewood [5] developed a beautiful method to prove the existence of a zeroof the Riemann zeta-function ζ ( s ) on the critical line in a short interval. The significanceof their method is that it is amenable to generalization. We start by defining the functions f ( s ) , γ ( s ) and W ( t ) as f ( s ) = e πi ( − s ) (cid:18) √△ π (cid:19) s Γ( s ) ζ Q ( s ) = γ ( s ) ζ Q ( s )and W ( t ) = f (cid:18)
12 + it (cid:19) . From the functional equation (1.1), it follows that W ( t ) is real for real values of t . Thus,the real zeros of W ( t ) coincide with the zeros of ζ Q ( s ) on the critical line.First, let us assume that W ( t ) has no zero in the interval [ T − H, T + H ] , with T ε ≤ H ≤ T / . Let H = HT − ǫ and consider the integral I = Z H − H W ( T + u )e − ( u/H ) du. Then by our assumption | I | = Z H − H | W ( T + u ) | e − ( u/H ) du. (2.1)If the equality in (2.1) is violated, this will establish the existence of an odd order zeroof W ( t ) in the interval [ T − H, T + H ]. This contradiction is achieved by estimating theintegral in (2.1) from below and above, provided H = T / ε .Estimation from below is the easy step, thanks to a general result of K. Ramachandra[8] which states that the first power mean of a generalized Dirichlet series satisfying certainconditions can not be too small. We need only a particular case of this theorem which isreadily available as Theorem 3 of [1], which we state as: NOTE ON THE GAPS BETWEEN ZEROS OF EPSTEIN’S ZETA-FUNCTIONS ON THE CRITICAL LINE3
Lemma 1.
Let B ( s ) = P ∞ n =1 b n n − s be any Dirichlet series satisfying the following condi-tions: (i) not all b n ’s are zero; (ii) the function can be continued analytically in σ ≥ a, | t | ≥ t , and in this region B ( s ) = O (( | t | + 10) A ) . Then for every ǫ > , we have Z T + HT | B ( σ + it ) | dt ≫ H for all H ≥ (log T ) ǫ , T ≥ T ( ǫ ) , and σ > a . Thus the lower bound | I | ≫ H (2.2)follows directly from Lemma 1.Estimation from above is the hard part. We start with writing the integral I as I = Z H − H e π ( T + u ) (cid:16) √△ π (cid:17) + i ( T + u ) Γ (cid:0) + i ( T + u ) (cid:1) × ζ Q (cid:0) + i ( T + u ) (cid:1) e − ( u/H ) du. (2.3)The zeta-function ζ Q (cid:0) + i ( T + u ) (cid:1) appearing in the integrand is now replaced with anappropriate approximate formula. Such a formula was derived in [4], Lemma 1. We statethis as Lemma 2.
Let t ≥ and t ≪ X ≪ t A , where A is an arbitrarily large positive constant.Then we have ζ Q (cid:0) + it (cid:1) = X n ≤ X r Q ( n ) n − / − it + (log 2) − X X As in [4], we extract, from the right hand side, a weighted sum of the form X n η ( n ) r Q ( n ) n − / − it , (2.6)where the weight function η ( n ) is supported in the interval [ T √△ / π − K, T √△ / π + K ], t ≍ T and t lies close to T . The object is to show that this sum is small in a certain sense.The remaining terms are evaluated by complex integration technique and we shall showthat their contribution is negligible.To begin with, the smooth weight function is η ( n ) is defined as η ( x ) = (cid:26) | x − T √△ / π | ≤ K/ , | x − T √△ / π | ≥ K and K is chosen to satisfy the relation HK = T ε . (2.7)Thus, using the trivial identity 1 = η ( n ) + (1 − η ( n )) in (2.5), we obtain I = X n ≤ T | n − T √△ / π | >K/ r Q ( n )(1 − η ( n )) n − / − iT Z H − H γ (1 / i ( T + u )) n − iu e − ( u/H ) du + Z H − H γ (1 / i ( T + u )) X | n − T √△ / π |≤ K r Q ( n ) η ( n ) n − / − i ( T + u ) e − ( u/H ) du +(log 2) − X T 0; and therefore, e − ( u/H ) = e − ( H − y )+2 iyH/ ( H ) ≪ e − ( H/H ) ≪ e − T ε where as (2.10) is bounded. On the other hand on the lower horizontal side of the rectangle,setting u = x − iH , − H ≤ x ≤ H , we observe that e − ( u/H ) ≤ e − ( x − ( H ) ) / ( H ) ≤ e ( H ) / ( H ) = 1and γ (1 / i ( T + u )) n − iu ≪ e H log ( T √ ∆ / πn ) = e − H log ( πn/T √ ∆ ) . (2.11)For n ≥ T √ ∆ / π + K/ 2, using the elementary inequality log a/b ≥ | a − b | / ( a + b ), we seethat H log (cid:16) πn/T √ ∆ (cid:17) ≥ H K/ T √ ∆ / π + K/ . Now, H K/ T √ ∆ / π + K/ ≥ T ε provided H K > T ε . The last inequality is guaranteed by the choice of K in (2.7). Therefore, from (2.11), weget γ (1 / i ( T + u )) n − iu ≪ e − T ε Collecting all the estimates above, we have for n ≥ T √ ∆ / π + K/ Z H − H γ (1 / i ( T + u )) n − iu e − ( u/H ) du ≪ He − T ε . Thus, S ≪ He − T ε | X T √△ / π + K/ ≤ n ≤ T r Q ( n )(1 − η ( n )) n − / − iT |≪ He − T ε X T √△ / π + K/ ≤ n ≤ T r Q ( n ) n − / ≪ He − T ε T ǫ ( T p △ / π + K/ − / ≪ . (2.12) Estimation of S . The estimation of S is similar to that of S . In this case, theintegral (2.9) is written as a contour integral over a rectangle with vertices ± H, ± H + iH . STEPHAN BAIER, SRINIVAS KOTYADA AND USHA KESHAV SANGALE Then the integral is bounded by He − T ε on the vertical sides, where as on the horizontalsides, putting u = x + iH , − H ≤ x ≤ H ; we get | γ (1 / i ( T + u )) n − iu | ≤ e − H log ( T √ ∆ / πn ) ≪ e − T ε . Therefore, | S | ≪ He − T ε | X ≤ n ≤ T √△ / π − K/ r Q ( n )(1 − η ( n )) n − / − iT |≪ He − T ε X ≤ n ≤ T √△ / π − K/ r Q ( n ) n − / ≪ He − T ε T / ε ≪ . (2.13) Estimation of S . The estimation of S follows the same pattern as that of S . To showthat the integral (2.9) is small, we want H log (cid:16) πn/T √ ∆ (cid:17) > T ǫ , that is H log (cid:16) πn − T √ ∆2 πn + T √ ∆ (cid:17) > T ǫ , or H log (cid:16) πT − T √ ∆2 πT + T √ ∆ (cid:17) > T ǫ , which is the same as H log (cid:16) − T √ ∆2 πT + T √ ∆ (cid:17) > T ǫ , which is true if H > T ε . This is indeed the case, since our choice of H is T ε ≤ H ≤ T / .Thus we conclude that | S | ≪ (log 2) − X T Therefore, | S | ≤ H sup | T − t |≤ H | ∞ X n =1 r Q ( n ) η ( n ) n − / − it | (2.15)The objective now is to show that ∞ X n =1 r Q ( n ) η ( n ) n − / − it ≪ (log T ) − (2.16)for a suitable choice of the parameter K . Then combining this with (2.8), (2.12), (2.13),(2.14) and (2.15), we have | I | ≪ H (log T ) − . This is a contradiction to (2.2).In [4], the crucial sum in (2.16) was first transformed into another sum (equation (3.6)of [4]) using a transformation formula. By partial summation, this sum got reduced to theestimation of the following expression (for notations see subsection 4.1): K / N − / T − / X N ≤ Q ∗ ( x,y ) ≤ N ′ e (cid:18) Q ∗ ( x, y ) (cid:18) h ∆ k − hk ∆ (cid:19) + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19) (2.17)This is equation (4.4) of [4]. The task is show that this term is O ( T − ε ). To establish this,the double exponential sum, appearing above, was estimated non-trivially in one variableusing van der Corput’s method and trivial estimate was taken in the other variable. Theauthors obtained the bound O ( K / T − / ), which is O ( T − ε ), provided K = T / − ε .In the present paper we estimate the above double exponential sum non-trivially in bothvariables. To show that (2.17) is O ( T − ε ), it is now enough to take K = T / − ε and thusthe gap H = T / ε follows. 3. Preliminary lemmas We will use the following well-known lemmas in the proof of our theorem. Lemma 3. (Generalized Weyl differencing) Let a < b be integers, λ be a natural numberand ξ ( n ) be a complex valued function such that ξ ( n ) = 0 if n ( a, b ] . If H is a positiveinteger then | X n ξ ( n ) | ≤ ( b − a ) + HH X | h | For the case λ = 1, this is Lemma 2.5 of [2]. The general case can be provedsimilarly. Lemma 4. (B-Process , Lemma 3.6 of [2] ) Suppose that f has four continuous derivativeson [ a, b ] , and that f ′′ < on this interval. Suppose further that [ a, b ] ⊆ [ N, N ] and that α = f ′ ( b ) and β = f ′ ( a ) . Assume that there is some F > such that f (2) ( x ) ≍ F N − , f (3) ( x ) ≪ F N − , and f (4) ( x ) ≪ F N − STEPHAN BAIER, SRINIVAS KOTYADA AND USHA KESHAV SANGALE for x in [ a, b ] . Let x ν be defined by the relation f ′ ( x ν ) = ν , and let φ ( ν ) = − f ( x ν ) + νx ν .Then X a ≤ n ≤ b e ( f ( n )) = X α ≤ ν ≤ β e ( − φ ( ν ) − / | f ′′ ( x ν | / + O (log( F N − + 2) + F − / N ) . (3.2) Lemma 5. (van der Corput’s bound, Theorem 2.2 of [2] ) Suppose that f is a real valuedfunction with two continuous derivatives on [ a, b ] where a < b are integers. Suppose alsothat there is some λ > and some α ≥ such that λ ≤ | f ′′ ( x ) | ≤ αλ on [ a, b ] . Then X a ≤ n ≤ b e ( f ( n )) ≪ α ( b − a ) λ / + λ − / . (3.3)4. Estimation of the double exponential sum Preparation and description of the method. First, we explain the meanings ofthe functions and variables occurring in (2.17). The function Q ∗ ( x, y ) denotes a certainpositive definite quadratic form (for the details of its definition, see equation (3.4) of [4]) Q ∗ ( x, y ) = a ∗ x + b ∗ xy + c ∗ y , a ∗ , b ∗ , c ∗ ∈ Z , (4.1)which is related to Q ( x, y ) and a positive integer k in a specific way. Throughout thesequel, we denote by d the discriminant of this form, i.e. d := ( b ∗ ) − a ∗ c ∗ < . The form Q ∗ ( x, y ) is defined in such a way that a ∗ > , c ∗ > 0. Further, as remarked in[4], | d | ≤ ∆ . The variables in (2.17) satisfy the following conditions. We suppose that 1 ≤ N ≤ N ′ ≤ N , N ≪ K , ∆ , h and k are positive integers satisfying ∆ | ∆, ( h ∆ , k ) = 1 and KT ≪| √ ∆ − hk |≤ πKT ∆ , and the sizes of k, h and the real number t are k, h ≍ p T /K, t ≍ T. As usual, h ∆ denotes a multiplicative inverse of h ∆ modulo k .Finally, the function φ ( x ) is defined as φ ( x ) = arsinh( x / ) + ( x + x ) / . Our goal is now to bound non-trivially the exponential sum in (2.17), i.e. the exponentialsum X x X y ∈I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19) , (4.2) NOTE ON THE GAPS BETWEEN ZEROS OF EPSTEIN’S ZETA-FUNCTIONS ON THE CRITICAL LINE9 where r := h ∆ k − hk ∆ and I ( x ) := { y ∈ R : N ≤ Q ∗ ( x, y ) ≤ N ′ } . In [4], the summation over y was evaluated using the following classical estimate for expo-nential sums (Lemma 4.1 of [4]). Lemma 6. Suppose that f is a real valued function with three continuous derivatives on [ a, b ] where a < b are integers. Suppose also that there is some λ > and some α ≥ suchthat λ ≤ | f ′′′ ( x ) | ≤ αλ on [ a, b ] . Then X a ≤ n ≤ b e ( f ( n )) ≪ α / ( b − a ) λ / + ( b − a ) / λ − / . The above lemma can be proved using Weyl differencing, Lemma 3, followed by applyingthe can der Corput bound, Lemma 5.In [4], the sum over x was treated trivially. In the present paper, we also want to exploitcancellations in the x -sum. To this end, we explicitly carry out Weyl differencing for thesum over y , then employ the B process, Lemma 4, re-arrange the summation and finallyapply van der Corput’s bound to the sum over x .It is easy to see that I ( x ) is empty unless x ∈ J , where J := " − √ c ∗ N ′ p | d | , √ c ∗ N ′ p | d | and that I ( x ) = I ( x ) ∪ I ′ ( x ) , where I ( x ) := "s max (cid:26) , Nc ∗ − | d | (2 c ∗ ) · x (cid:27) , s N ′ c ∗ − | d | (2 c ∗ ) · x and I ′ ( x ) := " − s N ′ c ∗ − | d | (2 c ∗ ) · x , − s max (cid:26) , Nc ∗ − | d | (2 c ∗ ) · x (cid:27) . Set J = { x ∈ J : x ≥ } = " , √ c ∗ N ′ p | d | . In the following, we estimate the partial sum X x ∈ J X y ∈ I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19) , where x is positive and y runs over the interval I ( x ). The remaining three partial sumswith a) x ≥ y ∈ I ′ ( x ), b) x < y ∈ I ( x ), c) x < y ∈ I ′ ( x ) can beestimated in a similar way.4.2. Application of Weyl differencing. We start with applying the Cauchy-Schwarzinequality, getting (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ J X y ∈ I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪√ N · X x ∈ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X y ∈ I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.3)where we use | J | ≪ √ N . Applying Lemma 3 with λ = 2, and using | J | ≪ √ N and | I ( x ) | ≪ √ N , we have X x ∈ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X y ∈ I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ X x ∈ J √ NM · X ≤| m |≤ M (cid:18) − | m | M (cid:19) · X y ∈ I m ( x ) e ( f x ( y + 2 m ) − f x ( y )) ≪ √ NM · X ≤ m ≤ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ J X y ∈ I ( x ) e ( f x ( y + m ) − f x ( y − m )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + N M + N / M , (4.4)where M ≤ N is any natural number, I m ( x ) := { y ∈ I ( x ) : y + 2 m ∈ I ( x ) } and f x ( y ) := Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19) . We have Q ∗ ( x, y + m ) − Q ∗ ( x, y − m ) = 2 m ( b ∗ x + 2 c ∗ y )and, using Taylor series expansion, φ (cid:18) πQ ∗ ( x, y + m )2 hk ∆ t (cid:19) − φ (cid:18) πQ ∗ ( x, y − m )2 hk ∆ t (cid:19) = 2 mg ′ x ( y ) + O (cid:0) m | g ′′′ x ( y ) | (cid:1) , where g x ( y ) := φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19) . We note that the quadratic Taylor term disappears due to our treatment. This is thereason why we chose λ = 2 in our application of Lemma 3. The absence of a quadratic NOTE ON THE GAPS BETWEEN ZEROS OF EPSTEIN’S ZETA-FUNCTIONS ON THE CRITICAL LINE11 term is advantageous for us because the cubic term can be handled easily, whereas thepresence of a quadratic term would lead to difficulties.Using φ ′ ( u ) = 1 √ u + O ( √ u ) , φ ′′ ( u ) = O (cid:18) u / (cid:19) and φ ′′′ ( u ) = O (cid:18) u / (cid:19) for | u | ≤ 1, we calculate that g ′ x ( y ) = π ( b ∗ x + 2 c ∗ y )2 hk ∆ t · q πQ ∗ ( x,y )2 hk ∆ t + O s πQ ∗ ( x, y )2 hk ∆ t = √ π ( b ∗ x + 2 c ∗ y ) p hk ∆ tQ ∗ ( x, y ) + O (cid:18) N K / T (cid:19) and g ′′′ x ( y ) = O (cid:18) K / N T (cid:19) . It follows that X y ∈ I ( x ) e ( f x ( y + m ) − f x ( y − m )) = X y ∈ I ( x ) e (2 mF x ( y )) + O (cid:18) mN / K / T + m K / N / (cid:19) , (4.5)where F x ( y ) := ( b ∗ x + 2 c ∗ y ) · r + √ t p πhk ∆ Q ∗ ( x, y ) ! . Application of the B process. Now we want to employ the B process, Lemma 4,to transform the exponential sum on the right-hand side of (4.5). To this end, we calculatethat F ′ x ( y ) =2 c ∗ r + √ t | d | x √ πhk ∆ Q ∗ ( x, y ) / ,F ′′ x ( y ) = − √ t | d | x ( b ∗ x + 2 c ∗ y )4 √ πhk ∆ Q ∗ ( x, y ) / ≍ K / N ,F ′′′ x ( y ) = O (cid:18) K / N / (cid:19) ,F ′′′′ x ( y ) = O (cid:18) K / N (cid:19) . (4.6)We also need to find the precise range in which F ′ x ( y ) lies, which we do in the following.By our assumptions on x and y , we have N ≤ Q ∗ ( x, y ) ≤ N ′ . For fixed x , we also have Q ∗ ( x, y ) ≥ | d | c ∗ · x . Hence a ( x ) ≤ F ′ x ( y ) ≤ b ( x ), where a ( x ) := 2 c ∗ r + √ t | d | x √ πhk ∆ N ′ / and b ( x ) :=2 c ∗ r + √ t | d | x √ πhk ∆ max { N, | d | x / (4 c ∗ ) } / =2 c ∗ r + min ( √ t | d | x √ πhk ∆ N / , (4 c ∗ ) / √ t p π | d | hk ∆ x ) . Now Lemma 4 yields X y ∈ I ( x ) e (2 mF x ( y ))= X ma ( x ) ≤ n ≤ mb ( x ) e (2 mF x ( y x,m,n ) − ny x,m,n − / p m | F ′′ x ( y x,m,n ) | + O (cid:18) log T + N / m / K / (cid:19) , (4.7)where y x,m,n ∈ I ( x ) is the solution of 2 mF ′ x ( y x,m,n ) = n .We compute that F x ( y x,m,n ) = s −| d | x + 4 c ∗ (cid:18) m √ t | d | x √ πhk ∆ ( n − mc ∗ r ) (cid:19) / · r + (cid:18) ( n − mc ∗ r ) t πhk ∆ | d | mx (cid:19) / ! and ny x,m,n = n c ∗ · − b ∗ x + s −| d | x + 4 c ∗ (cid:18) m √ t | d | x √ πhk ∆ ( n − mc ∗ r ) (cid:19) / . Putting together gives G m,n ( x ) := 2 mF x ( y x,m,n ) − ny x,m,n = b ∗ n c ∗ · x − c ∗ · s −| d | x + 4 c ∗ (cid:18) m √ t | d | x √ πhk ∆ ( n − mc ∗ r ) (cid:19) / × ( n − c ∗ mr ) − c ∗ m (cid:18) ( n − mc ∗ r ) t πhk ∆ | d | mx (cid:19) / ! = b ∗ n c ∗ · x + ( n − c ∗ mr )2 c ∗ | d | x · −| d | x + 4 c ∗ (cid:18) m √ t | d | x √ πhk ∆ ( n − mc ∗ r ) (cid:19) / ! / = b ∗ n c ∗ · x + 12 c ∗ · (cid:18) − ( n − c ∗ mr ) / | d | / x / + 4 c ∗ · m / t / (2 πhk ∆ ) / (cid:19) / . (4.8) NOTE ON THE GAPS BETWEEN ZEROS OF EPSTEIN’S ZETA-FUNCTIONS ON THE CRITICAL LINE13 Combining (4.3), (4.4), (4.5), (4.6), (4.7) and (4.8), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ J X y ∈ I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ NM · X ≤ m ≤ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ J X ma ( x ) ≤ n ≤ mb ( x ) e ( G m,n ( x )) p | F ′′ x ( y x,m,n ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O (cid:18) M N / + N M + M N / K / T + M N / K / + N M / K / (cid:19) . (4.9)4.4. Application of van der Corput’s bound. Let’s first work out what the trivialestimate for the double exponential sum above gives. We will see that we recover preciselythe result in [4] in this way. Clearly, | J | ≪ N / and b ( x ) − a ( x ) = O (cid:0) K / N − / (cid:1) . Together with (4.6), this implies NM · X ≤ m ≤ M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ J X ma ( x ) ≤ n ≤ mb ( x ) e ( G m,n ( x )) p m | F ′′ x ( y x,m,n ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) M / N / K / (cid:1) . (4.10)Choosing M := (cid:2) N / K − / (cid:3) to balance the O -term in (4.10) and the second O -term N /M in (4.9), using N ≪ K , and taking the square root, we deduce that K / N − / T − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ J X y ∈ I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) K / T / + K / T / (cid:19) . (4.11)So if K := T / − ε , the above is O ( T − ε ), as in [4].Now we estimate the said double exponential sum non-trivially, thus obtaining an im-provement over the result in [4]. First, we re-arrange summations, getting X x ∈ J X ma ( x ) ≤ n ≤ mb ( x ) e ( G m,n ( x )) p m | F ′′ x ( y x,m,n ) | = X n ∈ J m X A m ( n ) ≤ x ≤ B m ( n ) e ( G m,n ( x )) p m | F ′′ x ( y x,m,n ) | , (4.12)where J m = (cid:20) c ∗ mr, c ∗ mr + 2 c ∗ m √ tN ′ √ πhk ∆ N / (cid:21) ,A m ( n ) := ( n − mc ∗ r ) / · (2 πhk ∆ ) / N / ( m | d | ) / t / and B m ( n ) := min ( c ∗ N ′ ) / | d | / , ( n − mc ∗ r ) / · (2 πhk ∆ ) / N ′ / ( m | d | ) / t / , (4 c ∗ ) / mt / (2 π | d | hk ∆ ) / ( n − mc ∗ r ) ) . Our idea is to use the van der Corput bound, Lemma 5, to estimate the inner sum over x on the right-hand side of (4.12). To this end, we note that | J m | ≪ mK / N / and B m ( n ) − A m ( n ) ≪ N / (4.13)and compute that G ′′ m,n ( x ) = − ( n − c ∗ mr ) / | d | / c ∗ x / · (cid:18) ( n − c ∗ mr ) / | d | / x / + 4 c ∗ · m / t / (2 πhk ∆ ) / (cid:19) × (cid:18) − ( n − c ∗ mr ) / | d | / x / + 4 c ∗ · m / t / (2 πhk ∆ ) / (cid:19) − / . (4.14)Further, we observe that( n − c ∗ mr ) / | d | / c ∗ x / ≍ m / t / ( hk ∆ ) / N ≍ m / K / N and ( n − c ∗ mr ) / | d | / x / + 4 c ∗ · m / t / (2 πhk ∆ ) / ≍ m / t / ( hk ∆ ) / ≍ m / K / and hence G ′′ m,n ( x ) ≍ m / K / N · (cid:18) − ( n − c ∗ mr ) / | d | / x / + 4 c ∗ · m / t / (2 πhk ∆ ) / (cid:19) − / (4.15)for n and x in the relevant summation intervals.We first assume that n ∈ J ′ m ⊆ J m , where J ′ m := (cid:20) c ∗ mr, c ∗ mr + c ∗ m √ t √ πhk ∆ N ′ (cid:19) , in which case we compute that − ( n − c ∗ mr ) / | d | / x / + 4 c ∗ · m / t / (2 πhk ∆ ) / ≍ m / t / ( hk ∆ ) / ≍ m / K / and hence, using (4.15), G ′′ m,n ( x ) ≍ mK / N if A m ( n ) ≤ x ≤ B m ( n ) . (4.16) NOTE ON THE GAPS BETWEEN ZEROS OF EPSTEIN’S ZETA-FUNCTIONS ON THE CRITICAL LINE15 Now we apply partial summation to remove the factor 1 / p F ′′ x ( y x,m,n ) and Lemma 5 to get X n ∈ J ′ m X A m ( n ) ≤ x ≤ B m ( n ) e ( G m,n ( x )) p m | F ′′ x ( y x,m,n ) |≪ N / m / K / · mK / · (cid:18) mK / N (cid:19) / + mK / N / · (cid:18) mK / N (cid:19) − / ! ≪ mK / , (4.17)where we have used (4.13) and (4.16).Next, we assume that n ∈ J ′′ m ⊆ J m , where J ′′ m := J m \ J ′ ( m ) = (cid:20) c ∗ mr + c ∗ m √ t √ πhk ∆ N ′ , c ∗ mr + 2 c ∗ m √ tN ′ √ πhk ∆ N / (cid:21) (4.18)and note that in this case x ≍ N / if A m ( n ) ≤ x ≤ B m ( n ) . (4.19)If n ∈ J ′′ m ⊆ J m , an estimate of the form | G ′′ m,n ( x ) | ≍ λ doesn’t hold on the entire x -interval.To make Lemma 5 applicable, we split this interval into subintervals such that an estimateof this form holds on each of them, except for one subinterval which is so short that it canbe treated trivially. To this end, we write I m ( n ) := (4 c ∗ ) / mt / (2 π | d | hk ∆ ) / ( n − mc ∗ r ) − , (4 c ∗ ) / mt / (2 π | d | hk ∆ ) / ( n − mc ∗ r ) and I m ( n, δ ) := (4 c ∗ ) / mt / (2 π | d | hk ∆ ) / ( n − mc ∗ r ) − δ, (4 c ∗ ) / mt / (2 π | d | hk ∆ ) / ( n − mc ∗ r ) − δ and set S m ( n ) := I m ( n ) ∩ [ A m ( n ) , B m ( n )] and S m ( n, δ ) := I m ( n, δ ) ∩ [ A m ( n ) , B m ( n )] . We observe that the interval [ A m ( n ) , B m ( n )] can be split into O (log T ) intervals of theform S m ( n ) or S m ( n, δ ), where 1 ≪ δ ≪ √ N . (4.20)Estimating trivially gives X n ∈ J ′′ m X x ∈ S m ( n ) e ( G m,n ( x )) p m | F ′′ x ( y x,m,n ) | ≪ m / K / , (4.21)where we have used (4.6) and (4.13). If x ∈ S m ( n, δ ), then we compute using (4.18), (4.19)and the mean value theorem that − ( n − c ∗ mr ) / | d | / x / + 4 c ∗ · m / t / (2 πhk ∆ ) / ≍ δ · (cid:18) m √ t √ hk ∆ N / (cid:19) / x − / ≍ δm / K / N / . Hence, using (4.15), it follows that G ′′ m,n ( x ) ≍ mK / δ / N / . (4.22)Again using partial summation to remove the factor 1 / p F ′′ x ( y x,m,n ) and Lemma 5, wededuce that X n ∈ J ′′ m X x ∈ S m ( n,δ ) e ( G m,n ( x )) p m | F ′′ x ( y x,m,n ) |≪ N / m / K / · δmK / N / · (cid:18) mK / δ / N / (cid:19) / + mK / N / · (cid:18) mK / δ / N / (cid:19) − / ! ≪ mK / , (4.23)where we have employed | S m ( n, δ ) | ≤ δ , (4.13), (4.20) and (4.22). From (4.21) and (4.23),it follows that X n ∈ J ′′ m X A m ( n ) ≤ x ≤ B m ( n ) e ( G m,n ( x )) p m | F ′′ x ( y x,m,n ) | ≪ mK / log T. (4.24)Now using J m = J ′ m ∪ J ′′ m , (4.17) and (4.24), we get X n ∈ J m X A m ( n ) ≤ x ≤ B m ( n ) e ( G m,n ( x )) p m | F ′′ x ( y x,m,n ) | ≪ mK / log T. (4.25)Combining (4.9), (4.12) and (4.25), we deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ J X y ∈ I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) M N / + N M + M N / K / T + M N / K / + N M / K / + M N K / log T (cid:19) . (4.26)Choosing M := (cid:2) N / K − / (cid:3) to balance the second and last O -terms above, using N ≪ K ,and taking the square root, it follows that K / N − / T − / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x ∈ J X y ∈ I ( x ) e (cid:18) Q ∗ ( x, y ) · r + tπ · φ (cid:18) πQ ∗ ( x, y )2 hk ∆ t (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) K / T / log T + K / T / (cid:19) . This is O ( T − ε ), provided that K := T / − ε , which completes the proof. Acknowledgement. We express our gratitude to Professor Matti Jutila for pointing outsome typos and suggesting some changes in an earlier version this paper. We thank Prof.A. Ivi´c for encouragement and for some useful suggestions. The third author would liketo thank the Institute of Mathematical Sciences, Chennai for supporting her visits which NOTE ON THE GAPS BETWEEN ZEROS OF EPSTEIN’S ZETA-FUNCTIONS ON THE CRITICAL LINE17 enabled her to work on this project and would like to mention that this paper constitutespart of her Ph D thesis work. References [1] R. Balasubramanian, An improvement of a theorem of Titchmarsh on the mean square of | ζ (1 / it ) | , Proc. London Math. Soc. , (1978) 540–576.[2] S. W. Graham and G. Kolesnik, Van der Corput’s Method of Exponential sums, Cambridge Uni-versity Press , New York, 1991.[3] M. Jutila, Transformation Formulae for Dirichlet Polynomials, J. Number Theory , (1984), no.2, 135-156.[4] M. Jutila and K. 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Heath-Brown, The Clarendon Press, Oxford University Press, New York, 1986. Jawaharlal Nehru University Delhi, School of Physical Sciences, Delhi 110067, IndiaInstitute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai 600 113,IndiaSRTM University, Nanded, Maharashtra 431606, India E-mail address , Stephan Baier: email [email protected] E-mail address , Kotyada Srinivas: [email protected] E-mail address , Usha Sangale:, Usha Sangale: