Zeros of combinations of the Riemann Ξ -function and the confluent hypergeometric function on bounded vertical shifts
Atul Dixit, Rahul Kumar, Bibekananda Maji, Alexandru Zaharescu
ZZEROS OF COMBINATIONS OF THE RIEMANN Ξ -FUNCTIONAND THE CONFLUENT HYPERGEOMETRIC FUNCTION ONBOUNDED VERTICAL SHIFTS ATUL DIXIT, RAHUL KUMAR, BIBEKANANDA MAJI AND ALEXANDRU ZAHARESCU
Abstract.
In 1914, Hardy proved that infinitely many non-trivial zeros of theRiemann zeta function lie on the critical line using the transformation formula ofthe Jacobi theta function. Recently the first author obtained an integral represen-tation involving the Riemann Ξ-function and the confluent hypergeometric functionlinked to the general theta transformation. Using this result, we show that a seriesconsisting of bounded vertical shifts of a product of the Riemann Ξ-function andthe real part of a confluent hypergeometric function has infinitely many zeros onthe critical line, thereby generalizing a previous result due to the first and the lastauthors along with Roy and Robles. The latter itself is a generalization of Hardy’stheorem. Introduction
Ever since the appearance of Riemann’s seminal paper [22] in 1859, the zeros of theRiemann zeta function ζ ( s ) have been a constant source of inspiration and motivationfor mathematicians to produce beautiful mathematics. While the Riemann Hypoth-esis, which states that all non-trivial zeros of ζ ( s ) lie on the critical line Re( s ) = 1 / ζ ( s ) lie on the critical line Re( s ) = 1 /
2. Let N ( T ) denote the number of non-trivial zeros lying on the critical line such thattheir positive imaginary part is less than or equal to T . Hardy and Littlewood [14]showed that N ( T ) > AT ,where A is some positive constant. Selberg [24] remarkablyimproved this to N ( T ) > AT log T . Levinson [16] further improved this by provingthat more than one-third of the non-trivial zeros of ζ ( s ) lie on the critical line. Conrey[4] raised this proportion to more than two-fifths. This was later improved by Bui,Conrey and Young [3], Feng [11], Robles, Roy and one of the authors [21], with the Mathematics Subject Classification: Primary 11M06 Secondary 11M26Keywords: Riemann zeta function, theta transformation, confluent hypergeometric function,bounded vertical shifts, zeros. a r X i v : . [ m a t h . N T ] D ec EROS OF COMBINATIONS OF Ξ( t ) AND F current record, due to Pratt and Robles [18], being that 41 .
49% of the zeros lie onthe critical line.The proof of Hardy’s result in [13], which acted as a stimulus to the aforementioneddevelopments, is well-known for its beauty and elegance. One of the crucial ingredientsin his proof is the transformation formula satisfied by the Jacobi theta function. Thelatter is defined by ϑ ( λ ; τ ) := ∞ (cid:88) n = −∞ q n λ n , where q = exp( πiτ ) for τ ∈ H (upper half-plane) and λ = e πiu for u ∈ C . If we let τ = ix for x > u = 0, then the theta function becomes ϑ (1; ix ) = ∞ (cid:88) n = −∞ e − n πx =: 2 ψ ( x ) + 1 , so that ψ ( x ) = (cid:80) ∞ n =1 e − n πx . The aforementioned transformation formula employedby Hardy in his proof is due to Jacobi and is given by [26, p. 22, Equation 2.6.3] √ x (2 ψ ( x ) + 1) = 2 ψ (cid:18) x (cid:19) + 1 , (1.1)which can be alternatively written as √ a (cid:18) a − ∞ (cid:88) n =1 e − πa n (cid:19) = √ b (cid:18) b − ∞ (cid:88) n =1 e − πb n (cid:19) (1.2)for Re( a ) >
0, Re( b ) > ab = 1. The former is the same version of thetheta transformation using which Riemann [22] derived the functional equation ofthe Riemann zeta function ζ ( s ) in the form [26, p. 22, eqn. (2.6.4)] π − s Γ (cid:16) s (cid:17) ζ ( s ) = π − (1 − s )2 Γ (cid:18) − s (cid:19) ζ (1 − s ) . In fact, the functional equation of ζ ( s ) is known to be equivalent to the theta trans-formation. Another crucial step in Hardy’s proof is the identity2 π (cid:90) ∞ Ξ( t ) t + cosh( αt )d t = e − iα − e iα ψ (cid:0) e iα (cid:1) , (1.3)where Ξ( t ) is the Riemann Ξ-function defined byΞ( t ) = ξ (cid:0) + it (cid:1) with ξ ( s ) being the Riemann ξ -function ξ ( s ) := 12 s ( s − π − s Γ (cid:16) s (cid:17) ζ ( s ) . EROS OF COMBINATIONS OF Ξ( t ) AND F Equation (1.3) is easily seen to be equivalent to2 π (cid:90) ∞ Ξ( t/ t cos (cid:18) t log a (cid:19) d t = √ a (cid:18) a − ∞ (cid:88) n =1 e − πa n (cid:19) (1.4)by replacing t by 2 t and letting a = e iα , − π < α < π , in the latter.At this juncture, it is important to state that the above theta transformation, thatis (1.2), has a generalization [1, Equation (1.2)], also due to Jacobi, and is as follows.Let z ∈ C . If a and b are positive numbers such that ab = 1, then √ a (cid:32) e − z / a − e z / ∞ (cid:88) n =1 e − πa n cos( √ πanz ) (cid:33) = √ b (cid:32) e z / b − e − z / ∞ (cid:88) n =1 e − πb n cosh( √ πbnz ) (cid:33) . (1.5)It is then natural to seek for an integral representation equal to either sides of theabove transformation similar to how the expressions on either sides of (1.2) are equalto the integral on the left-hand side of (1.4). Such a representation was recentlyobtained by the first author [5, Thm. 1.2]. This result is stated below. Theorem 1.
Let ∇ ( x, z, s ) := µ ( x, z, s ) + µ ( x, z, − s ) ,µ ( x, z, s ) := x / − s e − z / F (cid:18) − s z (cid:19) , where F ( c ; d ; z ) := (cid:80) ∞ n =0 ( c ) n z n ( d ) n n ! is the confluent hypergeometric function, ( c ) n := (cid:81) n − i =0 ( c + i ) = Γ( c + n )Γ( c ) , for any c, d, z, s ∈ C . Then either sides of the transformationin (1.5) equals π (cid:90) ∞ Ξ( t )1 + t ∇ (cid:18) a, z, it (cid:19) d t. (1.6)The next question that comes naturally to our mind is, with the additional param-eter z in the above theorem at our disposal, can we generalize the proof of Hardy’sresult to obtain information on the zeros of a function which generalizes Ξ( t )? Eventhough one could possibly get a generalization of Hardy’s result this way, unfortu-nately it would not be striking since in the integrand of the generalization of the leftside of (1.3), one would have Ξ( t ) (cid:16) e αt F (cid:16) − it ; ; z (cid:17) + e − αt F (cid:16) it ; ; z (cid:17)(cid:17) , andthen saying that for z fixed in some domain, this function has infinitely many realzeros does not appeal much, for, it is already known that Hardy’s theorem impliesthat Ξ( t ) has infinitely many real zeros.However, even though this approach fails, one can still use the generalized thetatransformation in (1.5) and the integral (1.6) linked to it to obtain information about EROS OF COMBINATIONS OF Ξ( t ) AND F zeros of a function which generalizes Ξ( t ). This function is obtained through verticalshifts s → s + iτ of ζ ( s ) and of the confluent hypergeometric function. Many mathe-maticians have studied the behavior of ζ ( s ) on vertical shifts. See [12], [17], [19], [20],[25] and [27] for some papers in this direction.Let η ( s ) := π − s/ Γ (cid:18) s (cid:19) ζ ( s ) and ρ ( t ) := η (cid:18)
12 + it (cid:19) . These are related to Riemman’s ξ ( s ) and Ξ( t ) functions by ξ ( s ) = s ( s − η ( s ) andΞ( t ) = (cid:0) + it (cid:1) (cid:0) − + it (cid:1) ρ ( t ). It is known that η ( s ) is a meromorphic function of s with poles at 0 and 1. For real t , ρ ( t ) is a real-valued even function of t .Recently the first and the last authors along with Robles and Roy [9, Theorem 1.1]obtained the following result : Let { c j } be a sequence of non-zero real numbers so that (cid:80) ∞ j =1 | c j | < ∞ . Let { λ j } bea bounded sequence of distinct real numbers that attains its bounds. Then the function F ( s ) = (cid:80) ∞ j =1 c j η ( s + iλ j ) has infinitely many zeros on the critical line Re ( s ) = . Note that the restriction c j (cid:54) = 0 is not strict but only for removing redundancy sinceif c j = 0 for some j ∈ N , then the corresponding term does not contribute anythingtowards the function F ( s ). But if we let all but one elements in the sequence { c j } tobe equal to zero and the non-zero element, say c j (cid:48) to be 1 along with the corresponding λ j (cid:48) = 0, we recover Hardy’s theorem.The goal of the present paper is to generalize the above theorem, that is, [9, The-orem 1.1], which, in turn, as we have seen, generalizes Hardy’s theorem. Our mainresult is as follows. Theorem 2.
Let { c j } be a sequence of non-zero real numbers so that (cid:80) ∞ j =1 | c j | < ∞ .Let { λ j } be a bounded sequence of distinct real numbers such that it attains its bounds.Let D denote the region | Re( z ) − Im( z ) | < (cid:112) π − (cid:113) π Re( z )Im( z ) in the z -complexplane. Then for any z ∈ D , the function F z ( s ) := ∞ (cid:88) j =1 c j η ( s + iλ j ) (cid:26) F (cid:18) − ( s + iλ j )2 ; 12 ; z (cid:19) + F (cid:18) − (¯ s − iλ j )2 ; 12 ; ¯ z (cid:19)(cid:27) has infinitely many zeros on the critical line Re( s ) = 1 / . There are some typos in the proof of this result in [9]. Throughout the paper, lim α → π should bechanged to lim α → π − . The summation sign (cid:80) ∞ j =1 is missing on the left-hand sides of (3.6) and (3.9).The expression F (cid:0) + i ( t + λ j ) (cid:1) on the left-hand side of (3.11) should be replaced by F (cid:0) + it (cid:1) .Also, mθ M in (3.12), and p n θ M and q n θ M in (3.16) and (3.17) should be replaced by 2 mθ M , 2 p n θ M and 2 q n θ M . Finally, the integration on the right side of (3.22) should be performed over | t | > T rather than on ( T, ∞ ). EROS OF COMBINATIONS OF Ξ( t ) AND F The region D is sketched in Figure 1. The vertices of the square in the center aregiven by A = (cid:112) π − i (cid:112) π , B = (cid:112) π + i (cid:112) π , C = − (cid:112) π + i (cid:112) π and D = − (cid:112) π − i (cid:112) π .It is easy to see that when z = 0, this theorem reduces to the result of the first andthe last author along with Robles and Roy [9, Theorem 1.1] given above.This paper is organized as follows. In Section 2, we collect necessary tools andderive lemmas that are necessary in the proof of Theorem 2. Section 3 is then de-voted to proving Theorem 2. In Section 4, we give concluding remarks and possibledirections for future work. Figure 1.
Region D in the z -complex plane given by | Re( z ) − Im( z ) | < (cid:112) π − (cid:113) π Re( z )Im( z ).2. Preliminaries
We begin with a lemma which gives a bound on Ξ( t ). It readily follows by usingelementary bounds on the Riemann zeta function and Stirling’s formula on a verticalstrip which states that if s = σ + it , then for a ≤ σ ≤ b and | t | ≥ | Γ( s ) | = (2 π ) | t | σ − e − π | t | (cid:18) O (cid:18) | t | (cid:19)(cid:19) as t → ∞ . Lemma 1.
For t → ∞ , we have Ξ( t ) = O (cid:0) t A e − π t (cid:1) , where A is some positiveconstant. We will also need the following estimate for the confluent hypergeometric functionproved in [5, p. 398, Eqn. 4.19].
EROS OF COMBINATIONS OF Ξ( t ) AND F Lemma 2.
For z ∈ C and | s | → ∞ , F (cid:18) − s ; 12 ; z (cid:19) = e z cos (cid:16) z (cid:112) s + 1 / (cid:17) + O z (cid:16) | s + 1 / | − / (cid:17) . Lemma 3.
Let D be a collection of all those complex numbers z such that | Re( z ) − Im( z ) | < (cid:112) π − (cid:113) π Re( z )Im( z ) . Then D = (cid:110) z : | Re( z ) | < (cid:112) π , | Im( z ) | < (cid:112) π (cid:111) (cid:83) (cid:110) z :Re( z ) > (cid:112) π , Im( z ) < − (cid:112) π (cid:111) (cid:83) (cid:110) z : Re( z ) < − (cid:112) π , Im( z ) > (cid:112) π (cid:111) . Proof.
Let c = (cid:112) π , Re( z ) = x and Im( z ) = y . One can easily verify that z / ∈ D if x = ± c or y = ± c . According to the given hypothesis, all points z = x + iy of D satisfy | x − y | < c − xyc . We divide the domain into three parts. First, if | x | < c , i.e. x + c > x − c < x − y < c − xyc ⇔ x − c < (cid:16) − yc (cid:17) ( x − c ) ⇔ > (cid:16) − yc (cid:17) ⇔ y > − c. (2.1)Again, x − y > − c + xyc ⇔ x + c > yc ( x + c ) ⇔ c > y. (2.2)Thus combining (2.1) and (2.2), we have | y | < c . Therefore, the domain (cid:110) ( x, y ) : | x | < c, | y | < c (cid:111) is a sub-domain of D .Now if x > c , so that x ± c >
0. We have x − y < c − xyc ⇔ x − c < (cid:16) − yc (cid:17) ( x − c ) ⇔ y < − c, (2.3)and x − y > − c + xyc ⇔ x + c > yc ( x + c ) ⇔ y < c. (2.4)Therefore (2.3) and (2.4) imply that y < − c when x > c . Similarly it can be seenthat if x < − c , then y > c . (cid:3) The following lemma, due to Kronecker, will be used in the proof of the maintheorem. See [15, p. 376, Chapter XXIII] for proofs.
Lemma 4.
Let { nθ } denote the fractional part of nθ . If θ is irrational, then the setof points { nθ } is dense in the interval (0 , . EROS OF COMBINATIONS OF Ξ( t ) AND F Let ψ ( x, z ) := ∞ (cid:88) n =1 e − πn x cos( √ πxnz ) . (2.5)Replacing a = √ x in (1.5), we have √ x (cid:32) e − z / √ x − e z / ψ ( x, z ) (cid:33) = 1 √ x (cid:32) e z / √ x − e − z / ψ (cid:18) x , iz (cid:19)(cid:33) so that ψ ( x, z ) = e − z / √ x ψ (cid:16) x , iz (cid:17) + e − z / √ x − . (2.6)In particular, if we let z = 0 then we recover (1.1).We now prove a lemma which is instrumental in the proof of Theorem 2. Lemma 5.
Let z ∈ D . Then the expressions √ δ e − z (cid:0) iδ (cid:1) ψ (cid:16) δ , iz (cid:113) iδ (cid:17) and √ δ e − z (cid:0) iδ (cid:1) ψ (cid:16) δ , iz (cid:113) iδ (cid:17) and their derivatives tend to zero as δ → along anyroute in | arg( δ ) | < π .Proof. We prove that lim δ → √ δ e − z (cid:0) iδ (cid:1) ψ (cid:32) δ , iz (cid:114) iδ (cid:33) = 0 . (2.7)The second part can be analogously proved. Now ψ (cid:32) δ , iz (cid:114) iδ (cid:33) = ∞ (cid:88) n =1 e − πn δ cos (cid:32) n √ π √ δ iz (cid:114) iδ (cid:33) = ∞ (cid:88) n =1 e − πn δ cosh (cid:18) n √ πz √ i + δ δ (cid:19) = ∞ (cid:88) n =1 e − πn δ (cid:32) e n √ πz √ i + δ δ + e − n √ πz √ i + δ δ (cid:33) . (2.8)Since the function ψ ( x, z ) is analytic as a function of x in the right half plane Re( x ) > z for any z ∈ C , it suffices to show that each term of (2.8) goesto zero as δ goes to 0 + .Therefore,lim δ → + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ δ e − z (cid:0) iδ (cid:1) e − πn δ (cid:32) e n √ πz √ i + δ δ + e − n √ πz √ i + δ δ (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) EROS OF COMBINATIONS OF Ξ( t ) AND F = lim δ → + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − z √ δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e Re (cid:16) − z i δ (cid:17) − πn δ (cid:16) e n √ π δ Re( z √ i + δ ) + e − n √ π δ Re( z √ i + δ ) (cid:17) = lim δ → + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − z √ δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) e − δ ( πn +Re( z i ) − n √ π Re( z √ i + δ ) ) + e − δ ( πn +Re( z i )+2 n √ π Re( z √ i + δ ) ) (cid:17) . Note that Re ( z i ) = − z )Im( z ) and Re( z √ i + δ ) = √ (Re( z ) − Im( z )), since δ → + . For n >
1, the above limit goes to zero because of the presence of n in theexponentials. So we will be done if we can show that it goes to zero for n = 1 too.To ensure this happens, the condition to be imposed on z is π − z )Im( z ) ± √ π (Re( z ) − Im( z )) > ⇒ (cid:12)(cid:12) Re( z ) − Im( z ) (cid:12)(cid:12) < (cid:114) π − (cid:114) π Re( z )Im( z ) , The points which satisfy the above inequality are nothing but those which lie in theregion D . This proves (2.7). Since exp (cid:0) − δ (cid:1) goes to zero faster than δ r for any r > δ → + , derivatives of all orders of √ δ e − z (cid:0) iδ (cid:1) ψ (cid:16) δ , iz (cid:113) iδ (cid:17) also tend to zeroas δ → ∞ . This completes the proof of the lemma. (cid:3) Now let ψ ( α, z ) := e ( i − λ j ) α (cid:16) e − z / e z / ψ (cid:0) e iα , z (cid:1)(cid:17) . (2.9)We prove the following result. Lemma 6.
Let z ∈ D and − π < α < π . Then the m th derivative of the function ψ ( α, z ) with respect to α , tends to − (cid:0) i − λ j (cid:1) m e π ( i − λ j ) sinh (cid:16) z (cid:17) as α → π − .Proof. Observe that for X := e iα , lim α → π − X = i . Note that X tends to i in acircular path where both x and y coordinates are positive. We now let i + δ tend to i as δ goes to zero along any route in | arg( δ ) | < π . We first show that ψ ( i + δ, z ) goesto − / δ go to zero as δ → | arg( δ ) | < π . One can easily check thatlim α → π − (cid:18) dd α (cid:19) m ψ ( e iα , z ) = (2 i ) m lim X → i m (cid:88) j =1 a j X j (cid:18) dd X (cid:19) j ψ ( X, z )= (2 i ) m lim δ → m (cid:88) j =1 a j ( i + δ ) j (cid:18) dd δ (cid:19) j ψ ( i + δ, z ) , (2.10)where a j ’s are positive integers depending on j . EROS OF COMBINATIONS OF Ξ( t ) AND F Using the definition (2.5) of ψ ( x, z ), we have ψ ( i + δ, z ) = ∞ (cid:88) n =1 e − n π ( i + δ ) cos (cid:0)(cid:112) π ( i + δ ) nz (cid:1) = ∞ (cid:88) n =1 ( − n e − n πδ cos (cid:0)(cid:112) π ( i + δ ) nz (cid:1) = (cid:88) n even e − n πδ cos (cid:0)(cid:112) π ( i + δ ) nz (cid:1) − (cid:88) n odd e − n πδ cos (cid:0)(cid:112) π ( i + δ ) nz (cid:1) =2 ψ (cid:18) δ, √ i + δ √ δ z (cid:19) − ψ (cid:18) δ, √ i + δ √ δ z (cid:19) . Invoking the transformation formula (2.6) of ψ ( x, z ), we get ψ ( i + δ, z ) = 1 √ δ e − z (cid:0) iδ (cid:1) ψ (cid:32) δ , iz (cid:114) iδ (cid:33) − √ δ e − z (cid:0) iδ (cid:1) ψ (cid:32) δ , iz (cid:114) iδ (cid:33) − . Along with Lemma 5, this implies that e − z / + e z / ψ ( i + δ, z ) tends to − sinh (cid:16) z (cid:17) and its derivatives go to zero as δ → | arg( δ ) | < π , thatis, e − z / + e z / ψ ( e iα , z ) goes to − sinh (cid:16) z (cid:17) , and, due to (2.10), its derivatives go tozero as α → π − . With the help of this result,lim α → π − (cid:18) dd α (cid:19) m ψ ( α, z )= lim α → π − (cid:88) ≤ j,k ≤ mj + k =2 m (cid:18) mj (cid:19) (cid:18) dd α (cid:19) j e ( i − λ j ) α · (cid:18) dd α (cid:19) k (cid:32) e − z / e z / ψ ( e iα , z ) (cid:33) = lim α → π − (cid:88) ≤ j< m, Theorem 1 implies e − z / π (cid:90) ∞ Ξ( t )1 + t (cid:18) a − it/ F (cid:16) − it z (cid:17) + a it/ F (cid:16) it z (cid:17)(cid:19) d t = √ a (cid:32) e − z / a − e z / ∞ (cid:88) n =1 e − πa n cos( √ πanz ) (cid:33) . Replace t by 2 t and a by e iα , − π < α < π , and then add and subtract the term e − z / e iα/ on the right hand side of the resulting equation to arrive at e − z / π (cid:90) ∞ Ξ( t ) + t (cid:18) e αt F (cid:16) − it z (cid:17) + e − αt F (cid:16) it z (cid:17)(cid:19) d t = 2 e − z / cos α/ − e iα/ (cid:32) e − z / e z / ψ (cid:0) e iα , z (cid:1)(cid:33) . The integrand on the left side is an even function of t . Hence e − z / π (cid:90) ∞−∞ ρ ( t ) e αt F (cid:16) − it z (cid:17) d t = − e − z / cos α/ e iα/ (cid:32) e − z / e z / ψ (cid:0) e iα , z (cid:1)(cid:33) . Replacing t by t + λ j , we find (cid:90) ∞−∞ e αt ρ ( t + λ j ) F (cid:16) − i ( t + λ j )4 ; 12 ; z (cid:17) d t = πe − αλ j (cid:32) − α/ e z / e iα/ (cid:32) e − z / e z / ψ (cid:0) e iα , z (cid:1)(cid:33)(cid:33) = − π (cid:20) e iα − αλ j + e − iα − αλ j − e z / e iα − αλ j (cid:18) e − z / + e z / ∞ (cid:88) n =0 e − n πe αi cos (cid:0) √ πne iα z (cid:1) (cid:19)(cid:21) . (3.1) EROS OF COMBINATIONS OF Ξ( t ) AND F Differentiating both sides 2 m times with respect to α , one gets (cid:90) ∞−∞ t m e αt ρ ( t + λ j ) F (cid:16) − i ( t + λ j )4 ; 12 ; z (cid:17) d t = − π (cid:20)(cid:18) i − λ j (cid:19) m e αi − αλ j + (cid:18) i λ j (cid:19) m e − αi − αλ j − e z / ∂ m ∂α m (cid:18) e αi − αλ j (cid:18) e − z / e z / ∞ (cid:88) n =0 e − n πe αi cos (cid:0) √ πne iα z (cid:1) (cid:19)(cid:19)(cid:21) . Let i − λ j = r j e iθ j . Without loss of generality, let 0 < θ j < π . Then (cid:90) ∞−∞ t m e αt ρ ( t + λ j ) F (cid:18) − i ( t + λ j )4 ; 12 ; z (cid:19) d t = − πe − αλ j (cid:18) r mj e i ( α mθ j ) + r mj e i (cid:16) − α πm − mθ j (cid:17) (cid:19) + 4 πe z / ∂ m ∂α m (cid:32) e αi − αλ j (cid:32) e − z / e z / ∞ (cid:88) n =0 e − n πe αi cos (cid:0) √ πne iα z (cid:1)(cid:33)(cid:33) = − πe − αλ j r mj cos (cid:18) α mθ j (cid:19) + 4 πe z / ∂ m ∂α m ψ ( α, z ) , (3.2)where ψ ( α, z ) is defined in (2.9). Using Lemmas 1 and 2, we see that | ρ ( t ) | (cid:28) | t | A e − π | t | , and (cid:12)(cid:12)(cid:12)(cid:12) Re (cid:18) F (cid:18) − it z (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) z e | z | √ | t | / as | t | → ∞ , where A is some positive constant. Since { λ j } is bounded, one sees that ∞ (cid:88) j =1 c j ρ ( t + iλ j )Re (cid:18) F (cid:18) − i ( t + λ j )4 ; 12 ; z (cid:19)(cid:19) (cid:28) z | t | A (cid:48) e − π | t | + | z | (cid:113) | t | ∞ (cid:88) j =1 | c j | as | t | → ∞ . Along with the fact that (cid:80) ∞ j =1 c j converges absolutely, this implies thatthe above series is uniformly convergent, as a function of t , on any compact intervalof ( −∞ , ∞ ).Take real parts on both sides of (3.2), multiply both sides by c j , sum over j , andthen interchange the order of summation and integration, which is justified from theuniform convergence of the above series and the fact that α < π/ 4, so as to obtain (cid:90) ∞−∞ t m e αt ∞ (cid:88) j =1 c j ρ ( t + iλ j )Re (cid:18) F (cid:18) − i ( t + λ j )4 ; 12 ; z (cid:19)(cid:19) d t = − π ∞ (cid:88) j =1 c j e − αλ j r mj cos (cid:18) α mθ j (cid:19) + 4 π Re (cid:34) e z ∞ (cid:88) j =1 c j ∂ m ∂α m ψ ( α, z ) (cid:35) . (3.3) EROS OF COMBINATIONS OF Ξ( t ) AND F Now using the notation F z (cid:0) + it (cid:1) for the series on the left-hand side of (3.3) asdefined in the statement of Theorem 2 and letting α → π − on both sides, we see thatlim α → π − (cid:90) ∞−∞ t m e αt F z (cid:18) 12 + it (cid:19) d t = − π ∞ (cid:88) j =1 c j e − π λ j r mj cos (cid:18) π mθ j (cid:19) − π Re (cid:34) e z ∞ (cid:88) j =1 c j (cid:18) i − λ j (cid:19) m e π ( i − λ j ) sinh (cid:18) z (cid:19)(cid:35) = − π ∞ (cid:88) j =1 c j e − π λ j r mj (cid:26) cos (cid:18) π mθ j (cid:19) + Re (cid:20) e i ( π +2 mθ j ) e z sinh (cid:18) z (cid:19)(cid:21)(cid:27) , (3.4)where in the penultimate step we used Lemma 6.Note that if z is real or purely imaginary, the right-hand side of (3.4) becomes (cid:18) e z sinh (cid:18) z (cid:19)(cid:19) (cid:40) − π ∞ (cid:88) j =1 c j e − π λ j r mj cos (cid:18) π mθ j (cid:19)(cid:41) , and thus the logic to prove that the above expression changes sign infinitely often issimilar to that in [9].Now let us assume that z is a complex number lying in the region D that is neitherreal nor purely imaginary. Then from (3.4),lim α → π − (cid:90) ∞−∞ t m e αt F z (cid:18) 12 + it (cid:19) d t = − π ∞ (cid:88) j =1 c j e − π λ j r mj (cid:26) cos (cid:18) π mθ j (cid:19) (cid:18) (cid:18) e z sinh (cid:18) z (cid:19)(cid:19)(cid:19) − sin (cid:18) π mθ j (cid:19) Im (cid:18) e z sinh (cid:18) z (cid:19)(cid:19) (cid:27) . (3.5)Now let u z := 1 + Re (cid:18) e z sinh (cid:18) z (cid:19)(cid:19) and v z := Im (cid:18) e z sinh (cid:18) z (cid:19)(cid:19) . (3.6)From (3.5) and (3.6),lim α → π − (cid:90) ∞−∞ t m e αt F z (cid:18) 12 + it (cid:19) d t = − πw z ∞ (cid:88) j =1 c j e − π λ j r mj cos (cid:18) π β z + 2 mθ j (cid:19) , (3.7) EROS OF COMBINATIONS OF Ξ( t ) AND F where w z := (cid:112) u z + v z and β z := cos − (cid:18) u z w z (cid:19) . Since u z and v z are real, the quantities w z and β z are real too. We now show thatthe series on the right side of (3.7) changes sign infinitely often.By the hypothesis, there exists a positive integer M such that | λ M | = max j {| λ j |} , u and λ M (cid:54) = λ j for M (cid:54) = j. Then the series on the right-hand side of (3.7) (without the constant term in thefront) can be written as c M r mM e − πλM cos (cid:18) π β z + 2 mθ M (cid:19) (1 + E ( X, z ) + H ( X, z )) , (3.8)where E ( X, z ) := (cid:88) j (cid:54) = Mj ≤ X c j c M e − π ( λ j − λ M ) (cid:18) r j r M (cid:19) m cos( π + β z + 2 mθ j )cos( π + β z + 2 mθ M ) , (3.9)as well as H ( X, z ) := (cid:88) j (cid:54) = Mj>X c j c M e − π ( λ j − λ M ) (cid:18) r j r M (cid:19) m cos( π + β z + 2 mθ j )cos( π + β z + 2 mθ M ) , (3.10) X being a real number that is sufficiently large.We now claim that there exists a subsequence of natural numbers such that foreach value m in it, the inequality | cos( π + β z + 2 mθ M ) | ≥ c holds for some positiveconstant c .Note that i − λ M = r M e iθ M for 0 < θ M < π . Then r M > r j for M (cid:54) = j. (3.11)Observe thatcos (cid:16) π β z + 2 mθ M (cid:17) = cos (cid:18) π β z + 2 π (cid:18) mθ M π (cid:19)(cid:19) = cos (cid:18) π β z + 2 π (cid:22) mθ M π (cid:23) + 2 π (cid:26) mθ M π (cid:27)(cid:19) = cos (cid:18) π β z + 2 π (cid:26) mθ M π (cid:27)(cid:19) . In the remainder of the proof we construct two subsequences { p n } and { q n } of N such that { p n θ M π } and { q n θ M π } tend to some specific numbers inside the interval (0 , EROS OF COMBINATIONS OF Ξ( t ) AND F resulting incos (cid:18) π β z + 2 π (cid:26) mθ M π (cid:27)(cid:19) > (cid:18) π β z + 2 π (cid:26) mθ M π (cid:27)(cid:19) < , where for the first inequality, m takes values from the sequence { p n } with n ≥ N where N is large enough, and for the second, m takes values from { q n } with n ≥ N .To that end, we divide the proof of the claim into two cases. First consider thecase when θ M π is irrational. This case itself is divided into five subcases dependingupon the location of β z in the interval [0 , π ]. In all these cases, Kronecker’s lemma,that is, Lemma 4 plays an instrumental role. Case 1: Let 0 ≤ β z < π .Take j to be a large enough natural number and consider the subsequence { p n } of N such that (cid:8) p n θ M π (cid:9) → j +1 and so that π < lim n →∞ (cid:0) π + β z + 2 π (cid:8) p n θ M π (cid:9)(cid:1) < π . Thisensures that cos (cid:0) π + β z + 2 π (cid:8) mθ M π (cid:9)(cid:1) > n ≥ N for some N ∈ N large enough.It is also clear that if we let m run through the subsequence { q n } of N such that (cid:8) q n θ M π (cid:9) → , then cos (cid:0) π + β z + 2 π (cid:8) mθ M π (cid:9)(cid:1) < n ≥ N for some N ∈ N largeenough.In the subcases that follow, the argument is similar to that in Case 1, and hence ineach such case we only give the two subsequences { p n } and { q n } that we can let m run through so that cos (cid:0) π + β z + 2 π (cid:8) mθ M π (cid:9)(cid:1) > (cid:0) π + β z + 2 π (cid:8) mθ M π (cid:9)(cid:1) < n ≥ N for some N ∈ N large enough. Case 2: Let π ≤ β z < π .Choose { p n } to be such that (cid:8) p n θ M π (cid:9) → and { q n } to be such that (cid:8) q n θ M π (cid:9) → . Case 3: Let π ≤ β z < π .Here we can select { p n } so that (cid:8) p n θ M π (cid:9) → and { q n } so that (cid:8) q n θ M π (cid:9) → j (cid:48) +1 ,where j (cid:48) ∈ N is large enough so that π < lim n →∞ (cid:0) π + β z + 2 π (cid:8) q n θ M π (cid:9)(cid:1) < π . Case 4: Let π ≤ β z < π .Choose { p n } to be such that (cid:8) p n θ M π (cid:9) → and { q n } to be such that (cid:8) q n θ M π (cid:9) → . Case 5: Let π ≤ β z < π .Here we can allow { p n } to be such that (cid:8) p n θ M π (cid:9) → and { q n } to be such that (cid:8) q n θ M π (cid:9) → . EROS OF COMBINATIONS OF Ξ( t ) AND F From the above construction it is clear that, according to the location of β z , wecan always find a positive real number c such that (cid:12)(cid:12)(cid:12) cos (cid:16) π β z + 2 mθ M (cid:17)(cid:12)(cid:12)(cid:12) ≥ c, (3.12)when m runs over the sequence { p n } ∪ { q n } for n ≥ N , where N is large enough.If m runs over the above mentioned sequence then (3.10), (3.11) and (3.12) imply | H ( X, z ) | ≤ c | c M | (cid:88) j (cid:54) = Mj>X | c j | e − π ( λ j − λ M ) . (3.13)By our hypothesis { λ j − λ M } is also a bounded sequence, so that e − π ( λ j − λ M ) < A for some positive constant A . Therefore from (3.13), | H ( X, z ) | ≤ A c | c M | (cid:88) j (cid:54) = Mj>X | c j | . (3.14)Since (cid:80) ∞ j =1 | c j | is convergent, this implies | H ( X, z ) | = O (1) for X large enough.Now C X := max j ≤ X (cid:110) | r j || r M | (cid:111) is finite, in fact, (3.11) implies C X < 1. Similarly using(3.9) and (3.11), it can be shown that when m runs over the same sequence, | E ( X, z ) | ≤ A C mX c | c M | (cid:88) j (cid:54) = Mj ≤ X | c j | , (3.15)where A is some constant, independent of m . Since C X < 1, we conclude that E ( X, z ) → m → ∞ through the above sequence { p n } ∪ { q n } .It is to be noted that when m runs over the sequence that we have constructed,cos (cid:0) π + β z + 2 mθ M (cid:1) changes sign infinitely often. Thus, from (3.8), (3.14) and (3.15),it is clear that the right hand side of (3.7) changes sign infinitely often for infinitelymany values of m .Our aim is to prove that the function F z ( s ) has infinitely many zeros on the criticalline Re( s ) = 1 / 2. Suppose not. Then F z (cid:0) + it (cid:1) never changes sign for | t | > T forsome T large. First, consider F z (cid:0) + it (cid:1) > | t | > T . Define L z,m ( T ) := lim α → π − (cid:90) | t |≥ T F z (cid:18) 12 + it (cid:19) t m e αt d t. Since the integrand of the above integral is positive, for any T > T ,lim α → π − (cid:90) T ≤| t |≤ T F z (cid:18) 12 + it (cid:19) t m e αt d t ≤ lim α → π − (cid:90) | t |≥ T F z (cid:18) 12 + it (cid:19) t m e αt d t = L z,m ( T ) . EROS OF COMBINATIONS OF Ξ( t ) AND F Therefore, (cid:90) T ≤| t |≤ T F z (cid:18) 12 + it (cid:19) t m e π t d t ≤ L z,m ( T ) . Now if T tends to ∞ , (cid:90) T ≤| t | F z (cid:18) 12 + it (cid:19) t m e π t d t ≤ L z,m ( T )so that (cid:90) ∞−∞ F z (cid:18) 12 + it (cid:19) t m e π t d t ≤ L z,m ( T ) + (cid:90) T − T F z (cid:18) 12 + it (cid:19) t m e π t d t. Since the integrand on the right hand side is an analytic function of t in [ − T, T ], (cid:82) ∞−∞ F z (cid:0) + it (cid:1) t m e π t d t is convergent. Using [23, p. 149, Theorem 7.11], for example,it can be checked that that the integrand on the left hand side of (3.7) is uniformlyconvergent, with respect to α , on 0 ≤ α < π . Then (3.7) implies (cid:90) ∞−∞ t m e π t F z (cid:18) 12 + it (cid:19) d t = − πw z ∞ (cid:88) j =1 c j e − π λ j r mj cos (cid:18) π β z + 2 mθ j (cid:19) , for every m ∈ N . As per our construction of the sequences { p n } and { q n } , there existinfinitely many m ∈ { p n } ∪ { q n } large enough such that the right hand side of theabove identity is negative, and hence (cid:90) T ≤| t | F z (cid:18) 12 + it (cid:19) t m e π t d t < − (cid:90) | t |≤ T F z (cid:18) 12 + it (cid:19) t m e π t d t< T m (cid:90) | t |≤ T (cid:12)(cid:12)(cid:12)(cid:12) F z (cid:18) 12 + it (cid:19) e π t (cid:12)(cid:12)(cid:12)(cid:12) d t ≤ BT m , (3.16)where B := B ( T ) is independent of m .By our assumption on F z (cid:0) + it (cid:1) , we can find a positive number δ = δ ( T ) suchthat F z (cid:0) + it (cid:1) > δ for all t ∈ (2 T, T + 1) . Hence (cid:90) T ≤| t | F z (cid:18) 12 + it (cid:19) t m e π t d t ≥ (cid:90) T +12 T δt m e π t d t ≥ δ (cid:90) T +12 T t m d t = δ (cid:18) (2 T + 1) m +1 m + 1 − (2 T ) m +1 m + 1 (cid:19) ≥ δ (2 T ) m . (3.17) EROS OF COMBINATIONS OF Ξ( t ) AND F From (3.16) and (3.17), δ (2 T ) m ≤ (cid:90) T ≤| t | F z (cid:18) 12 + it (cid:19) t m e π t d t < BT m for infinitely many large m ∈ { p n } ∪ { q n } . This implies that2 m δ < B (3.18)holds for infinitely many m ∈ { p n } ∪ { q n } . However this is impossible since m can bechosen to be arbitrarily large. So our assumption that F z (cid:0) + it (cid:1) > | t | > T isnot true.Similar contradiction can be reached at when F z (cid:0) + it (cid:1) < | t | > T and θ M π isirrational.Lastly if F z (cid:0) + it (cid:1) > t > T and F z (cid:0) + it (cid:1) < t < − T (or vice-versa),we differentiate (3.1) 2 m + 1 times with respect to α . If θ M π is irrational, one canconstruct the subsequences { p n } and { q n } of N , similarly as done in the first case,that is, according to the location of β z .For example, if 0 ≤ β z < π , one can find j ∈ N large enough such that π + β z + π j < π . Now if θ M π < j +1 , then by Kronecker’s lemma (Lemma 4), there exists a sequence { p n } such that { p n θ M π } → j +1 − θ M π , so that cos (cid:0) π + β z + (2 p n + 1) θ M (cid:1) > 0. Nowif θ M π > j +1 , by Kronecker’s lemma again one can find a sequence { p n } such that { p n θ M π } − → j +1 − θ M π . Since π β z + 2 π (2 p n + 1) θ M π = π β z + 2 π (cid:18)(cid:22) p n θ M π (cid:23) + 1 + (cid:26) p n θ M π (cid:27) − θ M π (cid:19) , by periodicity,cos (cid:18) π β z + 2 π (2 p n + 1) θ M π (cid:19) = cos (cid:18) π β z + 2 π (cid:18)(cid:26) p n θ M π (cid:27) − θ M π (cid:19)(cid:19) . Thus, cos (cid:18) π β z + 2 π (cid:18)(cid:26) p n θ M π (cid:27) − θ M π (cid:19)(cid:19) = cos (cid:18) π β z + 2 π j +1 (cid:19) > n ≥ N , where N ∈ N is large enough.Similarly we can find a subsequence { q n } of N such that (cid:8) q n θ M π (cid:9) → − θ M π , thencos (cid:0) π + β z + 2 π (cid:8) q n θ M π (cid:9)(cid:1) < n ≥ N , where N ∈ N is large enough.Similarly corresponding to other locations of β z , one can obtain correspondingsubsequences { p n } and { q n } of N such that cos (cid:0) π + β z + (2 p n + 1) θ M (cid:1) > (cid:0) π + β z + (2 q n + 1) θ M (cid:1) < n ≥ N , where N ∈ N is large enough. One canthen use them to obtain a contradiction similar to that in (3.18) by using argumentsimilar to that given in equations (3.12) through (3.18). EROS OF COMBINATIONS OF Ξ( t ) AND F Thus, we conclude that F z (cid:0) + it (cid:1) changes sign infinitely often. This proves The-orem 2 when θ M /π is irrational.It only remains to prove Theorem 2 in the case when θ M /π is rational. We reduceit to the previous case, that is, when θ M /π is irrational, by performing the followingtrick. Fix a small positive real number (cid:15) and consider the function F (cid:15) ,z ( s ) := F z ( s + i(cid:15) ). Then F (cid:15) ,z is a vertical shift of F z , and the statement that F z hasinfinitely many zeros on the critical line is equivalent to the statement that F (cid:15) ,z hasinfinitely many zeros on the critical line. Moreover, F (cid:15) ,z ( s ) satisfies the conditionsfrom the statement of Theorem 2, with the same value of z , the same sequence ofcoefficients c j , and with the sequence { λ j } replaced by { λ j + (cid:15) } . Therefore, up tothis point, the proof of the theorem would work even with F z replaced by F (cid:15) ,z . Inthe course of doing this, the angle θ M , however, changes to, say, θ M (cid:48) . But since wehave the liberty of choosing (cid:15) , we choose it in such a way that the angle θ M (cid:48) becomesan irrational multiple of π . With this choice of (cid:15) and the analysis done so far, it isclear that the function F (cid:15) ,z ( s ) has infinitely many zeros on the critical line. Thus F z ( s ) also has infinitely many zeros on the critical line when θ M /π is rational. Thiscompletes the proof of Theorem 2 in all cases.4. Concluding remarks In this work, we saw an application of the general theta transformation (1.5) and theintegral (1.6) equal to each of its expressions towards proving that a certain functioninvolving the Riemann Ξ-function and the confluent hypergeometric function hasinfinitely many zeros on the critical line, thereby vastly generalizing Hardy’s theorem.There are plethora of new modular-type transformations, that is, the transformationsgoverned by the relation a → b , where ab = 1, that have linked to them certain definiteintegrals having Ξ( t ) under the sign of integration. See the survey article [6], [8], [10],and [2, Section 15], for example. Recently in [7, Theorems 1.3, 1.5], a higher leveltheta transformation and the integral involving the Riemann Ξ-function linked to itwas obtained. It may be interesting to see up to what extent one can extend Hardy’sidea to obtain new interesting results. One thing is clear - when one has Ξ ( t ) underthe sign of integration, the sign change argument as in the case of Ξ( t ) cannot beused. Nevertheless, the higher level theta transformation in [7, Theorems 1.3, 1.5] hastwo additional parameters in it, so it would be interesting to see what informationcould be extracted from it. Acknowledgements The first author’s research is supported by the SERB-DST grant RES/SERB/MA/P0213/1617/0021 whereas the third author is a SERB National Post Doctoral Fellow(NPDF) supported by the fellowship PDF/2017/000370. Both sincerely thank SERB-DST for the support. EROS OF COMBINATIONS OF Ξ( t ) AND F References [1] B.C. Berndt, C. Gugg, S. Kongsiriwong and J. Thiel, A proof of the general theta transforma-tion formula , in Ramanujan Rediscovered: Proceedings of a Conference on Elliptic Functions,Partitions, and q-Series in memory of K. 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Rudin, Principles of Mathematical Analysis , 3rd ed., McGraw-Hill, New York, 1976.[24] A. Selberg, On the zeros of Riemann’s zeta-function , Skr. Norske Vid. Akad. Oslo (1942), no.10, 1–59.[25] J. Steuding and E. Wegert, The Riemann zeta function on arithmetic progressions , Experiment.Math. (3) (2012), 235–240.[26] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Clarendon Press, Oxford, 1986.[27] M. van Frankenhuijsen, Arithmetic progressions of zeros of the Riemann zeta function , J. Num-ber Theory (2) (2005), 360–370. Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj,Gandhinagar 382355, Gujarat, India E-mail address : [email protected], [email protected], [email protected] Department of Mathematics, University of Illinois, 1409 West Green Street, Ur-bana, IL 61801, USA andSimion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1–764,RO–014700 Bucharest, Romania. E-mail address ::