ZEROS OF COMBINATIONS OF THE RIEMANN Ξ-FUNCTION AND THE CONFLUENT HYPERGEOMETRIC FUNCTION ON BOUNDED VERTICAL SHIFTS ATUL DIXIT, RAHUL KUMAR, BIBEKANANDA MAJI AND ALEXANDRU ZAHARESCU Abstract. In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral represen- tation involving the Riemann Ξ-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann Ξ-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby generalizing a previous result due to the first and the last authors along with Roy and Robles. The latter itself is a generalization of Hardy's theorem. 1. Introduction Ever since the appearance of Riemann's seminal paper [22] in 1859, the zeros of the Riemann zeta function ζ(s) have been a constant source of inspiration and motivation for 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=2, has defied all attempts towards its proof as of yet, the beautiful area of analytic number theory has been blossomed to what it is today because of these attempts. One of the major breakthroughs in this area occurred with G. H. Hardy [13] proving that infinitely many non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1=2. Let N0(T ) denote the number of non-trivial zeros lying on the critical line such that their positive imaginary part is less than or equal to T . Hardy and Littlewood [14] showed that N0(T ) > AT ,where A is some positive constant. Selberg [24] remarkably improved this to N0(T ) > AT log T . Levinson [16] further improved this by proving that 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 11M26 Keywords: Riemann zeta function, theta transformation, confluent hypergeometric function, bounded vertical shifts, zeros. 1 ZEROS OF COMBINATIONS OF Ξ(t) AND 1F1 2 current record, due to Pratt and Robles [18], being that 41:49% of the zeros lie on the critical line. The proof of Hardy's result in [13], which acted as a stimulus to the aforementioned developments, is well-known for its beauty and elegance. One of the crucial ingredients in his proof is the transformation formula satisfied by the Jacobi theta function. The latter is defined by 1 X 2 #(λ; τ) := qn λn; n=−∞ where q = exp(πiτ) for τ 2 H (upper half-plane) and λ = e2πiu for u 2 C. If we let τ = ix for x > 0 and u = 0, then the theta function becomes 1 X 2 #(1; ix) = e−n πx =: 2 (x) + 1; n=−∞ P1 −n2πx so that (x) = n=1 e . The aforementioned transformation formula employed by Hardy in his proof is due to Jacobi and is given by [26, p. 22, Equation 2.6.3] p 1 x(2 (x) + 1) = 2 + 1; (1.1) x which can be alternatively written as 1 p 1 p 1 X 2 2 1 X 2 2 a − e−πa n = b − e−πb n (1.2) 2a 2b n=1 n=1 for Re(a2) > 0, Re(b2) > 0 and ab = 1. The former is the same version of the theta transformation using which Riemann [22] derived the functional equation of the Riemann zeta function ζ(s) in the form [26, p. 22, eqn. (2.6.4)] − s s − (1−s) 1 − s π 2 Γ ζ(s) = π 2 Γ ζ(1 − s): 2 2 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 identity Z 1 2 Ξ(t) − 1 iα 1 iα 2iα cosh(αt)dt = e 2 − 2e 2 e ; (1.3) 2 1 π 0 t + 4 where Ξ(t) is the Riemann Ξ-function defined by 1 Ξ(t) = ξ 2 + it with ξ(s) being the Riemann ξ-function 1 − s s ξ(s) := s(s − 1)π 2 Γ ζ(s): 2 2 ZEROS OF COMBINATIONS OF Ξ(t) AND 1F1 3 Equation (1.3) is easily seen to be equivalent to Z 1 1 2 Ξ(t=2) 1 p 1 X 2 2 cos t log a dt = a − e−πa n (1.4) π 1 + t2 2 2a 0 n=1 iα π π by replacing t by 2t and letting a = e ; − 4 < α < 4 , in the latter. At this juncture, it is important to state that the above theta transformation, that is (1.2), has a generalization [1, Equation (1.2)], also due to Jacobi, and is as follows. Let z 2 C. If a and b are positive numbers such that ab = 1, then −z2=8 1 ! p e 2 X 2 2 p a − ez =8 e−πa n cos( πanz) 2a n=1 p z2=8 1 ! e 2 X 2 2 p = b − e−z =8 e−πb n cosh( πbnz) : (1.5) 2b n=1 It is then natural to seek for an integral representation equal to either sides of the above transformation similar to how the expressions on either sides of (1.2) are equal to the integral on the left-hand side of (1.4). Such a representation was recently obtained by the first author [5, Thm. 1.2]. This result is stated below. Theorem 1. Let r(x; z; s) := µ(x; z; s) + µ(x; z; 1 − s); 2 2 1 − s 1 z µ(x; z; s) := x1=2−se−z =8 F ; ; ; 1 1 2 2 4 n where F (c; d; z) := P1 (c)nz is the confluent hypergeometric function, (c) := 1 1 n=0 (d)nn! n Qn−1 Γ(c+n) i=0 (c + i) = Γ(c) , for any c; d; z; s 2 C. Then either sides of the transformation in (1.5) equals Z 1 t 1 Ξ( 2 ) 1 + it 2 r a; z; dt: (1.6) π 0 1 + t 2 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's result to obtain information on the zeros of a function which generalizes Ξ(t)? Even though 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 left αt 1−2it 1 z2 −αt 1+2it 1 z2 side of (1.3), one would have Ξ(t) e 1F1 4 ; 2 ; 4 + e 1F1 4 ; 2 ; 4 , and then saying that for z fixed in some domain, this function has infinitely many real zeros does not appeal much, for, it is already known that Hardy's theorem implies that Ξ(t) has infinitely many real zeros. However, even though this approach fails, one can still use the generalized theta transformation in (1.5) and the integral (1.6) linked to it to obtain information about ZEROS OF COMBINATIONS OF Ξ(t) AND 1F1 4 zeros of a function which generalizes Ξ(t). This function is obtained through vertical shifts 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 s 1 η(s) := π−s=2Γ ζ(s) and ρ(t) := η + it : 2 2 1 These are related to Riemman's ξ(s) and Ξ(t) functions by ξ(s) = 2 s(s − 1)η(s) and 1 1 1 Ξ(t) = 2 2 + it − 2 + it 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 1: P1 Let fcjg be a sequence of non-zero real numbers so that j=1 jcjj < 1. Let fλjg be a bounded sequence of distinct real numbers that attains its bounds. Then the function P1 1 F (s) = j=1 cjη(s + iλj) has infinitely many zeros on the critical line Re(s) = 2 . Note that the restriction cj 6= 0 is not strict but only for removing redundancy since if cj = 0 for some j 2 N, then the corresponding term does not contribute anything towards the function F (s). But if we let all but one elements in the sequence fcjg to be equal to zero and the non-zero element, say cj0 to be 1 along with the corresponding λj0 = 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 main result is as follows.