  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
representation involving the Riemann $\Xi$-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 $\Xi$-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.

Number Theory;Classical Analysis and ODEs