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In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemannzeta function lie on the critical line using the transformation formula of theJacobi theta function. Recently the first author obtained an integralrepresentation involving the Riemann $\Xi$-function and the confluenthypergeometric function linked to the general theta transformation. Using thisresult, we show that a series consisting of bounded vertical shifts of aproduct of the Riemann $\Xi$-function and the real part of a confluenthypergeometric function has infinitely many zeros on the critical line, therebygeneralizing a previous result due to the first and the last authors along withRoy and Robles. The latter itself is a generalization of Hardy's theorem.

Zeros of combinations of the Riemann $\Xi$-function and the confluent hypergeometric function on bounded vertical shifts