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We prove that the auxiliary function R (s) has the integral representation \ R (s) =-2ˢ ^se^{ i s/4} (s) ₀^ y^s1-e^- y²+ y1-e^{2 y}\, dyy, =e^ i/4, s>0, \ valid for >0. The function in the integrand 1-e^- y²+ y1-e^{2 y} is entire. Therefore, no residue is added when we move the path of integration.
Juan Arias de Reyna (Tue,) studied this question.
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