Gram lines and the average of the real part of the Riemann zeta function

The contours =Λ(s) = 0 of the function which satisfies ζ(1− s) = Λ(s)ζ(s) cross the critical strip on lines which are almost horizontal and straight, and cut the critical line alternately at Gram points and points where ζ(s) is imaginary. The real part of ζ(s), when averaged in a modified manner, for fixed values of σ over the values on the “Gram lines”, satisfies a relation which extends a theorem of Titchmarsh, (namely that the average of ζ(s) over the Gram points is 2), to the right hand side of the critical strip.