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

Abstract

The contours ξ 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 which cut the critical line alternately at Gram points and points where ζ(s) is imaginary. When suitably averaged the real part of ζ(s) satisfies a relation which greatly extends a theorem of Titchmarsh, (namely that the average of ζ(s) over the Gram points has the value 2), to the open right-hand half plane σ > 0.