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

Broughan, Kevin; Barnett, A.

doi:10.1090/S0025-5718-2011-02565-2

Gram lines and the average of the real part of the Riemann zeta function
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by Kevin A. Broughan and A. Ross Barnett PDF

Math. Comp. 81 (2012), 1669-1679 Request permission

Abstract:

The contours $\Im \Lambda (s)=0$ of the function which satisfies $\zeta (1-s)=\Lambda (s)\zeta (s)$ 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 $\zeta (s)$ is imaginary. When suitably averaged the real part of $\zeta (s)$ satisfies a relation which greatly extends a theorem of Titchmarsh, (namely that the average of $\zeta (s)$ over the Gram points has the value 2), to the open right-hand half plane $\sigma >0$.

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