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Mathematics of Computation

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

Authors: Kevin A. Broughan and A. Ross Barnett
Journal: Math. Comp. 81 (2012), 1669-1679
MSC (2010): Primary 11M06
Published electronically: December 7, 2011
MathSciNet review: 2904597
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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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Additional Information

Kevin A. Broughan
Affiliation: University of Waikato, Hamilton, New Zealand

A. Ross Barnett
Affiliation: University of Waikato, Hamilton, New Zealand

Keywords: Gram points, Gram lines, Riemann zeta function
Received by editor(s): November 28, 2010
Received by editor(s) in revised form: March 25, 2011, and April 15, 2011
Published electronically: December 7, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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