A New Reason for Doubting the Riemann Hypothesis

Abstract

Assuming computations of the Riemann zeta function exhibit its true behavior, we get, under the Riemann hypothesis, a bound for a linear combination of odd order derivatives of Hardy’s Z-function evaluated at T + a and T – a where $T\pm a$ are some well chosen inflection points of Z. This bound, which only holds for $T\pm a$ beyond the computational capabilities of modern computers, suggests that Riemann hypothesis is not true. The key element in our argument is an identity which links the zeroes of a function f defined on the interval $[-a,a]$ and the values of its derivatives of odd order at $\pm a.$

Keywords:

• Riemann hypothesis
• distribution of zeroes
• Hardy’s function

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 11M26