An investigation of the non-trivial zeros of the Riemann zeta function

Abstract

The zeros of the Riemann zeta function outside the critical strip are the trivial zeros. While many zeros of the Riemann zeta function are located on the critical line ℜpsq " 1{2, the non-existence of zeros in the remaining part of the critical strip ℜpsq P s0, 1r remains to be proven. The Riemann zeta functional leads to a relationship between the zeros of the Riemann zeta function on either sides of the critical line. Given s a complex number and s its complex conjugate, if s is a zero of the Riemann zeta function in the critical strip ℜpsq P s0, 1r, then 1 ´s is also a zero. As the Riemann hypothesis states that all non-trivial zeros lie on the critical line ℜpsq " 1{2, it is enough to show there are no zeros on either of the intervals ℜpsq P s0, 1 2 r or s 1 2 , 1r, to say the Riemann hypothesis is true.

A lower bound for the modulus of the Dirichlet eta function on partition $\mathcal{P}$ from 2-D principal component analysis

“…As the Dirichlet eta function and the Riemann zeta function share the same zeros in the critical strip, the Riemann zeta function has no zeros in the strip ℜpsq P s 1 2 , 1r. Let us recall proposition 8 in [3]: Given s a complex number and s its complex conjugate, if s is a complex zero of the Riemann zeta function in the strip ℜpsq P s0, 1r, then we have: ζpsq " ζp1 ´sq.…”