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.
The aim of the present manuscript is to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines ℜpsq " α. An approach based on a two-dimensional principal component analysis matching the dimensions of the complex plane, which is built on a parametric ellipsoidal shape, has been undertaken to achieve this result. This lower bound, which is expressed as @s P C s.t. ℜpsq ě 1 2 , |ηpsq| ě 1 ´?2 2 α where η is the Dirichlet eta function, has implications for the Riemann hypothesis as |ηpsq| ą 0 for any s P C such that ℜpsq P s 1 2 , 1r.
The Etherington's distance-duality equation is the relationship between the luminosity distance of standard candles and the angular-diameter distance. This relationship has been validated from astronomical observations based on the X-ray surface brightness and the Sunyaev-Zel'dovich effect of galaxy clusters. In the present study, we propose a derivation of the Etherington's reciprocity relation in the dichotomous cosmology.
The present manuscript is about application of Itô's calculus to the moment-generating function of the lognormal distribution. While Taylor expansion fails when applied to the moments of the lognormal due to divergence, various methods based on saddle-point approximation conjointly employed with integration methods have been proposed. By the Jensen's inequality, the MGF of the lognormal involves some convexity adjustment, which is one of the aspects under consideration thereof. A method based on zero-entropy principle is proposed part of this study, which deviations from the benchmark by infinitesimal epsilons is attributed to an asymmetry of the reciprocal. As applied to systems carrying vibrating variables, the partial offset by the reciprocal of an action, is a principle meant to explain a variety of phenomena in fields such as quantum physics.
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