Functions and Analysis

1809 Submissions

[4] viXra:1809.0557 [pdf] replaced on 2018-10-15 21:39:23

Disproof of the Riemann Hypothesis

Authors: Jonathan W. Tooker
Comments: 1 Page. Everyone makes mistakes but only a fool fails to distinguish errors from errata.

We present a disproof by direct contradiction. We use an elementary representation of the Riemann zeta function to show that there are infinitely many non-trivial zeros of zeta off the critical line. All of these zeros are in the neighborhood of infinity and we define that neighborhood.
Category: Functions and Analysis

[3] viXra:1809.0481 [pdf] submitted on 2018-09-24 03:45:57

The Riemann Hypothesis

Authors: Michael Atiyah
Comments: 5 Pages.

The Riemann Hypothesis is a famous unsolved problem dating from 1859. This paper will present a simple proof using a radically new approach. It is based on work of von Neumann (1936), Hirzebruch (1954) and Dirac (1928).
Category: Functions and Analysis

[2] viXra:1809.0234 [pdf] replaced on 2018-11-06 23:43:46

Proof of the Limits of Sine and Cosine at Infinity

Authors: Jonathan W. Tooker
Comments: 71 Pages. Greatly improved in v5

We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition of the derivative. After studying the transformation law for the variation between representations of complex numbers, we show that the new representation has special properties which allow for a consistent modification to the transformation law for the variation which preserves the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. Then we use the newly defined modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.
Category: Functions and Analysis

[1] viXra:1809.0171 [pdf] submitted on 2018-09-08 15:03:38

On the Infinite Product for the Ratio of k-th Power and Factorial

Authors: Edigles Guedes
Comments: 3 Pages.

I derive an infinite product for the ratio of k-th power and factorial.
Category: Functions and Analysis