viXra manuscript No. (will be inserted by the editor) Rigorous proof for Riemann hypothesis that rigidly comply with Principle of Maximum Density for Integer Number Solutions John Ting June 17, 2021 Abstract The 1859 Riemann hypothesis conjectured all nontrivial zeros in Riemann zeta function are uniquely located on sigma = 1/2 critical line. Derived from Dirichlet eta function [proxy for Riemann zeta function] are, in chronological order, simplified Dirichlet eta function and Dirichlet Sigma-Power Law. Computed Zeroes from the former uniquely occur at sigma = 1/2 resulting in total summation of fractional exponent (–sigma) that is twice present in this function to be integer –1. Computed Pseudo-zeroes from the later uniquely occur at sigma = 1/2 resulting in total summation of fractional exponent (1 – sigma) that is twice present in this law to be integer 1. All nontrivial zeros are, respectively, obtained directly and indirectly as one specific type of Zeroes and Pseudo-zeroes only when sigma = 1/2. Thus, it is proved that Riemann hypothesis is true whereby this function and law must rigidly comply with Principle of Maximum Density for Integer Number Solutions. Keywords Dirichlet Sigma-Power Law · Pseudo-zeroes · Riemann hypothesis · Zeroes Mathematics Subject Classification (2010) 11M26 · 11A41 Contents 1 Introduction . 1 2 Notation and sketch of the proof . 2 3 The Completely Predictable and Incompletely Predictable entities . 10 4 The exact and inexact Dimensional analysis homogeneity for Equations . 11 5 Gauss Areas of Varying Loops and Principle of Maximum Density for Integer Number Solutions . 11 6 Simple observation on Shift of Varying Loops in z(s + ıt) Polar Graph and Principle of Equidistant for Multiplicative Inverse............................................................. 14 7 Riemann zeta function, Dirichlet eta function, simplified Dirichlet eta function and Dirichlet Sigma-Power Law . 18 8 Conclusions . 20 Declaration of interests . 20 Acknowledgment . 20 References . 20 1 Introduction Riemann hypothesis is an intractable open problem in Number theory that was proposed in 1859 by famous German mathematician Bernhard Riemann (September 17, 1826 - July 20, 1866). This hypothesis conjectured 1 all nontrivial zeros in Riemann zeta function are uniquely located on s = 2 critical line. By applying Euler Corresponding Author E-mail: [email protected] (Alma Mater: University of Queensland) Dr. John Yuk Ching Ting, Dental and Medical Surgery, 729 Albany Creek Road, Albany Creek QLD 4035, Australia 2 Ting 1 2 3 Fig. 1: INPUT for s = 2 , 5 , and 5 . z(s) has CIS of CP trivial zeros located at s = all negative even numbers 1 and [conjectured] CIS of IP NTZ located at s = 2 given by various t values. formula to Dirichlet eta function [proxy for Riemann zeta function], we obtain simplified Dirichlet eta function 1 whereby its computed Zeroes uniquely occur at s = 2 resulting in total summation of fractional exponent (–s) that is twice present in this function to be integer –1. Dirichlet Sigma-Power Law is the solution from performing integration on simplified Dirichlet eta function whereby its computed Pseudo-zeroes uniquely occur 1 at s = 2 resulting in total summation of fractional exponent (1 – s) that is twice present in this law to be integer 1. All nontrivial zeros are, respectively, obtained directly and indirectly as one specific type of Zeroes 1 and Pseudo-zeroes only when s = 2 . Then [non-existent] virtual nontrivial zeros and virtual Pseudo-nontrivial zeros cannot be obtained directly and indirectly as a type of virtual Zeroes and virtual Pseudo-zeroes when 1 s 6= 2 . Based on Theorem 1 – 3 with associated Lemma 1 – 2 below, it is proved that Riemann hypothesis is true whereby this function and law must rigidly comply with Principle of Maximum Density for Integer Number Solutions. In addition, they also serendipitously obey trigonometric identities and manifest Principle of Equidistant for Multiplicative Inverse. 2 Notation and sketch of the proof Notation. CFS: countable finite set CIS: countable infinite set UIS: uncountable infinite set CP: Completely Predictable – see section 3 on definition for CP entities IP: Incompletely Predictable – see section 3 on definition for IP entities DA: Dimensional analysis – see section 4 on definitions for exact and inexact DA homogeneity NTZ: nontrivial zeros z(s): Riemann zeta function h(s): Dirichlet eta function sim-h(s): simplified-Dirichlet eta function DSPL: Dirichlet Sigma-Power Law Symbolically named after German mathematician Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859), the word “Law” utilized in DSPL represent a convenient terminology to describe this function – viz, there is resemblance to Zipf’s law via power law functions in s from s = s + it being exponent of a power function as similar format to ns , logarithm scale use, and harmonic z(s) series connection. Respectively, we use the terms 1 Zeroes (as three types of Gram points) and Pseudo-zeroes (as three types of Pseudo-Gram points) at s = 2 to collectively refer to corresponding f (n)’s and F(n)’s x-axis intercept points, y-axis intercept points and Origin intercept points. Respectively, we use the terms virtual Zeroes (as two types of virtual Gram points) and virtual 1 Pseudo-zeroes (as two types of virtual Pseudo-Gram points) at s 6= 2 to collectively refer to corresponding f (n)’s and F(n)’s x-axis intercept points and y-axis intercept points [with absent Origin intercept points]. Riemann hypothesis and closely related two types of Gram points 3 Geometrical and mathematical definitions for Gram points and virtual Gram points. Figure 1 depicts com- plex variable s (= s ±ıt) as INPUT with x-axis denoting real part Refsg associated with s, and y-axis denoting 1 1 1 imaginary part Imfsg associated with t. Then critical line: s = 2 ; non-critical lines: s 6= 2 viz, 0 < s < 2 and 1 1 1 2 < s < 1; and critical strip: 0 < s < 1. Both unique s = 2 value and non-unique s 6= 2 values 2 Set all s values whereby Set all s values = s j s is a real number, and 0 < s < 1. With including its complex conjugate, s = s ± it is present in all our well-defined f(n) and F(n) whereby these are continuous [complex] functions al- ways defined for any arbitrarily chosen intervals [a,b]. Given f(n) = 0 and F(n) = 0 as relevant derived equations, they dependently generate corresponding types of IP entities. All these IP entities will inherently be correctly assigned to appropriate independent mutually exclusive CIS constituted by t values as transcendental numbers except for the first Gram[y=0] point (and first virtual Gram[y=0] point) given by t = 0. Origin intercept points, x-axis intercept points and y-axis intercept points are geometrical definitions for IP entities of Gram[x=0,y=0] points, Gram[y=0] points and Gram[x=0] points. These geometrical definitions are equivalent to corresponding 1 mathematical definitions given by the relevant equations below for s = 2 : The Origin intercept points consisting of Gram[x=0,y=0] points or NTZ are computed directly from equa- 1 tions h(s) = 0 and sim-h(s) = 0; and indirectly from equation DSPL = 0 when s = 2 . The x-axis intercept points consisting of Gram[y=0] points or (traditional) ‘usual’ Gram points are computed directly from equa- 1 tion Gram[y=0] points-sim-h(s) = 0; and indirectly from equation Gram[y=0] points-DSPL = 0 when s = 2 . The y-axis intercept points consisting of Gram[x=0] points are computed directly from equation Gram[x=0] 1 points-sim-h(s) = 0; and indirectly from equation Gram[x=0] points-DSPL = 0 when s = 2 . Relevant functions and equations are unique mathematical objects that can be classified as three types of inifinite series: Harmonic series, Alternating harmonic series or Alternating series with trigonometric terms. We perform crucial de novo analysis on these functions and equations by noting their manifested intrinsic properties. Without loss of validity in our correct and complete set of mathematical arguments, we adopt the convention of providing focused analysis predominantly on appropriately chosen Alternating series with trigonometric terms throughout our presentation. The complex f(n) z(s) is a Harmonic series that does not converge in the critical strip. The complex f(n) h(s) is an Alternating harmonic series that converge in the critical strip. Then h(s) must act as proxy function for z(s) in this strip. Derived as Euler formula application to h(s) is the complex f(n) sim-h(s), and derived as R sim − h(s)dn is the complex F(n) DSPL. Both sim-h(s) and DSPL are Alternating series with trigonometric terms that converge in the critical strip. The f(n) h(s) will converge infinitely often to a zero value as h(s) = 0 equation giving rise to all NTZ or Gram[x=0,y=0] points. This event will only happen when h(s) is substituted with one unique s value which is 1 conjectured to be s = 2 by Riemann hypothesis. Being an Alternating harmonic series, we inherently cannot derive valid functions to obtain corresponding equations Gram[y=0] points-h(s) = 0 and Gram[x=0] points-h(s) = 0 that will enable mathematical computations of Gram[y=0] points as x-axis intercept points and Gram[x=0] points as y-axis intercept points. Then, computed Zeroes are mathematically defined as h(s) = 0 and sim-h(s) 1 = 0 when variable s = 2 ; computed virtual Zeroes are mathematically defined as h(s) 6= 0 and sim-h(s) = 0 1 1 when variable s 6= 2 ; computed Pseudo-Zeroes are mathematically defined as DSPL = 0 when variable s = 2 ; 1 and computed virtual Pseudo-zeroes are mathematically defined as DSPL = 0 when variable s 6= 2 .