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On Perron’s Formula and the Prime Numbers

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Advances in Pure Mathematics 2024年第6期14卷 487-494页

作者： Michael M. Anthony Department of Engineering Enertron Inc. Hohenwald Tennessee USA

The riemann hypothesis is intimately connected to the counting functions for the primes. In particular, Perron’s explicit formula relates the prime counting function to fixed points of iterations of the explicit formula with particular relations involving the trivial and non-trivial roots of the riemann Zeta function and the Primes. The aim of the paper is to demonstrate this relation at the fixed points of iterations of explicit formula, defined by functions of the form limT∈Ν→∞fT(zw)=zw,where, zwis a real number.

关键词： Perron Fixed Points Iterations Number Theory riemann hypothesis Iterations Invariance Primes

riemann hypothesis, Catholic Information and Potential of Events with New Techniques for Financial and Other Applications

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Advances in Pure Mathematics 2021年第5期11卷 524-572页

作者： Prodromos Char. Papadopoulos Aristotle University National Hellenic Research Foundation Giannitsa (or Yiannitsa) Greece

In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions () which will true for every element of a set A. We will study the riemann hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function ζ(s) of riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, …, 9.

关键词： Twin Problem Twin’s Problem Unsolved Mathematical Problems Prime Number Problems Millennium Problems riemann hypothesis riemann’s hypothesis Number Theory Information Theory Probabilities Statistics Management Financial Applications Arithmetical Analysis Optimization Theory Stock Exchange Mathematics Approximation Methods Manifolds Economical Mathematics Random Variables Space of Events Strategy Games Probability Density Stock Market Technical Analysis Forecasting

Approach to a Proof of the riemann hypothesis by the Second Mean-Value Theorem of Calculus

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Advances in Pure Mathematics 2016年第13期6卷 972-1021页

作者： Alfred Wünsche Humboldt-Universitü t Berlin Institut für Physik Nichtklassische Strahlung (MPG) Berlin Germany

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functionswith an integral representation of the form with a real-valued function which is non-increasing and decreases in infinity more rapidly than any exponential functions , possesses zeros only on the imaginary axis. The riemann zeta function as it is known can be related to an entire functionwith the same non-trivial zeros as . Then after a trivial argument displacement we relate it to a function with a representation of the form where is rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iy by the second mean-value theorem. Besides this theorem we apply the Cauchy-riemann differential equation in an integrated operator form derived in the Appendix B. All this means that we prove a theorem for zeros of on the imaginary axis z=iy for a whole class of function which includes in this way the proof of the riemann hypothesis. This whole class includes, in particular, also the modified Bessel functions for which it is known that their zeros lie on the imaginary axis and which affirms our conclusions that we intend to publish at another place. In the same way a class of almost-periodic functions to piece-wise constant non-increasing functions belong also to this case. At the end we give shortly an equivalent way of a more formal description of the obtained results using the Mellin transform of functions with its variable substituted by an operator.

关键词： riemann hypothesis riemann Zeta Function Xi Function Gauss-Bonnet Theorem Mellin Transformation

Solutions to Beal’s Conjecture, Fermat’s Last Theorem and riemann hypothesis

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Advances in Pure Mathematics 2016年第10期6卷 638-646页

作者： A. C. Wimal Lalith de Alwis School of Four Incalculables and Tathagatha’s Society No. 299 Moratuwa Sri Lanka

A Simple Mathematical Solutions to Beal’s Conjecture and Fermat’s Marginal Conjecture in his diary notes, Group Theoretical and Calculus Solutions to Fermat’s Last theorem & Integral Solution to riemann hypothesis ... 详细信息

关键词： Beal’s Conjecture Fermat’s Last Theorem riemann hypothesis

The Proof of riemann hypothesis, the Key to the Door Is the Periodicity

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Applied Mathematics 2021年第8期12卷 723-765页

作者： Jinliang Wang Research Institute of ESMD Method and Its Applications College of Science Qingdao University of Technology Qingdao China

The riemann hypothesis is a well-known mathematical problem that has been in suspense for 162 years. Its difficulty lies in the fact that it is involved in an infinite integral which includes infinite series with complex variables. To detour this is in vain, since all the messages are hid in it. To unscramble them, there is a totally new idea, that is, the “periodicity”! By investigating the numerical approximate values of zero points, an explicit distribution law on the critical line was found. To accord with this, a periodic form for the real part of Xi function was constructed and rigidly proved. The riemann hypothesis can be divided into three progressive propositions. The first proposition (the number of zero points in the critical strip satisfies a certain estimation) had been proved in 1905. The second proposition (the number of zero points on the critical line satisfies the same estimation as in the critical strip) is ever in suspense. It can be solved perfectly with the newly found “periodicity”. The third proposition (all the nontrivial zero points are on the critical line), that is, the riemann hypothesis, is also true. The proof is a combination of the symmetry, monotonicity, periodicity of the Xi function and the extremum principle of the harmonic functions. It is the moment to draw full stop for this suspending problem.

关键词： riemann hypothesis riemann Zeta Function Distribution Law of Zero Point Periodicity Monotonicity Extremum Principle

About the riemann hypothesis

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Journal of Applied Mathematics and Physics 2016年第3期4卷 561-570页

作者： Jinhua Fei Changling Company of Electronic Technology Baoji China

The riemann hypothesis is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems. It is also one of the Clay Mathematics Institute’s Millennium Prize Problems. Some mathematicians consider it the most important unresolved problem in pure mathematics. Many mathematicians made a lot of efforts;they don’t have to prove the riemann hypothesis. In this paper, I use the analytic methods to deny the riemann hypothesis;if there’s something wrong, please criticize and correct me.

关键词： riemann hypothesis Disavowal

All Zeros of the riemann Zeta Function in the Critical Strip Are Located on the Critical Line and Are Simple

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Advances in Pure Mathematics 2023年第6期13卷 402-411页

作者： Frank Stenger University of Utah Salt Lake City USA

In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the riemann zeta function located on the critical line are simple.

关键词： riemann hypothesis Fourier Transforms Schwarz Reflection Principle Cauchy-riemann Equations Trapezoidal-Midordinate Quadrature

The Towering Zeta Function

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Advances in Pure Mathematics 2016年第5期6卷 351-392页

作者： Michael M. Anthony Enertron Corp. Hohenwald TN USA

Over a century and half has passed when Bernhard riemann hypothesized that the non-trivial roots of the riemann zeta function ζ(s) all lie on the half-line . In this paper the Zeta function is iterated as a power tower and its properties are applied as an approach to an indication that the riemann hypothesis might be true. It is known that complex valued Power towers converge under certain conditions to exponential power towers of entire functions. These properties can be used to resolve the riemann hypothesis.

关键词： riemann hypothesis Zeta Power Towers Convergence Exponential Iterations

Some Conjectures on the Divisor Function

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Journal of Mathematics and System Science 2020年第2期10卷 13-22页

作者： Masatoshi Nakano Kesennuma College of Technology 1-174 Ohtougeyama Kesennuma-shiMiyagi 988-0812Japan

We propose the following conjecture on(σ)n the sum-of-divisors function:log(e'nlog logn-σ(n))/log(e'nlog log n)will increase strictly and converge to1 when n runs from the colossally abundant numbers to infinity This conjecture is a sufficient condition for the and converge to1 when n nuns from the colossally abundant numbers to infinity.This conjecture is a sufficient condition for the Ricemann hypothesis by Robin's theorem,and it is confirmed for n from10^(4 )up to 10^(103078) Further,we present two additional riemann hypothesis by Robin's theorem,and it is confirmed forn from 10^(4) up to 10^(103078) Further,we present two additional conjectures that are related to Robin's theorem.

关键词： riemann hypothesis Robin’s theorem colossally abundant number divisor function.