The Complexity of Mathematics

Riemann Zeta ◽

Goldbach’S Conjecture ◽

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

The Complexity of Mathematics

10.20944/preprints202002.0379.v7 ◽

2020 ◽

Author(s):

Frank Vega

Keyword(s):

Number Theory ◽

Complexity Theory ◽

Riemann Hypothesis ◽

Open Problems ◽

Mathematical Proofs ◽

Pure Mathematics ◽

Riemann Zeta ◽

Goldbach’S Conjecture ◽

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true or this has an infinite number of counterexamples using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

A Proof to the Riemann Hypothesis

10.31219/osf.io/gbnpz ◽

2020 ◽

Author(s):

Sourangshu Ghosh

Keyword(s):

Number Theory ◽

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Prime Numbers ◽

Complex Numbers ◽

Riemann Zeta

In this paper, we shall try to prove the Riemann Hypothesis which is a conjecture that the Riemann zeta function hasits zeros only at the negative even integers and complex numbers with real part ½. This conjecture is very importantand of considerable interest in number theory because it tells us about the distribution of prime numbers along thereal line. This problem is one of the clay mathematics institute’s millennium problems and also comprises the 8ththe problem of Hilbert’s famous list of 23 unsolved problems. There have been many unsuccessful attempts in provingthe hypothesis. In this paper, we shall give proof to the Riemann Hypothesis.

The Complexity of Number Theory

10.20944/preprints202002.0379.v3 ◽

2020 ◽

Author(s):

Frank Vega

Keyword(s):

Infinite Number ◽

Riemann Hypothesis ◽

Unsolved Problem ◽

Short Proof ◽

History Of Mathematics ◽

Difficult Problem ◽

Pure Mathematics ◽

History Of ◽

Goldbach’S Conjecture ◽

The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A principal complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). This proof is based on the assumption that if some language belongs to NSPACE(S(n)), then the unary version of that language belongs to NSPACE(S(log n)) and vice versa. However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Twin prime conjecture is true. Moreover, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal's conjecture is true. Since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem. In mathematics, the Riemann hypothesis is consider to be the most important unsolved problem in pure mathematics. If PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Riemann hypothesis is true.

The Riemann Hypothesis is most likely true

10.33774/coe-2021-978nv ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev Function ◽

Riemann Zeta ◽

Sum Of Divisors

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems and it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

Corrigendum to "Refinements of Goldbach's conjecture, and the Generalized Riemann Hypothesis"

Functiones et Approximatio Commentarii Mathematici ◽

10.7169/facm/1229696543 ◽

2008 ◽

Vol 38 (2) ◽

pp. 235-237

Author(s):

Andrew Granville

Keyword(s):

Riemann Hypothesis ◽

Generalized Riemann Hypothesis ◽

Goldbach’S Conjecture ◽

The Riemann Hypothesis Is True

10.33774/coe-2021-wd17h-v4 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev Function ◽

Riemann Zeta ◽

Sum Of Divisors

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. This problem has remained unsolved for many years. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n>5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1}>e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}>2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is true.

The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

Symmetry ◽

10.3390/sym13122410 ◽

2021 ◽

Vol 13 (12) ◽

pp. 2410

Author(s):

Janyarak Tongsomporn ◽

Saeree Wananiyakul ◽

Jörn Steuding

Keyword(s):

Asymptotic Formula ◽

Zeta Function ◽

Riemann Zeta Function ◽

Real Line ◽

Riemann Hypothesis ◽

Critical Line ◽

Nontrivial Zeros ◽

Riemann Zeta ◽

Riemann’S Zeta Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

Farey Sequences and the Riemann Hypothesis

10.20944/preprints201607.0003.v5 ◽

2016 ◽

Author(s):

Darrell Cox

Keyword(s):

Zeta Function ◽

Riemann Hypothesis ◽

Analytic Number Theory ◽

Square Root ◽

Nontrivial Zeros ◽

Landau Theorem ◽

Farey Sequences ◽

Riemann Zeta ◽

Function Zeros

Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.

The Riemann Hypothesis Is True

10.33774/coe-2021-wd17h-v5 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev Function ◽

Riemann Zeta ◽

Sum Of Divisors

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. This problem has remained unsolved for many years. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n>5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1}>e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}>2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is true.