A proof of the prime number summation formula without assuming the Riemann hypothesis

1970 ◽  
Vol 4 (3) ◽  
pp. 384-394
Author(s):  
I. C. Chakravartty
Keyword(s):  
Prime Number ◽  
Riemann Hypothesis ◽  
Summation Formula ◽  
Number Summation

A proof of the prime number summation formula without assuming the Riemann hypothesis

1970 ◽  
Vol 4 (1-2) ◽  
pp. 275-276
Author(s):  
I. C. Chakravartty
Keyword(s):  
Prime Number ◽  
Riemann Hypothesis ◽  
Summation Formula ◽  
Number Summation

scholarly journals Proposed Proof of the Riemann Hypothesis

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Prime Number ◽  
Riemann Hypothesis ◽  
Chebyshev Function

For every prime number $q_{n}$, we define the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$, where $\theta(x)$ is the Chebyshev function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. This is known as the Nicolas inequality. The Nicolas criterion states that the Riemann hypothesis is true if and only if the Nicolas inequality is satisfied for all primes $q_{n} > 2$. We prove indeed that the Nicolas inequality is satisfied for all primes $q_{n} > 2$. In this way, we show that the Riemann hypothesis is true.

scholarly journals The Riemann Hypothesis

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Prime Number ◽  
Riemann Hypothesis ◽  
Sum Of Divisors

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 $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We prove in another paper that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. Using this result, we show there is a contradiction just assuming the possible smallest counterexample $n > 5040$ of the Robin inequality. In this way, we prove that the Robin inequality is true for all $n > 5040$ and thus, the Riemann Hypothesis is true.

scholarly journals Robin Criterion on Divisibility

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Riemann Hypothesis ◽  
Sum Of Divisors

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 $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We show that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. We prove that the Robin inequality holds when $\frac{\pi^{2}}{6} \times \log\log n' \leq \log\log n$ for some $n>5040$ where $n'$ is the square free kernel of the natural number $n$. The possible smallest counterexample $n > 5040$ of the Robin inequality complies that necessarily $(\log n)^{\beta} < 1.2592\times\log(N_{m})$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$ and $\beta = \prod_{i = 1}^{m} \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}$ when $n$ is an Hardy-Ramanujan integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$.

scholarly journals The Riemann Hypothesis

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Prime Number ◽  
Riemann Hypothesis ◽  
Sum Of Divisors

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 $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We prove in another paper that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. Using this result, we show there is a contradiction just assuming the possible smallest counterexample $n > 5040$ of the Robin inequality. In this way, we prove that the Robin inequality is true for all $n > 5040$ and thus, the Riemann Hypothesis is true.

scholarly journals The average multiplicative order of a finitely generated subgroup of the rationals modulo primes

2016 ◽  
Vol 12 (08) ◽  
pp. 2147-2158
Author(s):  
Cihan Pehlivan
Keyword(s):  
Prime Number ◽  
Explicit Expression ◽  
Riemann Hypothesis ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Euler Product ◽  
Finitely Generated ◽  
Numerical Computations ◽  
Multiplicative Order ◽  
Reduction Group

Let [Formula: see text] be a finitely generated subgroup of [Formula: see text]. Let [Formula: see text] be a prime number for which the reduction group [Formula: see text] is a well-defined subgroup of the multiplicative group [Formula: see text], and denote the order of [Formula: see text] by [Formula: see text]. Assuming the Generalized Riemann Hypothesis, we study the average of [Formula: see text] and the average of the powers of [Formula: see text] as [Formula: see text] ranges over prime numbers. The problem was considered in the case of rank 1 by Pomerance and Kurlberg. In the case when [Formula: see text] contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.

scholarly journals Large deviations of the limiting distribution in the Shanks–Rényi prime number race

2012 ◽  
Vol 153 (1) ◽  
pp. 147-166 ◽  
Author(s):  
YOUNESS LAMZOURI
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Prime Number ◽  
Riemann Hypothesis ◽  
Linear Independence ◽  
Limiting Distribution ◽  
Absolutely Continuous ◽  
Independence Hypothesis ◽  
Vector Valued Function ◽  
Vector Valued

AbstractLet q ≥ 3, 2 ≤ r ≤ φ(q) and a1, . . ., ar be distinct residue classes modulo q that are relatively prime to q. Assuming the Generalized Riemann Hypothesis (GRH) and the Linear Independence Hypothesis (LI), M. Rubinstein and P. Sarnak [11] showed that the vector-valued function Eq;a1, . . ., ar(x) = (E(x;q,a1), . . ., E(x;q,ar)), where $E(x;q,a)= ({\log x}/{\sqrt{x}})(\phi(q)\pi(x;q,a)-\pi(x))$, has a limiting distribution μq;a1, . . ., ar which is absolutely continuous on $\mathbb{R}^r$. Furthermore, they proved that for r fixed, μq;a1, . . ., ar tends to a multidimensional Gaussian as q → ∞. In the present paper, we determine the exact rate of this convergence, and investigate the asymptotic behavior of the large deviations of μq;a1, . . ., ar.

scholarly journals The Nicolas criterion for the Riemann Hypothesis

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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 $\frac{1}{2}$. For every prime number $p_{n}$, we define the sequence $X_{n} = \prod_{q \leq p_{n}} \frac{q}{q-1} - e^{\gamma} \times \log \theta(p_{n})$, where $\theta(x)$ is the Chebyshev 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 $X_{n} > 0$ holds for all primes $p_{n} > 2$. For every prime number $p_{k} > 2$, $X_{k} > 0$ is called the Nicolas inequality. We prove that the Nicolas inequality holds for all primes $p_{n} > 2$. In this way, we demonstrate that the Riemann hypothesis is true.

scholarly journals The Complete Proof of the Riemann Hypothesis

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Prime Number ◽  
Riemann Hypothesis ◽  
Complete Proof ◽  
Sum Of Divisors

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 $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We prove that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. Using this result, we show there is a contradiction just assuming the possible smallest counterexample $n > 5040$ of the Robin inequality. In this way, we prove that the Robin inequality is true for all $n > 5040$ and thus, the Riemann Hypothesis is true.

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