The Riemann Zeta Function and the Prime Number Theorem

2020 ◽  
pp. 187-236
Author(s):  
Tarlok Nath Shorey
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Prime Number Theorem ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Summability by Dirichlet convolutions

1967 ◽  
Vol 63 (2) ◽  
pp. 393-400 ◽  
Author(s):  
S. L. Segal
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Prime Number Theorem ◽  
Summation Method ◽  
Cesàro Means ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position ◽  
Equivalent Method

Ingham (3) discusses the following summation method:A series ∑an will be said to be summable to s ifwhere, as usual, [x] indicates the greatest integer ≤ x. (An equivalent method was introduced somewhat earlier by Wintner (8), but the notation (I) for the above method and the attachment to Ingham's name seem to have become usual following [(1), Appendix IV].) The method (I) is intimately connected with the prime number theorem and the fact that the Riemann zeta-function ζ(s) has no zeros on the line σ = 1. Ingham proved, among other results, that (I) is not comparable with convergence but, nevertheless, for every δ > 0, (I) ⇒ (C, δ) and for every δ, 0 < δ < 1, (C, −δ) ⇒ (I), where the (C, k) are Cesàro means of order k.

A Tauberian Theorem and Analogues of the Prime Number Theorem

1968 ◽  
Vol 20 ◽  
pp. 362-367 ◽  
Author(s):  
T. M. K. Davison
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Tauberian Theorem ◽  
Prime Number Theorem ◽  
Negative Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

In 1945 Ingham (3) proved the following Tauberian theorem: if ƒ is a non-decreasing, non-negative function on [1, ∞) and1then ƒ(x) ∼ cx. His proof is based on the non-vanishing of the Riemann zeta-function, ζ (s), on the line , and uses Pitt's form of Wiener's Tauberian theorem; (see, e.g., 5, Theorem 109, p. 211).

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.

DIRICHLET TRUNCATIONS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP POSSESS ZEROS NEAR EVERY VERTICAL LINE

2008 ◽  
Vol 04 (04) ◽  
pp. 653-662 ◽  
Author(s):  
HANY M. FARAG
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Prime Number Theorem ◽  
Vertical Line ◽  
Elementary Geometry ◽  
Critical Strip ◽  
Inverse Mapping ◽  
Kronecker’S Theorem ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We study the zeros of the finite truncations of the alternating Dirichlet series expansion of the Riemann zeta function in the critical strip. We do this with an (admittedly highly) ambitious goal in mind. Namely, that this series converges to the zeta function (up to a trivial term) in the critical strip and our hope is that if we can obtain good estimates for the zeros of these approximations it may be possible to generalize some of the results to zeta itself. This paper is a first step towards this goal. Our results show that these finite approximations have zeros near every vertical line (so no vertical strip in this region is zero-free). Furthermore, we give upper bounds for the imaginary parts of the zeros (the real parts are pinned). The bounds are numerically very large. Our tools are: the inverse mapping theorem (for a perturbative argument), the prime number theorem (for counting primes), elementary geometry (for constructing zeros of a related series), and a quantitative version of Kronecker's theorem to go from one series to another.

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.

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