scholarly journals Fourier coefficients associated with the Riemann zeta-function

2016 ◽  
Vol 8 (1) ◽  
pp. 16-20
Author(s):  
Yu.V. Basiuk ◽  
S.I. Tarasyuk
Keyword(s):  
Fourier Series ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Fourier Coefficients ◽  
Critical Line ◽  
Series Method ◽  
Fourier Series Method ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We study the Riemann zeta-function $\zeta(s)$ by a Fourier series method. The summation of $\log|\zeta(s)|$ with the kernel $1/|s|^{6}$ on the critical line $\mathrm{Re}\; s = \frac{1}{2}$ is the main result of our investigation. Also we obtain a new restatement of the Riemann Hypothesis.

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

Symmetry ◽  
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 ◽  
The Riemann Zeta Function ◽  
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.

On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis

2018 ◽  
Vol 61 (3) ◽  
pp. 622-627
Author(s):  
Helmut Maier ◽  
Michael Th. Rassias
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Crucial Role ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Dirichlet Polynomials ◽  
Riemann Zeta

AbstractA crucial role in the Nyman–Beurling–Báez-Duarte approach to the Riemann Hypothesis is played by the distancewhere the infimum is over all Dirichlet polynomialsof length N. In this paper we investigate under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.

scholarly journals On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kenta Endo ◽  
Shōta Inoue
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Riemann Zeta Function ◽  
Open Problem ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Iterated Integrals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Iterated Integral

AbstractWe consider iterated integrals of {\log\zeta(s)} on certain vertical and horizontal lines. Here, the function {\zeta(s)} is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of {\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis. Moreover, we show that, for any {m\geq 2}, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

scholarly journals The Riemann Hypothesis is most likely true

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

scholarly journals Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line

2018 ◽  
Vol 72 (3) ◽  
pp. 500-535 ◽  
Author(s):  
Louis-Pierre Arguin ◽  
David Belius ◽  
Paul Bourgade ◽  
Maksym Radziwiłł ◽  
Kannan Soundararajan
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Short Interval ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Small values of the Riemann zeta function on the critical line

2015 ◽  
Vol 169 (3) ◽  
pp. 201-220 ◽  
Author(s):  
Justas Kalpokas ◽  
Paulius Šarka
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Null trajectories for the symmetrized Hurwitz zeta function

2006 ◽  
Vol 463 (2077) ◽  
pp. 303-319 ◽  
Author(s):  
Ross C McPhedran ◽  
Lindsay C Botten ◽  
Nicolae-Alexandru P Nicorovici
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Hurwitz Zeta Function ◽  
Trigonometric Functions ◽  
Asymptotic Results ◽  
End Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Pass Through

We consider the Hurwitz zeta function ζ ( s , a ) and develop asymptotic results for a = p / q , with q large, and, in particular, for p / q tending to 1/2. We also study the properties of lines along which the symmetrized parts of ζ ( s , a ), ζ + ( s , a ) and ζ − ( s , a ) are zero. We find that these lines may be grouped into four families, with the start and end points for each family being simply characterized. At values of a =1/2, 2/3 and 3/4, the curves pass through points which may also be characterized, in terms of zeros of the Riemann zeta function, or the Dirichlet functions L −3 ( s ) and L −4 ( s ), or of simple trigonometric functions. Consideration of these trajectories enables us to relate the densities of zeros of L −3 ( s ) and L −4 ( s ) to that of ζ ( s ) on the critical line.

Sign in / Sign up

Export Citation Format

Share Document