On the distribution of zeros of derivatives of the Riemann ξ-function

2020 ◽  
Vol 32 (1) ◽  
pp. 1-22
Author(s):  
Amita Malik ◽  
Arindam Roy
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Real Number ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Density Estimate ◽  
Distribution Of Zeros ◽  
Riemann Zeta ◽  
Derivatives Of

AbstractFor the completed Riemann zeta function {\xi(s)}, it is known that the Riemann hypothesis for {\xi(s)} implies the Riemann hypothesis for {\xi^{(m)}(s)}, where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence {(\alpha\gamma_{m})}, where α is any fixed non-zero real number and {\gamma_{m}} runs over the imaginary parts of the zeros of {\xi^{(m)}(s)}. We also obtain a zero density estimate and an explicit formula for the zeros of {\xi^{(m)}(s)}. In particular, all our results hold uniformly for {0\leq m\leq g(T)}, where the function {g(T)} tends to infinity with T and {g(T)=o(\log\log T)}.

scholarly journals Investigation of a Control Function Determining the Location of the Zeros as a Possible Proof of the Riemann Hypothesis.

2022 ◽  
Author(s):  
Miroslav Sukenik
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Control Function ◽  
Distribution Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Zero Points

The article examines the control function in relation to the distribution of Zeros on thecritical line x = 0,5. To confirm this hypothesis, it will be necessary to perform a large number ofstatistical analyzes of the distribution of non-trivial zero points of the Riemann Zeta function.

ON THE EXPLICIT FORMULA IN THE THEORY OF PRIME NUMBERS

2012 ◽  
Vol 08 (03) ◽  
pp. 589-597 ◽  
Author(s):  
XIAN-JIN LI
Keyword(s):  
New York ◽  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Li's Criterion

In [Complements to Li's criterion for the Riemann hypothesis, J. Number Theory77 (1999) 274–287] Bombieri and Lagarias observed the remarkable identity [1 - (1 - 1/s)n] + [1 - (1 - 1/(1 - s))n] = [1 - (1 - 1/s)n]⋅[1 - (1 - 1/(1 - s))n], and pointed out that the positivity in Li's criterion [The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory65 (1997) 325–333] has the same meaning as in Weil's criterion [Sur les "formules explicites" de la théorie des nombres premiers, in Oeuvres Scientifiques, Collected Paper, Vol. II (Springer-Verlag, New York, 1979), pp. 48–61]. Let λn = ∑ρ[1 - (1 - 1/ρ)n] for n = 1, 2, …, where ρ runs over the complex zeros of the Riemann zeta function ζ(s). In this note, a certain truncation of λn is expressed as Weil's explicit formula [Sur les "formules explicites" de la théorie des nombres premiers, in Oeuvres Scientifiques, Collected Paper, Vol. II (Springer-Verlag, New York, 1979), pp. 48–61] for each positive integer n. By using the Bombieri and Lagarias' identity, we prove that the positivity of these truncations implies the Riemann hypothesis. If these truncations have suitable upper bounds, we prove that all nontrivial zeros of the Riemann zeta function lie on the critical line.

Biased behavior of weighted Mertens sums

2019 ◽  
Vol 16 (03) ◽  
pp. 547-577
Author(s):  
Emre Alkan
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Prime Numbers ◽  
Summation Formulas ◽  
The Riemann Zeta Function ◽  
Convexity Properties ◽  
Riemann Zeta ◽  
Derivatives Of

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at [Formula: see text].

On estimates of the Mertens function

2019 ◽  
Vol 15 (02) ◽  
pp. 327-337
Author(s):  
Biswajyoti Saha ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Zeta Function ◽  
Real Number ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Additional Hypothesis ◽  
The Riemann Zeta Function ◽  
Number Formula ◽  
Sum Formula ◽  
Riemann Zeta

Assuming the simplicity of the zeros of the Riemann zeta function [Formula: see text], Gonek and Hejhal studied the sum [Formula: see text] for real number [Formula: see text] and conjectured that [Formula: see text] for any real [Formula: see text]. Assuming Riemann hypothesis and [Formula: see text], Ng [11] proved that the Mertens function [Formula: see text]. He also pointed out that with the additional hypothesis of [Formula: see text] one gets [Formula: see text]. Here we show that [Formula: see text] for any real number [Formula: see text], under similar hypotheses.

scholarly journals Counting distinct zeros of the Riemann zeta-function

10.37236/1195 ◽  
1994 ◽  
Vol 2 (1) ◽  
Author(s):  
David W. Farmer
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Simple Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Derivatives Of

Bounds on the number of simple zeros of the derivatives of a function are used to give bounds on the number of distinct zeros of the function.

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.

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