RAMANUJAN SUMS AND THE BURGESS ZETA FUNCTION

2012 ◽  
Vol 08 (08) ◽  
pp. 2069-2092 ◽  
Author(s):  
EMRE ALKAN
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Riemann Hypothesis ◽  
Mellin Transform ◽  
Bernoulli Numbers ◽  
Weighted Averages ◽  
Summatory Function ◽  
Ramanujan Sums

The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis.

scholarly journals A generalized Davenport expansion

2021 ◽  
pp. 1-5
Author(s):  
Alexander E. Patkowski
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Riemann Zeta Function ◽  
Fourier Expansion ◽  
Mellin Transform ◽  
Fractional Part ◽  
Arithmetic Functions ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.

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 The Riemann Hypothesis Is 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

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.

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.

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