The second moment of Sn(t) on the Riemann hypothesis

We prove an asymptotic formula for the second moment (up to height T) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to T2-ε, for any ε > 0.

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

Symmetry ◽

10.3390/sym13122410 ◽

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 ◽

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.

The Riemann Hypothesis is most likely true

10.33774/coe-2021-978nv ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev 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.

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

International Journal of Number Theory ◽

10.1142/s1793042118500252 ◽

2018 ◽

Vol 14 (02) ◽

pp. 371-382

Author(s):

K. Paolina Koutsaki ◽

Albert Tamazyan ◽

Alexandru Zaharescu

Keyword(s):

Asymptotic Formula ◽

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Linear Combinations ◽

The Real ◽

Unique Integer ◽

Riemann Zeta ◽

Derivatives Of

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.

An estimate of the second moment of a sampling of the Riemann zeta function on the critical line

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2014.01.040 ◽

2014 ◽

Vol 415 (1) ◽

pp. 121-134

Author(s):

Sihun Jo ◽

Minsuk Yang

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Critical Line ◽

Second Moment ◽

The Riemann Hypothesis Is True

10.33774/coe-2021-wd17h-v4 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev 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.

The Riemann Hypothesis Is True

10.33774/coe-2021-wd17h-v5 ◽

2021 ◽

Author(s):

Frank Vega

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Complex Numbers ◽

Natural Numbers ◽

Chebyshev 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.

On the Riemann Hypothesis for the Zeta Function

10.20944/preprints202105.0616.v2 ◽

2021 ◽

Author(s):

Xiao-Jun Yang

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Equivalence Theorems ◽

In this paper we address some variants for the products of Hadamard and Patterson. We prove that all zeros of the Riemann $\Xi$--function are real. We also prove that the Riemann hypothesis is true. The equivalence theorems associated with the Riemann zeta--function are obtained in detail.

A Proof to the Riemann Hypothesis

10.31219/osf.io/gbnpz ◽

2020 ◽

Author(s):

Sourangshu Ghosh

Keyword(s):

Number Theory ◽

Zeta Function ◽

Riemann Zeta Function ◽

Riemann Hypothesis ◽

Prime Numbers ◽

Complex Numbers ◽

In this paper, we shall try to prove the Riemann Hypothesis which is a conjecture that the Riemann zeta function hasits zeros only at the negative even integers and complex numbers with real part ½. This conjecture is very importantand of considerable interest in number theory because it tells us about the distribution of prime numbers along thereal line. This problem is one of the clay mathematics institute’s millennium problems and also comprises the 8ththe problem of Hilbert’s famous list of 23 unsolved problems. There have been many unsuccessful attempts in provingthe hypothesis. In this paper, we shall give proof to the Riemann Hypothesis.

The generalized divisor problem and the Riemann hypothesis

Nagoya Mathematical Journal ◽

10.1017/s0027763000003585 ◽

1991 ◽

Vol 122 ◽

pp. 149-159 ◽

Cited By ~ 5

Author(s):

Hideki Nakaya

Keyword(s):

Zeta Function ◽

Natural Number ◽

Complex Number ◽

Riemann Zeta Function ◽

Multiplicative Function ◽

Riemann Hypothesis ◽

Divisor Problem ◽

Divisor Function ◽

Let dz(n) be a multiplicative function defined bywhere s = σ + it, z is a. complex number, and ζ(s) is the Riemann zeta function. Here ζz(s) = exp(z log ζ(s)) and let log ζ(s) take real values for real s > 1. We note that if z is a natural number dz(n) coincides with the divisor function appearing in the Dirichlet-Piltz divisor problem, and d-1(n) with the Möbious function.