All Questions
Tagged with analytic-number-theory riemann-hypothesis
90
questions
1
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0
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92
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Inequalities for the Dirichlet eta function at non-trivial Riemann zeta zeros.
I am interested in these inequalities for sufficiently large $n$:
$$\Large \left(\Re\left(\sum _{k=1}^n (-1)^{k+1} x^{\log (k) c}\right)\right)^2 \leq \left( \Re\left(x^{\log \left(n+\frac{1}{2}\right)...
- 7,438
0
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0
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50
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Applications of the de Bruijn-Newman constant outside of the Riemann Hypothesis
According to Wikipedia,
The de Bruijn–Newman constant, denoted by Λ and named after
Nicolaas Govert de Bruijn and Charles Michael Newman, is a
mathematical constant defined via the zeros of a certain ...
- 56
0
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0
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74
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Is $\phi_0$ equivalent to the Riemann hypothesis?
This is an extension (and more distilled version) of Extension of PDE's to critical strip, with new information. I am fairly sure that my constructions are an alternate description of the De Brujn ...
- 754
1
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1
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149
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Question about Salem's integral equation reformulation of Riemann hypothesis
Consider an integral equation:
$$\int_{-\infty}^{+\infty}\frac{e^{-\sigma y}f(y)}{e^{e^{x-y}}+1}dy=0$$, where $\sigma\in(\frac{1}{2},1)$
Salem proved that this equation has no bounded solution other ...
- 375
1
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1
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66
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Better bounds in the error term of the summatory function of Von-Mangoldt function and the Riemann Hypothesis
Theorem Let $f : \mathbb{N} \to \mathbb{C}$ be an arithmetic function and let $M(f, x) = \sum_{n \leq x} f(n)$ be the summatory function of $f$. If $M(f, x) = Ax^{\alpha} + O(x^{\theta})$, where $\...
- 163
1
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1
answer
111
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Is this inequality related to $|\log \zeta (s)|$ wrong?
This is a doubt in the Littlewood's first estimate(in which it is assumed that Riemann hypothesis is true) given in this book, page 433.
Let $s = \sigma + it$ and $\delta $ such that $\frac12 + \...
- 751
7
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4
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280
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Where are the zeros of a slightly perturbed Riemann Zeta function?
We seek to understand the locations of the zeros when we introduce a minor perturbation to the Riemann Zeta function:
${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}...
- 553
3
votes
0
answers
192
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Why is the Riemann explicit formula for primes (via the Riemann Hypothesis) any better than existing formulae?
Many people, such as here, don't consider Willan's formula for primes, and other such formulas given here, as meaningful formulae for computing primes. My understanding of the main criticisms are that ...
- 681
10
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1
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280
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Why do number theorists care so much about how well $\text{Li}(x)$ approximates $\pi(x)$ if it's not our best approximation?
An alleged primary motivator for the RH is so that we can bound the error term $|\text{Li}(x) - \pi(x)|$ by a factor of $O(\sqrt{x}\log x)$. However, I also learned about Riemann's explicit formula $R(...
- 681
3
votes
1
answer
139
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Turán proof that constant sign of Liouville function implies RH
In Mat.-Fys. Medd. XXIV (1948) Paul Turán gives what he says is a proof of the statement that if the summatory $L(x) = \sum_{n\leq x} \lambda(n)$ of the Liouville function $\lambda(n) = (-1)^{\Omega(n)...
- 346
-6
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1
answer
170
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On the zeros of the Riemann Zeta function
Let us consider the infinite set of non-trivial zeros of Riemann zeta function $\{\rho_n \}$ and the following product
$$
\prod_{n=1}^{\infty}\bigg(1-\frac{1}{\rho_n}\bigg)\bigg(1-\frac{1}{\overline{\...
- 2,040
2
votes
1
answer
165
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On an upper bound for the summatory Möbius function
Denote by $\mu$ the Möbius function and $\forall x\ge0$, $$M(x)=\sum_{n\le x}\mu(n)$$ the summatory Möbius function. It is know that $M(x)=o(x)$, which is a consequence of the prime number theorem. ...
- 486
1
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2
answers
147
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Number of non-trivial zeros of $\zeta(s)$ in a small substrip
I want to examine the number of non-trivial zeros of $\zeta(s)$ in the small substrip given by $a \leq \Re s \leq 1$, where $a>0$ is an absolute constant, and $U \leq \Im s \leq T$ by considering ...
user1036107
14
votes
1
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422
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If for the first $\|n\|$ primes $p_i, \left(\frac{p_i}n\right)=+1$, then $n$ is a square
Can we prove or disprove (perhaps under some standard hypothesis):
$$\text{If for the first }\|n\|\text{ primes }p_i, \left(\dfrac{p_i} n\right)=+1,\text{ then }n\text{ is a square.}$$
where $\|n\|$ ...
- 1,768
0
votes
0
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102
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Equal ordinate zeros for the Riemann-zeta function
Write $\rho=\beta + i\gamma$ and $\rho'=\beta'+i\gamma'$ for two distinct non-trivial zeros of the Riemann zeta-function. Is it known that $\gamma\neq\gamma'$? That is, does a proof exist that two ...
- 435