Articles of riemann hypothesis

An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem – If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal V_p(n)=1-(n^{p-1}\mod p)$$ then$$\lim_{p\rightarrow \infty}\mathcal V_p(n) = \operatorname{sinc}(2\pi \,n)$$ and for the pair correlation of the non-trivial zeroes of the Riemann $\zeta$-function, […]

Have all the zeros of the Riemann Zeta function real part smaller than 1?

I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$ have real part smaller than 1, but I don’t know how to prove it. If Re(z) > 1 than ${\zeta}( z ) = \sum_{n = 1}^{\infty} \left( \frac{1}{n} \right)^z$

About Riemann's Hypothesis.

Could Riemann’ Hypothesis be proven true using Robin’s Inequality and that a counter-example to Riemann’s Hypothesis can not have a divisor that is a prime number to the exponent 5 ,according to some of Robin’s Theories? Also I think it can be proven the product of two numbers A and B that are counter-examples to […]

Is $M(x)=O(x^σ)$ possible with $σ≤1$ even if the Riemann hypothesis is false?

The wiki page on Mertens conjecture and the Connection to the Riemann hypothesis says Using the Mellin inversion theorem we now can express $M$ in terms of 1/ζ as $$ M(x) = \frac{1}{2 \pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{x^s}{s \zeta(s)}\, ds $$ which is valid for $\color{blue}{1} < σ < 2$, and valid for $\color{red}{1/2} < σ […]

Riemann Hypothesis, is this statement equivalent to Mertens function statement?

All: I saw one form of Riemann Hypothesis, it says: $$ \lim ∑(μ(n))/n^σ $$ Converges for all σ > ½ Is this statement same as the order of Mertens function is less than square root of n ?

Distribution of Subsets of Primes

Primes may be divided in to sets: $p=4n\pm1$. Gauss showed, that if $p=4n+1$, it may be written also as $p=a^2+b^2$. From LagrangesFour-SquareTheorem, we know that $g(2)=4$, where 4 may be reduced to 3 except for numbers of the form $4^n(8k+7)$,… (every rational integer is the sum of a fixed number $g(n)$ of $n$th powers of […]

A question about an asymptotic formula

I’ve been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann’s hypothesis is true, but I was unable to find a journal reference for this. Does anybody know of any journal reference or any other source where I can find this conditional result?

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, any results on reverse direction ? If we assume Goldbach Conjecture holds true, can we get any results about GRH ?

counterexample to RH; how big would it have to be?

If the Riemann hypothesis is false, then there has to be a first counterexample for $\zeta(z)=0$ in the critical strip with $\Re(z) \ne \frac{1}{2}$. For such a counterexample, how large would $T=|\Im(z)|$ have to be? On the one hand, we’ve only computed the first 10 trillion or so zeros, so it could be the very […]

How do you prove that $M(N)=O(N^{1/2+\epsilon})$ from the Riemann Hypothesis?

I understand that if $M(N)=O(N^\sigma)$, then $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}$ and therefore $$ \frac{1}{s\zeta(s)} = \int_0^\infty M(x) x^{-(s+1)} dx $$ for $s>\sigma$, and that having $\sigma=1/2+\epsilon$ for every $\epsilon>0$ will thus prove the RH. The Wikipedia article on the Mertens Conjecture states that the reverse also holds, but I don’t understand the details of argument: Using the […]