Articles of riemann hypothesis

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 […]

Riemann Hypothesis Proof – legit? The above paper claims to prove the Riemann Hypothesis, but seems somewhat suspect in its length, broken English, and the fact that it was published in an Applied Maths journal. I don’t have the background knowledge of Riemann’s hypothesis, and so I am having trouble reasoning through the paper, so I thought I’d ask […]

Proof of Riemann Hypothesis

This proof was released this year: Where is the mistake? I just found it and was wondering how obviously wrong it is.

Is classifying one dimensional generalized quasicrystals worthwhile strategy to approach RH?

Works done: After fruitlessly poring over books on zeta functions, it seems Freeman Dyson’s sotto voce nudge to classify generalized one-dimensional quasicrystals is a way to go. As he writes: Question: Will this be a worthwhile strategy to pursue where a big picture akin to Atiyah-Singer index theorem needs to be made for symmetry?

What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens function, error terms of Prime Number Theorem, and Farey sequences. Those equivalent statements do not use Riemann Zeta function directly, but provide additional insights about Riemann Hypothesis from […]

Are these known telescoping series for $\zeta\left(\frac12\right)$?

There are many known telescoping series for $\zeta(s)$ and I was playing with the following two: $$\displaystyle \zeta(s) = \frac{1}{(s-1)} \left(\sum _{n=1}^{\infty } \left( {\frac {n}{(n+1)^{s}}} – \frac{n-s}{n^s}\right) \right), \qquad \Re(s)>0$$ and $$\displaystyle \zeta(s) = \frac{1}{(s-1)} \left(\sum _{n=1}^{\infty } \left( {\frac {n-1+s}{n^{s}}} – \frac{n-1}{(n-1)^s}\right) \right), \qquad 0<\Re(s)<1$$ After adding the two together and then dividing […]