# Questions tagged [riemann-hypothesis]

Questions on the Riemann hypothesis, a conjecture on the behavior of the complex zeros of the Riemann $\zeta$ function. You might want to add the tag [riemann-zeta] to your question as well.

334 questions
Filter by
Sorted by
Tagged with
48 views

### Is the Riemann zeta function the only meromorphic function satisfying the Riemann hypothesis? [closed]

It is well known that Riemann zeta function is a meromorphic function satisfying the Riemann hypothesis in the complex plane. I want to ask if the Riemann zeta function is the only meromorphic ...
34 views

### On the error term of an aproximation to $\sum_{k\le n}\frac{\mu(k)}{k}$ through Legendre's formula

I was looking for information about the Riemann Hypothesis, and I found that the Riemann Hypothesis is equivalent to the statement $$\sum_{k\le n}\frac{\mu(k)}{k}=O(n^{-1/2+\varepsilon})$$ The well ...
53 views

### On the sum $\sum_{k\le n}\frac{\mu(k)}{k}$

I was looking for information about the Riemann Hypothesis, and I found that the Riemann Hypothesis is equivalent to the statement $$\sum_{k\le n}\frac{\mu(k)}{k}=O(n^{-1/2+\varepsilon})$$ If we ...
49 views

79 views

### Does the Riemann Zeta Function contemplate complex conjugates? [closed]

When there are two consecutive non-trivial zeros in the critical strip, should they always be complex conjugate? Should their imaginary parts be mirror symmetric? For example, in this attached ...
40 views

### Continuum hypothesis and Riemann hypothesis [duplicate]

In the video: https://www.youtube.com/watch?v=O4ndIDcDSGc it is stated that a way to prove the Riemann hypothesis would be to prove that it is undecidable (I assume within ZFC). Then, since if the ...
75 views

### Questions on the zeros of a couple of functions which overlap the non-trivial zeros of $\zeta(s)$

The following question is related to the following two derived formulas for the Riemann zeta function $\zeta(s)$ where formula (2) below is a simplification of formula (1) below. Both formulas below ...
89 views

### Riemann hypothesis cannot be proved to be unprovable?

I'm coming off the numberphile video: https://youtu.be/O4ndIDcDSGc In the very end, they say that 'Proving that Riemann hypothesis is unprovable would be a proof of the truth of Riemann hypothesis' ...
116 views

### Applications of Riemann Hypothesis outside number theory

I'm trying to write a survey article about Riemann Hypothesis, especially about its corollaries and analogies in other fields. I found that there are tons of results in number theory (especially about ...
80 views

### What is known on the sum $\sum_{N\le n\le 2N}\dfrac{\mu(n)n^{it}}{\sqrt{n}}$?

I am curious what are the known bounds on the truncated Dirichlet series $\sum_{N\le n\le 2N}\dfrac{\mu(n)n^{it}}{\sqrt{n}}$. Naively, we may upper bound this sum by some constant times $\sqrt{N}$. ...
45 views

### Relation between one dimensional Schrodinger's operator and Riemann hypothesis

At the end of lecture ( from conference "Perspectives on Riemann hypothesis" '18 ) S.J. Patterson mentioned about work done by 'Christa Mirgel' ( University of Gottingen) on one dimensional ...
50 views

59 views

107 views

### Questions on convergence of formula for $\zeta(s)$

This question assumes definition (1) below and relationship (2) below. With respect to the integral in (2) below, I selected $\frac{1}{2}$ as the lower integration bound because this is the ideal ...
91 views

### Distribution of Primes (elementary)

Disclaimer: I'm a High Schooler, with only a knowledge of Elementary Number Theory, Geometry/Trig, and Calculus I. I'm trying to understand the distribution of prime numbers. I've watched a few ...
116 views

182 views

### The connection between the Riemann hypothesis and the harmonic series

I have heard most people say that the harmonic series, given by $\displaystyle\sum_{n=1}^{\infty}{n^{-1}}$, is central or somehow related to a proof of the Riemann hypothesis. I have been trying to ...
638 views

### A Special Observation on Prime Numbers and $\pi (n)$

Update Please check recently posted on M.O $\eth(n)$ is a little algorithm I made, which may appear to be quite complex, so I will start with an example middle of the post. Questions are at the end ...
63 views

### How to get to formula for derivative of zeta/zeta

I reading a book for a Seminar, where I have to present some parts out of it. In the book the other says the formula below can be proved by partial integration, but it is not clear for me, hwo that ...
121 views

### The Riemann zeta function for complex conjugates

If $\zeta(x)=a+ib$ and $\zeta(y)=a-ib$, is there a single equation that relates $\zeta(x)$ to $\zeta(y)$?
91 views

### Formula for the reciprocal of the Hurwitz zeta function

The following equation is for the reciprocal of the zeta function at $Re(s) > 1$:  \frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}} = s \int_{1}^{\infty} \frac{M(x)}{x^{1+s}}dx, \ ...