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.

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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 ...
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1answer
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 ...
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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 ...
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49 views

What is known about the number of consecutive zeros of the Mertens function?

The Mertens function M(x) is defined as follows. $$\quad M(x)=\sum\limits_{n\le x}\mu(n)$$ The following table lists the consecutive zeros of $M(x)$ of length $l\ge 2$ for $0<x\le 1000$. $$\...
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2answers
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Does the Riemann hypothesis guarantee that integer factorization is difficult?

In an exchange of comments at Is there any mathematical conjecture that is successfully applied in the real world in spite of being yet unproven?, user R.J. Etienne claims that RH guarantees that ...
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1answer
158 views

An equivalent of the Riemann Hypothesis

Define $$I= \int_{-\infty}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2} \mathrm{d}t.$$ Balazard, Saias and Yor showed that the Riemann Hypothesis is equivalent to the statement that $I=...
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Question on convergence of an averaging function related to the non-trivial zeta zero counting function

This question is related to the non-trivial zeta zero counting function defined in (1) below and its asymptotic defined in (2) below. (1) $\quad N(t)=\sum\limits_{0<\Im(\rho)\le t}1$ (2) $\quad\...
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98 views

Riemann Hypothesis

Riemann Hypothesis is equivalent to the integral equation $\int_{-\infty}^{\infty} \frac{\log \mid \zeta (1/2+it)\mid }{1+4t^2} \ dt$ =0 What does this mean? Does it mean that Riemann Hypothesis is ...
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Derivation of Riemann Explicit formula from Von Mangoldt formula

To drive Riemann explicit formula for counting of primes $$J(x)=li(x)-\sum_{ρ}li(x^ρ )-log⁡(2)+\int_x^∞\frac{dt}{t(t^2-1) log⁡(t) }\qquad (1)$$ One can start from Von Mangoldt formula $$ψ(x)=x -\...
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1answer
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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 ...
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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 ...
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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 ...
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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' ...
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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 ...
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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 $\sqrt{N}$. ...
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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 ...
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On Lagarias's Equivalent Formulation of the Riemann Hypothesis

A result by Lagarias is that the Riemann Hypothesis is equivalent to: $$ \sigma(n) \leq H_n + \exp(H_n)\log(H_n)$$ Where $\sigma(n)$ is the sum of the divisors of $n$, and $H_n = \sum_{k=1}^n \...
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1answer
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Inference rules based on conjectures as the Riemann hypothesis.

When studying mathematics some fifty years ago I was amazed of how few inference rules that actually was used to prove theorems: modus ponens, reductio ad absurdum and tertium non datur, as far as I ...
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1answer
164 views

Riemann Hypothesis and the Zeta Function

I have been reading about the RH recently and I understood most of it until now. However, the biggest problem I'm having is to know what are the forms of the Riemann zeta function for the 3 main ...
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What are these Riemannian Ghosts?

So for New Year's Eve I'm here with some shining stars straight from the Riemann zeta function! The following plots show the first 5000 non trivial zeros "winding" around with different frequencies. ...
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Question on approximation of number of non-trivial zeros of $\zeta(s)$ with $0<\Im(\rho)\le t$

The asymptotic growth of $f(x)$ defined in (1) below is $\overset{\text{~}}{f}(x)$ defined in (2) below, and $\overset{\text{~}}{f}(\frac{t}{2 \pi})$ very closely approximates the asymptotic growth of ...
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1answer
73 views

Which equation did Riemann use when proving $\zeta(-2)=0$

Which equation did Riemann use when proving that $\zeta(-2)=0$. I know that the first trivial zero lies on the point $(-2,\,0)$ and would like to prove it using the same equation. Any help is much ...
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Are the non-trivial zeros of $\zeta(s)$ related specifically to the primes?

This question assumes the following definitions. (1) $\quad \psi(x)=\sum\limits_{n\le x}\Lambda(n)\qquad\text{(second Chebyshev function)}$ (2) $\quad M(x)=\sum\limits_{n\le x}\mu(n)\qquad\text{(...
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Where does this multiplicative and additive analogy breakdown?

Background We make use of bra-ket notation from quantum mechanics: $$ \hat K(t) = (a^\dagger)^{t-1} |1 \rangle \langle 1 | + ( a^\dagger)^{2(t-1)}|2 \rangle\langle 2 | + ( a^\dagger)^{3(t-1)}|3 \...
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Prove or disprove this inequality involving the Dirichlet inverse of the Euler totient function

Let: $a(n)=\sum\limits_{d \mid n} \mu(d)d$ Prove or disprove that there exists a constant $c$ such that the inequality: $$\sum\limits_{r=2}^{n} \frac{\sum\limits_{m=r}^{n} a(\gcd (m,r))}{r} \geq c\...
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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 ...
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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 ...
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1answer
116 views

If Riemann Hypothesis is “just barely” true, does it mean we can't have high hopes for its generalization? [closed]

Here is what I understand of the meaning of generalization. Suppose that the statement $p$ is true for a set $S$. Then $q$ is a generalization of $p$ if it is true for a set $S'$ such that $S\subset S'...
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List of books on the study of number theory. [duplicate]

I am requesting a list of books to learn number theory by myself that would take me from an introduction in the field to being an expert. I am most interested in understanding unsolved problems in ...
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While Exploring The Riemann Zeta Function, I Stumbled Upon Some Odd Graphs Which I can't Understand

First, I am not an expert mathematician at all, but I was just curious to explore and understand this topic a little better. I might had some mistakes along the way so if you find any, please point ...
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1answer
102 views

Why are there so many equations for the Riemann zeta function and how do you go about calculating it when it times to actually crunch some numbers

For example if you look at this graph the real and imaginary parts along the critical line $x$ is plugged into what equation? It can't be the normal function: $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{...
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Possible progress on the Riemann hypothesis?

Let $\pi(x)$ be the number of primes not exceeding $x$ and $Li(x) = $. Define $Li(x)=\lim_{\epsilon \rightarrow 0^{+}}\Big(\int_{0}^{1-\epsilon} + \int_{1+\epsilon}^{x}\Big) \frac{dt}{\log t} \mathrm{...
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1answer
139 views

Why is this not a valid argument for the Riemann hypothesis?

I am not a trained mathematician, I just enjoy playing around. The Riemann zeta function with the famous Riemann hypothesis has always fascinated me, and as an amateur it's fun to try to manipulate ...
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A probably wrong proof of the Riemann Hypothesis, but where is the mistake?

There has been a paper doing rounds on Facebook for the past several days, claiming a proof of the Riemann hypothesis. I feel sure that the argument is flawed, but can't see where exactly. It goes as ...
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1answer
41 views

How to describe the error of the asymptotic approximation of the prime counting function

I've been reading about this asymptotic approximation of the prime counting function $\pi(x)$:$$π(x)=Li(x)+O(\sqrt{x}\ log(x))$$ What does this tell me about the error of this approximation? If the ...
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2answers
167 views

Riemann hypothesis and prime distribution

What exactly does the Riemann hypothesis imply for the prime numbers? Since the explicit formula is independent of the Riemann hypothesis, what would it actually mean for the primes if all the ...
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61 views

Decomposition of Non-Trivial Zeroes of Riemann Zeta Function (Product of Integrals)

In my search for a solution to the Riemann Hypothesis, I entered the first non-trivial zero to the Zeta function into the product of integrals definition; specifically the Gamma Function. Using ...
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1answer
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Can it be proven that the left hand side of Robin's Inequality is bounded by some function?

Based on the shape of the curve shown in this post and the possible relationship shown here, I am willing to venture a conjecture: $$\frac{\sigma (n)}{e^{\gamma} n \log \log n}<1-\frac{0.242692}{\...
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1answer
111 views

Questions on a formula for the Mertens function

The Mertens function $M(x)$ is defined as follows. (1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)$ I've noticed the Merten's function can also be evaluated as follows which is related to OEIS entry ...
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What is the size of the kth Superabundant Number?

Utilizing the data generated by T.D. Noe for the first $10^6$ terms of A004394, a plot of $k$ against $log_{10}(S(k))$ where S(K) is the $k^{th}$ Superabundant Number shows a strong linear ...
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What approach provides the largest known verification of the Riemann Hypothesis?

The Riemann Hypothesis is true if and only if Robin's Inequality is true for all n>5040. It has also been shown by Akbary and Friggstad that the smallest counterexample greater than 5040, if it exists,...
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93 views

zeta of imaginary numbers

So the riemman hypothesis was dealing with the real part of the s be 1/2 and I was wondering if it is possible to take zeta of imaginary numbers and if it is, how would you proceed to do so?
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Questions on Riemann Hypothesis prediction and explicit formula for $\hat{\Phi}(x)=\sum\limits_{n\le x}\frac{\phi(n)}{n}$

I've primarily been able to find information on the the summatory Euler totient function $\Phi(x)$ defined in (1) below, but I'm more interested in the $\hat{\Phi}(x)$ defined in (2) below. (1) $\...
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2answers
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 ...
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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 ...
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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 ...
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1answer
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 $\zeta(y)$?
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1answer
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, \ ...
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47 views

Some inequalities similar to Lagarias inequality

I have found some inequalities which seem to be equivalent to Lagarias inequality. It would be very nice if someone takes the time to proofread this in detail and give constructive feedback. Thanks ...
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130 views

RH & explicit formula for the number of primes $ \le x$

Does the RH have to be true in order for Riemann's explicit formula for the number of primes <= x to hold? The formula is (copied from wikipedia:https://en.wikipedia.org/wiki/Explicit_formulae_(L-...

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