# Questions tagged [riemann-hypothesis]

Questions about the famous conjecture from Riemann saying that the non-trivial zeroes of the Riemann Zeta function all lie on the so-called critical line $\Re(s)=\dfrac{1}{2}$, its various generalizations and the different approaches towards its solution.

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### What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?

From Terry Tao's post here there is the statement: "Conversely, if one can somehow establish a bound of the form $$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$ ...
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### Books on complex analysis for self learning that includes the Riemann zeta function?

I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following: Analytic number theory : the connection between complex analysis and ...
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### Chebyshev's bias-conjecture and the Riemann Hypothesis

Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?
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### Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?

The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that $$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes ...
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### An integral involving the argument of the Gamma function and the Riemann Hypothesis

Evaluate $$I=\int_{0}^{\infty} \frac{t\arg \Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$ where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$ Note that $I$ converges ...
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### A computation in Conrey's paper on Riemann zeta function

I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed ...
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### What is the automorphic interpretation of the Weil conjectures over finite fields

I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me. A loose interpretation of the Langland's philosophy is that to any ...
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### Does the Riemann Xi function possess the universality property?

Here is the question.   Does the Riemann Xi function possess the universality property,  or something similar to Voronin 's universality property?  This is the reason why the answer to this question ...
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### Where can i find LeClair's 2007/8 papers on the Riemann Hypothesis?

I'm reading Schumayer's excellent expository article, ''Physics of the Riemann Hypothesis'', and on pp. 21-22 in the article, Schumayer refers to certain two papers of LeClair, published in 2007 and ...
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### On a quantum Riemann Hypothesis

Robin's theorem (1984) states that $$\sigma(n) < e^\gamma n \log \log n$$ for all $n > 5040$ if and only if the Riemann hypothesis is true. Recall that $γ$ is the Euler–Mascheroni ...
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### How many zeta zeros are needed to accurately calculate five digits for π(1000000), where π(x) is the prime counting function?

John Derbyshire in his book PRIME OBSESSION says on page 343: "I’ll round off with a complete calculation of $\pi(1,000,000)$, the number of primes up to one million, using Riemann’s formula -- ...
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### Confusion about Montgomery's Pair Correlation Conjecture

This question will be based roughly on the Bourgade Keating review on Zeta function and eigenvalue asymptotics (BK): https://link.springer.com/chapter/10.1007/978-3-0348-0697-8_4 To set up the ...
### Does $M(x)=O(\sqrt{x})$ if and only if the De Bruijn-Newman constant is negative?
The Riemann hypothesis is equivalent to the assertion that the De Bruijn-Newman constant $\Lambda$ , as defined in https://www.sciencedirect.com/science/article/pii/S0001870809001133/pdf?md5=...