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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Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple. I have often heard of the statement that the SZC is stronger than the Riemann ...
user257465's user avatar
5 votes
2 answers
774 views

Exact formula for partial sums of Liouville function $L(n)$ (OEIS sequence A002819)

I am wondering if it is possible to get a useful exact formula, or at least some useful asymptotics, for the partial sums of the Liouville function (OEIS sequence A002819) $$L(n)=\sum_{k=1}^n \lambda(...
Vincent Granville's user avatar
7 votes
0 answers
243 views

Computability assertions for Riemann zeta zeros

While looking for information about the Riemann zeta function, I kept running into the claim that there is an algorithm to decide whether or not a zero of the function is off the half-line. Is this ...
Pace Nielsen's user avatar
14 votes
1 answer
1k views

Normal numbers, Liouville function, and the Riemann Hypothesis

This is a question about whether or not some number $\lambda^*$ is normal in base 2. More specifically, I am wondering if $\lambda^*$ is not normal. Proving it is normal would be next to impossible, ...
Vincent Granville's user avatar
2 votes
0 answers
126 views

On primes of specified length and bit pattern

Denote $P(n,k)$ to be the number of primes between $2^n$ and $2^{n+1}-1$ having $k$ number of $1$s in its binary expansion between the $n+1$th binary digit and the least which is always $1$ if $n>1$...
Turbo's user avatar
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-4 votes
1 answer
390 views

Scaled Riemann zeta function with no zero in the critical strip

Update: I added $exp[i\theta_k(s)]$ in the definition of $\eta^*(s)$ to address some critical convergence issues. Thanks for the contributors who pointed to these issues. Prime numbers are denoted as $...
Vincent Granville's user avatar
2 votes
1 answer
435 views

Truncated Euler products, Dirichlet eta function, and convergence issues

Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as $$W(\sigma,...
Vincent Granville's user avatar
10 votes
1 answer
1k views

The (current) obstructions for a cohomological interpretation of the Riemann zeta function

I am interested in the idea of a cohomological interpretation of the Riemann hypothesis (suggested by Deninger/Connes). I am a beginner in étale cohomology, and I would like to ask the following ...
kindasorta's user avatar
  • 1,473
1 vote
0 answers
149 views

Prove that: $\sum _{c=1}^n \sum _{b=1}^n \sum _{a=1}^n \left(\left([b|c][b|a]\frac{\mu(b)b}{a}\right)-\frac{1}{a b\sqrt{c}}\right)<H_n+n$

In the OEIS there is the quote from Lowell Schoenfeld that the Riemann hypothesis is equivalent to: $$|\psi(n) - n| < \sqrt{n} \log^2(n)$$ From the Euler Maclaurin formula one gets: $$\sum _{c=1}^n ...
Mats Granvik's user avatar
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5 votes
1 answer
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Can the Lagarias inequality be written as a "kernel inequality"?

The Lagarias inequality, which is equivalent to the Riemann hypothesis, is: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n) =:L(n)$$ for all natural numbers $n$, where $\sigma=$ sum of divisors, $H_n=n$-th ...
mathoverflowUser's user avatar
4 votes
0 answers
222 views

To which value does this infinite sum of power series coefficients converge?

Context: In this and this paper, J. Arias de Reyna shows that the RH follows when: $$1.2663935... \le \sum_{n=1}^\infty A_n^2 \le 1.2723669...$$ where $A_n$ is the coefficient in the following power ...
Agno's user avatar
  • 4,179
0 votes
1 answer
247 views

How differently would we model the distribution of primes if prime gap is larger?

Cramer's conjecture based on his random model provides prime gaps are bound by $O(\log^2p_n)$ where the gap is between $(n+1)$th and $n$th prime. How differently would primes be modeled if gaps of $O(...
Turbo's user avatar
  • 13.7k
0 votes
0 answers
272 views

Axiom of determinacy as setting for studying rigs with $\operatorname{Aut}(\mathcal{M})\cong\operatorname {Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$?

I read years ago in the great French book "l'aventure des nombres" (Gilles Godefroy, Odile Jacob, 1997) that the axiom of determinacy was incompatible with the axiom of choice. The latter ...
Sylvain JULIEN's user avatar
2 votes
0 answers
327 views

Mertens Bound and the Riemann Hypothesis

Let $M(x)$ denote the Mertens function ($M(x)=\sum_{i=1}^{x}\mu(i)$ where $\mu(i)$ is the Möbius function) and let $\Lambda(i)$ denote the Mangoldt function ($\Lambda(i)$ equals $\log(p)$ if $i=p^{m}$ ...
Sourangshu Ghosh's user avatar
3 votes
0 answers
289 views

Lower bound of the modulus $|\eta(s)|$ of the Dirichlet Eta function if $0.6 < \Re(s) < 0.9$

Let $s=\sigma + it$, with $0.6 < \sigma < 1$ and $\sigma=\Re(s)$. I am trying to get good enough approximations for $\eta(s)$, hoping something useful might come out of it. I stumbled upon a ...
Vincent Granville's user avatar
4 votes
0 answers
508 views

On RH in the Clay Institute list

As everybody knows, the Riemann Hypothesis is one of the problems of the millenium raised by the Clay Institute. Looking at the "official formulation" of various problems, say for instance ...
Bazin's user avatar
  • 15.1k
4 votes
1 answer
160 views

A Hadamard product representation for Keiper's $\tau$-function?

In this paper J.B. Keiper defined the following function: $$\tau_k = \sum_{j=1}^k (-1)^j\,{k-1 \choose j-1} \sigma_{j+1} \qquad k \ge 1, k \in \mathbb{N} \tag{1}$$ where $\displaystyle\sigma_r = \sum_{...
Agno's user avatar
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0 votes
1 answer
346 views

On some property of the zeros of $\zeta(s)$ in the complex plane

This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well ...
Vincent Granville's user avatar
29 votes
1 answer
2k views

Riemann's attempts to prove RH

I read somewhere that Riemann believed he could find a representation of the zeta function that would allow him to show that all the non-trivial zeros of the zeta function lie on the critical line. I ...
Mustafa Said's user avatar
  • 3,679
1 vote
0 answers
564 views

Why didn't Robin prove the Riemann Hypothesis?

I'm reading Robin's paper, ''Grandes valeurs de la fonction somme des diviseurs et hypoth`ese de Riemann,'' J. Math. Pures Appl. (9) 63 (1984). In particular, Lemma 5 states that $\prod_{p\leq P} (1-p^...
Q_p's user avatar
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23 votes
1 answer
3k views

More mysteries about the zeros of the Riemann zeta function

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$. Update on 1/5/2020: I added the section "more interesting ...
Vincent Granville's user avatar
3 votes
1 answer
428 views

Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?

Robin's inequality $$\sigma_1(n)<e^\gamma n\log\log n$$ at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...
Turbo's user avatar
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2 votes
4 answers
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Prove that the real part of this limit converges to $\frac{1}{2}$

Let $s= 1/3 + 14i$. Prove that the real part of this limit converges to $\frac{1}{2}$: $$ \Re\lim_{n \rightarrow \infty} \left( \left[ 1- \left( \sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{...
Mats Granvik's user avatar
  • 1,133
2 votes
0 answers
590 views

An interesting sequence of numbers arising from the Riemann hypothesis

A very good coincidence occurred today with me. While just plotting random functions in Mathematica, I entered this command: ...
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0 votes
1 answer
353 views

Where can I find the problem by Lagarias?

Jeffrey Lagarias proved, unconditionally, that: $$ \sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1 $$ This was posed as a problem in: J. C. Lagarias, Problem 10949: A generous bound for divisor ...
The Company's user avatar
3 votes
1 answer
192 views

Error term for the summatory function of $k$-free numbers indicator and RH

I started to read this preprint: https://arxiv.org/abs/2010.03696 In it, the author states that $\sum_{n\leq x}\mu_{k}(n)=\zeta(k)^{-1}x+O(x^{1/k})$ and that under RH, the exponent in the error term ...
Sylvain JULIEN's user avatar
2 votes
1 answer
614 views

$\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?

In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf) we find the following result: If the Riemann hypothesis is true ...
The Company's user avatar
8 votes
1 answer
585 views

Riemann hypothesis for exponential sum

Recently I've heard about the Riemann hypothesis for one-variable exponential sums, which states as For a polynomial $f\in\mathbb{F}_{p^k}[x]$ of degree $d$ and a character $\chi$ of $(\mathbb{F}_{p^...
Jugendtraum's user avatar
3 votes
3 answers
486 views

Show that the ratio of limits converges to the nearest Riemann zeta zero except when the ratio is a singularity

Let $h(s,n)$ be: $$h(s,n)=\lim_{c\to 1} \, \frac{(-1)^{n-2}}{(n-2)!}\zeta (c)^{n-2} \sum _{k=1}^{n-1} \frac{(-1)^{k-1} \binom{n-2}{k-1}}{\zeta ((c-1) (k-1)+s)}$$ and let $g(s,n)$ be: $$g(s,n)=\lim_{c\...
Mats Granvik's user avatar
  • 1,133
1 vote
0 answers
171 views

Is $T(n)=\sum_{k=1}^{n}\frac{\lambda(k)\Lambda(k)}{k} \geq 0$ and what is the upper bound of $T(x)=\sum_{n\leq x} \lambda(n)\Lambda(n) $?

Let $\Lambda(n)$ denote the von Mangoldt function: $\Lambda(n)=\log p$ when $n=p^e$ is a prime power ($e\ge 1$) and $\Lambda(n)=0$ otherwise. and $\lambda(n)$ be Liouville Function, , I'm interested ...
zeraoulia rafik's user avatar
6 votes
0 answers
174 views

Is there a conjectured uniform Lindelof hypothesis for Hurwitz zeta functions

Consider $\zeta(s, a) = \sum_{n=1}^{\infty} (n+a)^{-s}$ (alternatively, consider its functional equation Dirichlet series $\sum_{n=1}e(a n) n^{-s}$). What is the expected growth-rate of $\zeta(1/2 + ...
Ralph Furman's user avatar
  • 1,243
7 votes
1 answer
787 views

Is there a collection of evidence and heuristic arguments against the Riemann hypothesis? [closed]

There is undoubtedly an overwhelming collection of evidence for the Riemann hypothesis. However, is there any evidence against it ?
12 votes
0 answers
538 views

On a revised quantum Riemann hypothesis

This post provides a revision of the disproved quantum Riemann hypothesis proposed 2 years ago in this post, where you can refer to have more details about the motivations, the notations and the ...
Sebastien Palcoux's user avatar
10 votes
0 answers
539 views

Toward a cyclotomic Riemann hypothesis

For an integer $n \ge 3$, consider the function $$u(n) = \frac{\sigma(n)}{n \log \log n}$$ with $\sigma$ the divisor function. Now consider the sequence (bounded below and decreasing) $$v_n = \sup_{m&...
Sebastien Palcoux's user avatar
0 votes
2 answers
930 views

An observation on the Riemann $\xi$ function

Anyone seen these conclusions about the Riemann xi function or see any errors here? With $\xi(s)$ the entire Landau Riemann xi function defined by the Hadamard product representation $$\xi(s) = (1/...
Tom Copeland's user avatar
  • 9,937
5 votes
3 answers
1k views

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 $$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$ ...
Mats Granvik's user avatar
  • 1,133
0 votes
0 answers
175 views

On the asymptotics of the Chebyshev psi function

Denote by $\psi(x)$ the Chebyshev psi function over prime powers. Assuming the RH, it can be shown via the Riemann explicit formula that $$\psi(x)-x \ll √x |\sum_{|\gamma| < T} \frac{1}{\rho} | +...
user156584's user avatar
2 votes
0 answers
529 views

Explicit formula for $n$th prime in terms of Riemann zeros:

We all know there exists an explicit Formula for prime counting function in terms of Riemann zeros. I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros? Or any other ...
bambi's user avatar
  • 375
5 votes
1 answer
367 views

Proving a specific case of Robin's Inequality

Edit: It turns out that this is equivalent to the RH which gives the idea that this might a a little difficult to show. As such we could consider an even simpler case in which the number $n$ is ...
wjmccann's user avatar
  • 315
7 votes
0 answers
343 views

What is known about "almost orthogonal vectors"?

Motivation: Suppose we have a kernel $k(a,b)$ defined over the natural numbers. Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...
user avatar
0 votes
1 answer
517 views

On provability of false statements in constructive mathematics [closed]

Lagarias "elementary" reformulation of Robin's theorem is that $$\mathrm{RH}\iff\sigma(n)\leq H_n+e^{H_n}\log(H_n)$$ holds for every $n\geq 1$, where $\sigma(n)$ is the sum of divisors function and $...
VS.'s user avatar
  • 1,816
4 votes
0 answers
799 views

Has any professional mathematician ever attempted to solve the Riemann hypothesis using only number theory? [closed]

I have often heard people saying that ''all attempts at solving the Riemann hypothesis using number theory have failed.'' But in the literature, i cannot find any failed ''purely number-theoretic'' ...
user avatar
6 votes
1 answer
602 views

Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers

Let $a(n)$ be the Dirichlet inverse of the Euler totient function: $$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$ and let the matrix $T(n,k)$ be: $$T(n,k)=a(\gcd(n,k)) \tag{2}$$ It has been ...
Mats Granvik's user avatar
  • 1,133
3 votes
0 answers
178 views

Largest observed value of $S(t)$

Let $S(t)$ be the deviation of the number of zeros of the Riemann zeta function up to height $t$ from the expectation. What is the largest observed value of $S(t)$ today? Here is a quote from a ...
Farzad Aryan's user avatar
10 votes
0 answers
533 views

Bounding $1/\zeta(s)$ given RH

Let $T\geq 0$. Assume RH(T+100), that is, assume that all non-trivial zeros $\rho$ of the Riemann zeta function with $|\Im(\rho)|\leq T+100$ satisfy $\Re(\rho)=1/2$. Can we then give a good upper ...
H A Helfgott's user avatar
  • 19.3k
9 votes
0 answers
494 views

On Riesz criteria for Riemann hypothesis:

Marcel Riesz defined a function : $R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$ The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$) For any $\varepsilon$ We have ...
TPC's user avatar
  • 690
4 votes
1 answer
843 views

On a possible equivalent of Riemann hypothesis

I've read in a Bombieri's paper on official problem statement of Riemann hypothesis for Clay Math institute's millennium problems, a statement and what I understood of it is the following : The ...
TPC's user avatar
  • 690
21 votes
1 answer
2k views

Is the Hilbert–Pólya intuition vindicated in the function field case?

The Hilbert–Pólya conjecture is the name given to the idea that the "reason" or "explanation" for the collinearity of the non-trivial zeros of the Riemann zeta function $\zeta(s)$ is that they are the ...
Timothy Chow's user avatar
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2 votes
1 answer
193 views

Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in the spirit of a formula from an article

In [1] the authors present an equivalence to the Riemann hypothesis that is the Theorem 6.2. On the other hand I know a statement from [2], in English this is the article Andrew Granville and Greg ...
user142929's user avatar
2 votes
0 answers
155 views

On the connection between $\pi(x)-Li(x)$ and $\theta(x)-x$

Let $\pi(x)$ be the number of primes $p$ not exceeding $x, \theta(x) = \sum_{p\leq x} \log p$ and $Li(x)$ be the logarithmic integral. Is it true that $$\pi(x)-Li(x) = \theta(x) - x + O(x^{1/2}\log^{...
Q_p's user avatar
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