All Questions
80 questions
157
votes
7
answers
74k
views
Consequences of the Riemann hypothesis
I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...
50
votes
5
answers
3k
views
Motivated account of the prime number theorem and related topics
Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
- 482
49
votes
3
answers
6k
views
The Hardy Z-function and failure of the Riemann hypothesis
David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...
- 13.1k
45
votes
4
answers
8k
views
Why is so much work done on numerical verification of the Riemann Hypothesis?
I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes.
I don't mean to ask a stupid question, ...
- 5,146
38
votes
2
answers
13k
views
What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
38
votes
4
answers
6k
views
Modular forms and the Riemann Hypothesis
Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $...
- 889
29
votes
4
answers
5k
views
Good uses of Siegel zeros?
The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...
- 291
28
votes
2
answers
3k
views
What are some consequences of zero free strip of the Riemann zeta function?
A weaker version of the Riemann hypothesis is the claim that if $\zeta(s) = 0$ then $Re(s) \leq 1 - h$ for some constant $h> 0$. What would the consequences be of a result of this type?
- 3,625
26
votes
5
answers
3k
views
Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?
The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here:
Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...
- 4,169
24
votes
1
answer
2k
views
How good is "almost all" when it comes to the Riemann Hypothesis?
Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part ...
- 231
18
votes
1
answer
2k
views
A question about Speiser's 1934 result on the Riemann hypothesis
A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) ...
- 243
17
votes
2
answers
2k
views
Is this equivalent to RH - Riemann hypothesis?
$$\pi = 3\prod_{\zeta(1/2+it) = 0}\frac{9+4t^2}{1+4t^2}\iff\text{RH is true}.$$
- 1,305
17
votes
3
answers
3k
views
Largest known zero of the Riemann zeta function
Numerical calculations on the zeroes of the Riemann zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) ...
- 16.2k
15
votes
3
answers
3k
views
On Robin's criterion for RH [closed]
\begin{equation}
\sigma(n) < e^\gamma n \log \log n
\end{equation}
In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984)....
- 731
15
votes
1
answer
1k
views
Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ?
Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ...
- 26k
14
votes
2
answers
1k
views
Effective Chebotarev without Artin's conjecture
$\DeclareMathOperator\Frob{Frob}$Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form
of the effective Chebotarev density theorem page 143, and prove it assuming both ...
- 26k
14
votes
1
answer
2k
views
Exceptional zeros and Liouville's $\lambda$ function
This originated from an textbook exercise (recently posted to math.stackexchange
https://math.stackexchange.com/questions/62883/quadratic-characters-and-liouvilles-function
with no success) but I ...
- 11.1k
14
votes
1
answer
1k
views
The location of the zeros of the "new" function $\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$
Defining the Psi function as $$\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$$ and by studying, from a numerical point of view, the location of the zeros in the complex plane up to $0<\Im(s)<...
- 161
12
votes
2
answers
1k
views
Prime differences and zero multiplicity
Concerning gaps between consecutive primes, Paul Erdős conjectured that:
$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$
Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
- 232
10
votes
1
answer
1k
views
How does Riemann hypothesis implies estimates?
In Iwaniec, Luo and Sarnak article (precisely (4.23)), it is said that GRH for $L(s, \mathrm{sym}^2(f))$, for a holomorphic cusp newform $f$ of level $N$ and weight $k$, implies
$$\sum_{p \nmid N} \...
- 551
10
votes
0
answers
570
views
Bounding $1/\zeta(s)$ given RH
Let $T\geq 0$. Assume RH(T+100), that is, assume that all non-trivial zeros $\rho$ of the Riemann zeta function with $|\Im(\rho)|\leq T+100$ satisfy $\Re(\rho)=1/2$. Can we then give a good upper ...
- 20.2k
10
votes
0
answers
2k
views
Questions on de Branges' work on the Riemann hypothesis
According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...
- 163
9
votes
2
answers
839
views
Reference and best bounds of $\sum_{n\leq x}\frac{\mu(n)}{n}$
Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me. Does ...
- 1,692
9
votes
0
answers
419
views
Numerical Evidence for Grand Riemann Hypothesis?
Let $L(s)$ be an $L$-function coming from Hecke characters or automorphic forms (e.g. modular form on GL(2), Maass form on GL(2), and higher-rank analogues).
Is there any numerical evidence for ...
- 3,804
8
votes
2
answers
2k
views
Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?
Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple.
I have often heard of the statement that the SZC is stronger than the Riemann ...
- 149
8
votes
1
answer
868
views
A question on an equivalence of RH
In page 6, RH Equivalence 5.3. An equivalence of the Riemann Hypothesis says that
$$\sum_{\rho} \frac{1}{|\rho|^2} =\sum_{\rho} \frac{1}{\rho (1{-}\rho)}= 2 + \gamma - \log 4\pi$$
where $\rho$ is ...
- 365
8
votes
1
answer
1k
views
A reformulation of the Riemann Hypothesis
I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.
Let's define $R(x,...
- 1,692
7
votes
1
answer
1k
views
Heuristic for Montgomery's conjecture
This is my third question on this site regarding Montgomery's conjecture -- and I apologize
if this is too much -- but I am still not understanding well why this conjecture is believed to be true.
...
- 26k
7
votes
1
answer
811
views
Is there a collection of evidence and heuristic arguments against the Riemann hypothesis? [closed]
There is undoubtedly an overwhelming collection of evidence for the Riemann hypothesis. However, is there any evidence against it ?
7
votes
1
answer
769
views
$\mathit{NP}$-hard statements which are $\mathit{NP}$-complete under the Riemann Hypothesis
$\newcommand\NP{\mathit{NP}}\newcommand\SAT{\mathit{SAT}}\newcommand\CH{\mathit{CH}}\newcommand\PSPACE{\mathit{PSPACE}}$Are there $\NP$-hard problems which are $\NP$-complete under the Riemann ...
- 13.9k
6
votes
1
answer
433
views
Asymptotic behavior of partial sums of Dirichlet series
Consider the Dirichlet series:
$$\sum_{n \geq 1} \frac{a_n}{n^s} = \frac{\zeta(s+1/3)}{\zeta(s)}$$
where $\zeta(s)$ is the Riemann zeta function.
Question: Assuming the Riemann Hypothesis (RH), how ...
- 611
6
votes
1
answer
2k
views
The connection between the Weil conjectures and Ramanujan's conjecture
I'm writing an essay about Ramanujan's conjecture and have some questions:
1 How is Ramanujan's conjecture connected with the Weil conjectures?
2 How could Ramanujan's conjecture be assumed true or ...
user108974
6
votes
0
answers
177
views
Is there a conjectured uniform Lindelof hypothesis for Hurwitz zeta functions
Consider $\zeta(s, a) = \sum_{n=1}^{\infty} (n+a)^{-s}$ (alternatively, consider its functional equation Dirichlet series $\sum_{n=1}e(a n) n^{-s}$). What is the expected growth-rate of $\zeta(1/2 + ...
- 1,243
5
votes
2
answers
1k
views
A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis
Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...
- 1,018
5
votes
1
answer
2k
views
Does $ M(x)=O(\sqrt{x}) $ if and only if the De Bruijn-Newman constant is negative?
The Riemann hypothesis is equivalent to the assertion that the De Bruijn-Newman constant $ \Lambda $ , as defined in https://www.sciencedirect.com/science/article/pii/S0001870809001133/pdf?md5=...
- 6,984
5
votes
2
answers
1k
views
Riemann Hypothesis and Euler product
It is conjecture that under certain conditions a L-function satisfies RH.
Among these conditions there is the necessity for the L-function to have an Euler product. (Some L-functions with a functional ...
- 1,199
5
votes
2
answers
872
views
Exact formula for partial sums of Liouville function $L(n)$ (OEIS sequence A002819)
I am wondering if it is possible to get a useful exact formula, or at least some useful asymptotics, for the partial sums of the Liouville function (OEIS sequence A002819)
$$L(n)=\sum_{k=1}^n \lambda(...
- 3,098
5
votes
1
answer
391
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 ...
- 315
5
votes
0
answers
241
views
Estimating $\left| \sum_{n \leq x, n \neq p} \frac{\Lambda(n)}{n^{\sigma + it} \log n} \frac{\log x/n}{ \log n} \right|$ on RH
I am having some issue verifying Lemma 2 of K. Soundarajan's paper Moments of the Riemann Zeta function. It states the following:
Assume RH. Let $T \leq t \leq 2T$, $2 \leq x \leq T^2$ and $\sigma \...
- 51
4
votes
2
answers
2k
views
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)?
- 1,305
4
votes
2
answers
423
views
Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH
In page 197, Equivalents of the Riemann Hypothesis Vol 1, the following statement caught my eye
There is an editorial comment in [102] that includes an observation by
the GCHQ Problem Solving Group. ...
- 313
4
votes
1
answer
629
views
Is $\sum_{n\leq x}{z^{\Omega(n)}} = O(x^{\frac12 + \varepsilon})$ equivalent to the Riemann hypothesis for all roots of unity $z\neq1$?
$\Omega(n)$ is the number of prime divisors of $n$, counted with multiplicity.
For $z=-1$, $z^{\Omega(n)} = \lambda(n)$ is the Liouville function, and it's known that $\sum_{n\leq x}\lambda(n) = O(n^{\...
- 3,319
4
votes
1
answer
928
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 ...
- 790
4
votes
1
answer
333
views
Double sum over zeros of Riemann zeta-function
In a paper by Saffari and Vaughan there appears a complicated-looking double sum
$$\Sigma_1=\sum_{\rho_1}\sum_{\rho_2}\frac{(1+\theta)^{\rho_1}-1}{\rho_1}\cdot \frac{(1+\theta)^{\bar{\rho_2}}-1}{\bar{\...
- 465
4
votes
1
answer
621
views
Seek a reference for Theorem 1.2 on p. 6 of the Riemann Hypothesis sourcebook of Borwein et. al
The book "The Riemann Hypothesis, A Resource for the Afficionado and Virtuoso Alike", Borwein, Choi, Rooney, Weirathmueller, Eds., states on its page six the following theorem (Theorem 1.2):
...
- 51
4
votes
0
answers
884
views
Has any professional mathematician ever attempted to solve the Riemann hypothesis using only number theory? [closed]
I have often heard people saying that ''all attempts at solving the Riemann hypothesis using number theory have failed.'' But in the literature, i cannot find any failed ''purely number-theoretic'' ...
user148542
4
votes
0
answers
624
views
Is there a hidden symmetry in the prime numbers distribution?
Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
Let'...
- 6,984
3
votes
1
answer
436
views
Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?
Robin's inequality
$$\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 ...
- 13.9k
3
votes
1
answer
276
views
Almost-Primes in Short Intervals
Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number ...
- 14.6k
3
votes
1
answer
309
views
Zeros of the derivative of $\xi$
In his paper on Zeros of derivatives of Riemann $\xi$-function on critical line Brian Conrey mention that
It can be shown that the Riemann hypothesis implies that all zeros of $\xi (s)$, the ...
- 61