All Questions
Tagged with analytic-number-theory riemann-hypothesis
27 questions with no upvoted or accepted answers
10
votes
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570
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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
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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
0
answers
414
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From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
- 3,098
9
votes
0
answers
419
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Numerical Evidence for Grand Riemann Hypothesis?
Let $L(s)$ be an $L$-function coming from Hecke characters or automorphic forms (e.g. modular form on GL(2), Maass form on GL(2), and higher-rank analogues).
Is there any numerical evidence for ...
- 3,804
6
votes
0
answers
177
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Is there a conjectured uniform Lindelof hypothesis for Hurwitz zeta functions
Consider $\zeta(s, a) = \sum_{n=1}^{\infty} (n+a)^{-s}$ (alternatively, consider its functional equation Dirichlet series $\sum_{n=1}e(a n) n^{-s}$). What is the expected growth-rate of $\zeta(1/2 + ...
- 1,243
5
votes
0
answers
241
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Estimating $\left| \sum_{n \leq x, n \neq p} \frac{\Lambda(n)}{n^{\sigma + it} \log n} \frac{\log x/n}{ \log n} \right|$ on RH
I am having some issue verifying Lemma 2 of K. Soundarajan's paper Moments of the Riemann Zeta function. It states the following:
Assume RH. Let $T \leq t \leq 2T$, $2 \leq x \leq T^2$ and $\sigma \...
- 51
4
votes
0
answers
624
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Is there a hidden symmetry in the prime numbers distribution?
Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
Let'...
- 6,984
3
votes
0
answers
4k
views
Intuition for the bias of the partial sums of the Liouville function
It's a well known result that the Dirichlet series of the Liouville function $ \lambda(n) $ is given by
$$ \sum_{k=1}^{\infty} \frac{\lambda(k)}{k^s} = \frac{\zeta(2s)}{\zeta(s)} $$
If we use Perron's ...
- 291
3
votes
0
answers
128
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What is the smallest sequence $a_k$ with nondecreasing $\frac{\sigma(a_k)-H_{a_k}}{\exp(H_{a_k})\log(H_{a_k})}$?
This is inspired by the Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH , an answer and some comments there.
For $n\geqslant2$ denote
$$
L(n):=\frac{\sigma(n)-H_n}{\exp(...
- 18.4k
3
votes
0
answers
315
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 ...
- 3,098
2
votes
0
answers
154
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Translation of an article of Littlewood
I want to read the English translation of an article of Littlwood titled "Quelques conséquences de l'hypothese que la fonction $ζ (s)$ de Riemann n'a pas de zéros dans le demi-plan $ℜs> 1/ 2$.&...
- 253
2
votes
0
answers
165
views
Tools to prove lower bounds in analytic number theory
Perron's formula and related methods are used to relate statements such as the Riemann hypothesis to upper bounds of functions occurring in analytic number theory. For example, Perron's formula is ...
- 1,018
2
votes
0
answers
128
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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 $n>1$...
- 13.9k
2
votes
0
answers
357
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 $i=p^{m}$ ...
- 177
2
votes
0
answers
537
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 ...
- 375
2
votes
0
answers
158
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
$$\pi(x)-Li(x) = \theta(x) - x + O(x^{1/2}\log^{...
- 1,019
2
votes
0
answers
270
views
On a sequence of L-functions having same zeros in critical strip and GRH
I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...
- 1,199
1
vote
0
answers
113
views
Are there any known statistics on the sign of the Stieltjes Constants?
The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
- 2,689
1
vote
0
answers
181
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Behavior of Dirichlet L-functions at the edge of the critical strip
Given a Dirichlet L-function $L(\chi, s)$ of a primitive character $\chi$, what is the asymptotic behavior of $L(\chi, 1+it)$ for real $t$? I am looking for as many answers for the same question. This ...
- 565
1
vote
0
answers
169
views
Normal numbers and law of the iterated logarithm
If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...
- 3,098
1
vote
0
answers
341
views
Riemann Explicit Formula
I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula:
$$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...
- 547
0
votes
0
answers
268
views
Do plots $(5)$ and $(6)$ go to infinity not at the same rate but at similar rates?
The following has been proven by joriki and GH from MO (see here): assuming that $n>1$, then the von Mangoldt function
$$
\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\...
- 1,183
0
votes
0
answers
157
views
Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?
In order to see what happens when taking the functional equation in this form:
$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$
$$\xi(s) = \xi(1 - s)$$
$$\pi^{-s/2}\ \Gamma\left(\...
- 1,183
0
votes
0
answers
603
views
Why didn't Robin prove the Riemann Hypothesis?
I'm reading Robin's paper, ''Grandes valeurs de la fonction somme des diviseurs et hypoth`ese de Riemann,'' J. Math. Pures Appl. (9) 63 (1984).
In particular, Lemma 5 states that
$\prod_{p\leq P} (1-p^...
- 1,019
0
votes
0
answers
185
views
On the asymptotics of the Chebyshev psi function
Denote by $\psi(x)$ the Chebyshev psi function over prime powers. Assuming the RH, it can be shown via the Riemann explicit formula that
$$\psi(x)-x \ll √x |\sum_{|\gamma| < T} \frac{1}{\rho} | +...
- 41
0
votes
0
answers
112
views
Explicit formula for k-central numbers
Given a positive integer $ n $ and assuming Goldbach's conjecture, let $r_{0}(n)$ denote the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are primes. Let $k_{0}(n)$ denote 'the ...
- 6,984
-3
votes
0
answers
136
views
Approximation on Dirichlet's arithmetic progression by means of central limit theorem
In this video lecture on
Number theory over function fields taught by Will Sawin
is presented a 'conceptional' reason for error estimation
$\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \}
=\frac{1}...
- 619