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489 views

Estimate on the prime-counting function $\psi(x)$.

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...
Joël's user avatar
  • 26k
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{\...
Daniel Johnston's user avatar
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): ...
user88693's user avatar
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'' ...
user avatar
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'...
Sylvain JULIEN's user avatar
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 ...
Turbo's user avatar
  • 13.9k
3 votes
3 answers
366 views

zeros of a complex function defined by integers

This is a crosspost from Math.SE. Is there a known increasing sequence of positive integers $\{\textbf{a}\} = a_0<a_1<a_2<.....$ such that all the zeros $z_k$ on $\Re[z]>0$ of the ...
hyportnex's user avatar
  • 179
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 ...
Ofir Gorodetsky's user avatar
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 ...
Tokita Ohma's user avatar
3 votes
3 answers
493 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
  • 1,183
3 votes
1 answer
426 views

On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...
Bertrand's user avatar
  • 1,199
3 votes
1 answer
709 views

Conditional bound on RH for $\Re\left(\sum_{p\leq\sqrt{x}}\frac{(1/2)}{p^{1+2it}}\right)$

I would like to prove that Assume RH. Let $T$ large, $2\leq x \leq T^2$ and $T\leq t \leq 2T$, then $$ \log|\zeta(1/2+it))|\leq \Re\left(\sum_{p\leq x}\frac{1}{p^{1/2+1/\log x+it}}\frac{\log(x/p)}{\...
asd's user avatar
  • 199
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 ...
Ege Erdil's user avatar
  • 291
3 votes
0 answers
128 views

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(...
მამუკა ჯიბლაძე's user avatar
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 ...
Vincent Granville's user avatar
2 votes
1 answer
249 views

Robin's criterion, Goldbach's conjecture and upper bound for $r_{0}(n)$

This question is a follow-up to both About Goldbach's conjecture and Question in Proof of Hardy Ramanujan theorem about $\omega(n) =\sum_{p|n} 1$. Can one derive from Robin's criterion for RH an ...
Sylvain JULIEN's user avatar
2 votes
1 answer
757 views

Does asymptotic Goldbach imply GRH?

It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
Sylvain JULIEN's user avatar
2 votes
1 answer
740 views

Does the Riemann hypothesis predict a bound for this prime-counting function?

Does the Riemann hypothesis predict an upper bound for $$\left|f(x)-\left(\operatorname{li}(x)-\frac{x}{\log x} \right)\right|,\quad x\ge 2\tag{1}$$ where $$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
Steven Clark's user avatar
  • 1,126
2 votes
1 answer
429 views

'Almost all' zeros of the Dirichlet L function lies 'near' the critical line?

Is there a well known result that states that as $t \to \infty$, 'almost all' zeros of any Dirichlet L function $L(s,\chi)$ lie in the region $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$...
Aritro Pathak's user avatar
2 votes
2 answers
475 views

Questions concerning the Fourier analysis of $ nx\ \%\ m$

Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e. $$x \equiv x\ \%\ m\pmod{m}$$ The plots of the functions $f_{nm}(x) = nx\ \%\ m$ exhibit characteristic patterns, especially periods of length $...
Hans-Peter Stricker's user avatar
2 votes
1 answer
212 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
2k views

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 ...
user144435's user avatar
2 votes
1 answer
377 views

Prime Number Theorem on APs under various conjectures

I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states: Unconditionally we have \begin{equation} \pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x \...
Stijn's user avatar
  • 338
2 votes
1 answer
197 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
2 answers
556 views

What is the physical interpretation of the Riemann Hypothesis? [closed]

Some propositions in math can be modeled as a physical system. Has anyone done this for RH?
liu's user avatar
  • 55
2 votes
1 answer
277 views

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 ...
OneTwoOne's user avatar
  • 105
2 votes
0 answers
154 views

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 $ℜs> 1/ 2$.&...
Melon_Musk's user avatar
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 ...
EGME's user avatar
  • 1,018
2 votes
0 answers
128 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
  • 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 $i=p^{m}$ ...
Sourangshu Ghosh's user avatar
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 ...
bambi's user avatar
  • 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^{...
Q_p's user avatar
  • 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 ...
Bertrand's user avatar
  • 1,199
1 vote
2 answers
236 views

Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions

For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, ...
Sylvain JULIEN's user avatar
1 vote
1 answer
558 views

On Robin's criterion for the Riemann Hypothesis

Statement 1 : (Robin) proved that if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\sum \limits_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c ...
Ahmad's user avatar
  • 185
1 vote
1 answer
229 views

Conjectured error term when counting square-free integers

It is well know that the density of positive square-free integers up to $x$ is asymptotically $x/\zeta(2)$. The error term $$ E(x)=\sum_{1\leq n \leq x } \mu(n)^2 -\frac{x}{\zeta(2)} $$ can easily ...
Dr. Pi's user avatar
  • 3,062
1 vote
1 answer
286 views

GRH and the Euler product

Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be ...
edward cornfoot's user avatar
1 vote
1 answer
283 views

Robin's inequality for odd numbers

In this article (Theorem 1.2) there is a proof for Robin's inequality for odd numbers, $\sigma(n)/n< e^{\gamma}\log(\log(n))$ where $\gamma$ is the Euler-Mascheroni constant and $\sigma(n)$ is the ...
Asanovic Tomas's user avatar
1 vote
1 answer
194 views

Asymptotics of cumulative Liouville function under RH versus simple random walk

The expectation values of the 1D simple random walk $S_n$ can be shown to have the asymptotic behavior of $$ \lim_{n\to\infty} \frac{a_n}{n^{1/2}} = \sqrt{\frac{2}{\pi}}, \tag{1}\label{1}$$ with $a_n =...
Raphael J.F. Berger's user avatar
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 $...
Sidharth Ghoshal's user avatar
1 vote
0 answers
181 views

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 ...
edward cornfoot's user avatar
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 ...
Vincent Granville's user avatar
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)...
William Chang's user avatar
0 votes
2 answers
669 views

Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...
Carl Schildkraut's user avatar
0 votes
1 answer
249 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
  • 13.9k
0 votes
1 answer
274 views

A convergence issue [Edited]

Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space $$l^2_{k^{-2}}:=\{z=\{z(k)\}_{k=1}^\infty:\sum\limits_{k=1}^\infty z(k)^2k^{-2}<\infty\}.$$ It is known that for some $x\in l^...
Subhajit Jana's user avatar
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{\...
Mats Granvik's user avatar
  • 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(\...
Mats Granvik's user avatar
  • 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^...
Q_p's user avatar
  • 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} | +...
user156584's user avatar