User roupam ghosh - MathOverflow

Growth of the reciprocal gamma function in the critical strip

2013-04-20T10:30:57Z

I was wondering if there are any results that studied the growth of $\left|\frac{1}{\Gamma(s)}\right|$ where $0 < \Re(s) < 1$ and as $\Im(s) \to \infty$? Any pointers to any results, papers, references will be highly appreciated.

Thanks.

A note by N. A. Carella on zero-free regions

2012-04-12T12:35:18Z

http://arxiv.org/abs/0908.4287 I could not find any reviews for it, but if true its a major claim, because it says that $\Re(\rho) < 21/40$ where $\rho$ is a zeta zero.

My question:

Are there any reviews of this paper, that reject or accept the claims made in this paper? Any references will be highly appreciated.

Euler Mascheroni Constant, Curious?

2012-08-14T06:52:19Z

I found this formula for the Euler-Mascheroni constant $\gamma$.

Just wondering whether such a formula already exists in literature? Also, wanted to know whether there are formulas that converge faster than this?

$$\gamma = \sum_{k = 1}^{\infty} \frac{1}{2^k k} - \sum_{k = 1}^{\infty} \frac{\zeta \left( 2 k + 1 \right)}{2^{2 k} \left( 2 k + 1 \right)} $$

UPDATE: Thanks for your reply quid. I just came across this while doing some calculations with the zeta function. The calculations are a bit too long to be posted, but in short it derives from $$\zeta(s) = \frac{s+1}{2(s-1)} + \frac{s}{8} - \frac{s(s+1)}{2\pi^2}\int_1^\infty \frac{(\tan^{-1}\cot(\pi x))^2}{x^{s+2}}dx$$.

Trying to debunk a claim

2012-06-23T13:48:01Z

Claim: Take any function $f(t) > 0$ for $t > 0$, such that $f(t) \to \infty$ as $t \to \infty$, then for $\sigma > 0$ $$|\zeta(\sigma + it)| = o(f(t))$$

Is there any already existing evidence, like papers or proofs or something that can debunk this?

As far as I know, under the Lindelof hypothesis $$|\zeta(\frac{1}{2} + it)| = o(t^\epsilon)$$ and Littlewood has already proved that under the Riemann hypothesis $$|\zeta(\frac{1}{2} + it)| = o\left(\exp\left(\frac{10\log t}{\log \log t}\right)\right)$$ both of which agree with the argument.

Also I know from this paper at http://arxiv.org/pdf/math/0612106v2.pdf that $$\int_0^T |\zeta(1/2 + it)|^{2k}dt \gg_k T (\log T)^{k^2}$$ which kind of gives me a hint that there must be an obvious lower bound which can probably show that the condition given above for $\zeta(\sigma + it)$ and $f(t)$ is invalid.

Looking for references.

Certain functional equations for the Riemann Zeta function?

2012-05-25T10:15:11Z

Referring to this question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry.

For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ and $\zeta(s+1)$, or $\zeta(s)$, $\zeta(s+1)$, and $\zeta(s+2)$ or in general $\zeta(s)$, $\zeta(s+1)$, $\zeta(s+2)$, ..., $\zeta(s+n)$ for $\Re(s) > 0$ and for $n \in \mathbb{N}$?

My main motivation behind asking this question is I have found such an equation, but I do not know whether such an equation exists in literature. Also, I do not want to appear as if I am promoting my formula here, but rather I am more interested in the works that have been done in such directions.

My formula goes as follows: For any $n \in \mathbb{N}$ and $\Re(s) > 0$ we have $$\frac{1}{s-1} + \sum_{r=1}^n \binom{n}{r} (-1)^r \left(\frac{\zeta(s+r-1)}{s+n-1} + \int_1^\infty\frac{\rho(x)^r}{x^{s+r}}\mathrm{d}x\right) = 0$$ where $\binom{n}{r}$ is the binomial coefficient and $\rho(x)$ is the fractional part of $x$

For example, putting $n=1$ we get the well known identity, $$\frac{1}{s-1} - \frac{\zeta(s)}{s} - \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x = 0$$ putting $n=2$, $$\frac{1}{s-1} - 2\left(\frac{\zeta(s)}{s+1} + \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x\right) + \left(\frac{\zeta(s+1)}{s+1} + \int_1^\infty\frac{\rho(x)^2}{x^{s+2}}\mathrm{d}x\right) = 0$$ and so on...

EDIT: Classic answer by Juan. This question is now solved.

On Robin's criterion for RH

2011-12-25T13:39:30Z

\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).

I could not get a hold of his paper. I am trying to understand how did he derived the inequality. Can anyone can outline the steps, how Robin derived this criterion?

Upper bound for real part of Riemann Zeta function zeros

2010-10-05T08:31:07Z

Hi,

I have been reading about Riemann Zeta function $\zeta(s)$ and have been thinking about it for some time. I did some calculations, and reached a conclusion where $\Re(\rho) \le \log_2(3) - 1$ as $\Im(\rho) \to \infty$ where $\rho$'s are the roots of Riemann Zeta function in the critical strip. Anyways, I know its not the place to discuss claimed proofs and similar stuff, but just to give a background of where I am coming from. So straight to the question.

Is there any similar result regarding upper bound ($< 1$) for the real part of the zeros zeta function as their imaginary parts tend to infinity?

Thanks

On the Nyman-Beurling equivalent form for RH

2011-01-26T07:40:21Z

Now, I am new to functional analysis. So please dont be harsh.

I was going through some papers by Balazard & Saias, Baez-Duarte, etc. that discussed and delved deep into details of approximating the characteristic function $\chi(0,1]$.

Now, what appears to me is, if we can construct a function, using functions of the form $f(x) = \sum c_k \rho(\frac{\theta_k}{x})$ where $\rho(x)$ is the fractional part of $x$ that stay constant in the interval (0,1] and satisfying $\sum c_k \theta_k = 0$, then we have a proof of RH.

But then I saw that they have also discussed the function $g(x) = - \sum \mu(n) \rho(1/nx) = \chi(t)$ which they say holds for every positive $t$ but in the sense of pointwise convergence, and then stated that in the words of Balazard and Saias's paper, "Unfortunately, this feeling is, at least to a certain extent, a mirage"

My question is:

Why is g(x) not the fulfilling the criteria of being that "function" which mathematicians are trying to find to serve as the approximation for $\chi(0,1]$? I would really appreciate any sort of clarification for this characteristic function that is being searched.

Thanks,

The papers that I am reading includes:

The Nyman-Beurling equivalent form for the Riemann hypothesis -- Michel Balazard and Eric Saias

Convergence and the Riemann hypothesis -- Jungseob Lee

A Note on Nyman-Beurling's equivalent formulation of the Riemann hypothesis -- Jean Francois Burnol

A Closure Problem related to the Riemann zeta funtion -- Arne Beurling

New versions of the Nyman-Beurling criterion for the Riemann hypothesis -- Luis Baez Duarte

Curious about a question on zeta zeros?

2010-10-15T03:04:47Z

I have Edwards and Titmarsch books on Riemann zeta function with me. I could not find (maybe I did not read through that carefully), but are there results similar to the form like the one given below:

Is there a non-trivial zero $\rho$ starting from which $(\Im(\rho)\log(2)/2\pi )$ runs only through integer values

I would love any elaboration or links on this topic.

Thanks,
Roupam Ghosh

Good books on Arithmetic Functions ?

2010-01-27T07:48:48Z

As, I was studying the Mobius Mu Function, and Gram Series... I got myself some pretty nice books:

Ribenboim - The New Book of Prime Number Records

Apostol - Introduction to Analytic Number Theory

Niven, Zuckerman, Montgomery - An Introduction to the Theory of Numbers

Inwaniec - Analytic Number theory

All of them dealt with the mobius mu function. But none of them dealt with the subject in details other than giving a few theorems and problems...

So I would like to know, If you guys know of some good books that deal exclusively with Arithmetic functions ?

Is the given expression, monotonically increasing or decreasing with increasing x?

2010-01-21T02:55:14Z

Is the given expression, monotonically increasing or decreasing with increasing x?

$\frac{1}{x \log(x)} \sum_{i=1}^{\infty} \frac{\mu(i)}{i} x^{1/i}$

EDIT: This is the derivative of the prime counting function $\pi(x)$ w.r.t. x, ie., $\frac{d\pi(x)}{dx} \sim \frac{1}{x \log(x)} \sum_{i=1}^{\infty} \frac{\mu(i)}{i} x^{1/i}$ Also, you might check Bruce C. Berndt's "Ramanujan's NoteBooks Part IV" page 123, equation 10.16, which gives = sign instead of $\sim$

A result on prime numbers

2010-01-01T15:20:49Z

Hi everyone

This is my first post... I do mathematics from home... ie., not attached with any institution... I have deduced some results...

$\lim \inf_{n\to\infty} \frac{d_n}{\log p_n} = 0$

and, for constants $A,B$

$\lim_{n\to\infty} \log p_n - \sum_{i=1}^{n-1} \frac{d_i}{p_{i+1}} = A$

$\lim_{n\to\infty} \log p_n - \sum_{i=1}^{n-1} \frac{d_i}{p_{i}} = B$

Where, $p_n$ is the nth prime... $d_n = p_{n+1} - p_n$

My question is: Do you think these results are good ?

Comment by Roupam Ghosh

2013-04-21T02:37:22Z

anton,Alexandre Thanks! :)

Comment by Roupam Ghosh

2012-08-14T14:31:55Z

Yep... pretty close to what I got... Here's the link for others to see... archive.org/stream/actamathematica24lefgoog#page/…

Comment by Roupam Ghosh

2012-08-14T12:47:15Z

Wow... I will look at that paper for sure! :) Thanks.

Comment by Roupam Ghosh

2012-08-14T12:46:36Z

I am using Mathematica, and "Sum[1./(k*2^k) - Zeta[2*k + 1]/((2 k + 1)*2^(2*k)), {k, 1, 100}]" gives me 0.5772156649015329

Comment by Roupam Ghosh

2012-08-14T10:50:39Z

+1 For the links. I am just wanting to know whether this formula is unique and is of any use from a computational perspective.

Comment by Roupam Ghosh

2012-06-23T15:38:48Z

Awesome... Let me probe on that a bit! :)

Comment by Roupam Ghosh

2012-05-28T14:55:51Z

@juan I just wrote and then deleted my comment and reposted it. Hope it didn't cause any confusion. :)

Comment by Roupam Ghosh

2012-05-28T14:54:30Z

Here's a direct link to the book by Landau mentioned by Juan. This takes you to paragraph 67. archive.org/stream/…

Comment by Roupam Ghosh

2012-05-28T10:29:47Z

Yes, I have noticed that too, there are no $\rho(x)^r$ terms. Thanks for your answer anyways!

Comment by Roupam Ghosh

2012-05-17T16:52:06Z

Things look easy when they are already solved!

Comment by Roupam Ghosh

2012-04-12T13:48:49Z

Thanks, Johan! I will take a look at that. :-)

Comment by Roupam Ghosh

2011-12-25T23:18:47Z

Thanks quid. Thats a pretty nice historical overview :)

Comment by Roupam Ghosh

2011-12-25T23:14:00Z

Thanks for the links! :)

Comment by Roupam Ghosh

2011-12-25T23:12:23Z

@Igor: Yes, I posted a similar question on math.SE. But I realized that this question is better suited here.

Comment by Roupam Ghosh

2011-12-25T16:54:24Z

I have gone through Lagarias' paper already. It seems to me that he has just treated Robin's inequality to give a better bound. But I never quite got how Robin managed to relate RH and sigma and got this inequality.