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Jan
7
revised $\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions
added ", which" and spacing
Jan
7
comment Two Representations of the Prime Counting Function
Thanks a lot. Concerning the last question, I thought about using a simple analogy $(\pi_{4n+3}-\pi_{4n+1})-\text{Li}(x)=\text{R}(x)-\sum_\rho \text{R}(x^\rho)$ with $\rho$ being the values you've kindly provided...?
Jan
6
comment Two Representations of the Prime Counting Function
Wow 331 sounds great. I was just out to find a table where these zeros are listed. Would you mind sharing the zeros of beta with me? And to be honest I still have one more question: Is it possible to write $\#\{\text{primes}\ 4n+3 \le x\} - \#\{\text{primes}\ 4n+1 \le x\}$(form [here](math.stackexchange.com/q/149755/19341)) in terms like $\operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho})$ ? I hope I don't bother you too much...
Jan
6
comment How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?
Hi @Eric ,one more question: Does $\int_{2}^{x}t^{-s-1}\pi(t)dt$ relate to a Mellin Transform $M_f(s) := \int \limits_{0}^\infty f(t)t^{s-1}\mathrm{d}t$? If so, what does that mean?
Jan
6
answered Rational Roots of Riemann's $\zeta$ Function
Jan
6
comment Two Representations of the Prime Counting Function
thank you very much. Again :-)
Jan
6
comment Two Steps away from the Hamilton Cycle
Hey @brian seriously why not make a photo of your drawing and post that? It can't be worse than PAINT...
Jan
5
comment Two Representations of the Prime Counting Function
But do your plots compare to Figure 4 in Prime Number Races. Beta functions are used in the case $\pi_{4n+3}-\pi_{4n+1}$ (p.19)...
Jan
4
comment Two Representations of the Prime Counting Function
It is from the linked paper here : math.stackexchange.com/q/149755/19341 . Sorry but i post from my tablet. I hate that...
Jan
4
revised Two Representations of the Prime Counting Function
added 544 characters in body
Jan
4
comment Is $\frac1\pi \arctan \frac\pi{\ln x}- \frac1{\ln x}$ related to the trivial solutions $\zeta(-2n)$?
Thanks again Raymond: I would be glad if you could have a look at this question on Two Representations of the Prime Counting Function as well. Cheers, draks...
Jan
4
accepted Is $\frac1\pi \arctan \frac\pi{\ln x}- \frac1{\ln x}$ related to the trivial solutions $\zeta(-2n)$?
Jan
4
revised Two Steps away from the Hamilton Cycle
added modified version of previous example and some more idea towards a general solution
Jan
3
revised Two Representations of the Prime Counting Function
gram added
Jan
3
asked Two Representations of the Prime Counting Function
Jan
3
revised Riesel and Gohl's Approximation of the Modified Prime Counting Function, $\pi_{0}$
fixed broken prime tex
Jan
3
comment Gram's series for integral equation
(i) What does $f(x)$ in $(1)$ correspond to in $(0)$, $\pi(e^t)$? (ii) Is $K(st)=1/(e^{st}-1)$? (iii) Why $g(s)/s$? Looks like you imply a logarithmic derivative somewhere...
Jan
3
revised Gram's series for integral equation
some typos, a little tex, two tags and a question mark
Jan
3
comment Aside from approximation functions, are there any proven functions that produce an exact $n$th prime?
@RaymondManzoni I love it...
Jan
3
comment Aside from approximation functions, are there any proven functions that produce an exact $n$th prime?
@RaymondManzoni I just realized that it was your answer back then...