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Results tagged with riemann-zeta-function
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The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
21
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On a pattern for upside-down Ramanujan pi formulas
Define,
$$\lambda_n =\frac{(\tfrac12)_n}{(1)_n} =\frac{(\tfrac12)_n}{n!} =\frac{\tbinom{2n}{n}}{2^{2n}} =\binom{n-\tfrac12}{n}$$
with Pochhammer symbol $(x)_n$ and binomial $\tbinom{n}{k}$. I noticed …
- 12.6k
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When does the function $F(x)=\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$...
We know from Ramanujan and Riemann that,
$$\pi(x) = \operatorname{li}(x) -\tfrac12\operatorname{li}(x^{1/2})-\tfrac13\operatorname{li}(x^{1/3})-\tfrac15\operatorname{li}(x^{1/5}) +\dots$$
with prime …
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Skewes' number and the ratio $\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$
(A complementary post is here.)
Given the prime counting function $\pi(x)$ and the logarithmic integral $\operatorname{li}(x)$, we have Table 1,
$$\begin{array}{|c|l|}
\hline
x&\operatorname{li}(x) …
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On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?
After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$.
I. Recurrences involving $\zeta(5)$
In Cohen's 2022 paper, ther …
- 12.6k
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answers
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About a Ramanujan-Sato formula of level 10, a recurrence, and $\zeta(5)$?
I. Level 6
This is a long shot, but I am curious where it leads. Given the Dedekind eta function $\eta(\tau),$ define,
$$\begin{aligned}
j_{6A}(\tau) &= \Big(\sqrt{j_{6B}(\tau)} - \frac{1}{\sqrt{j_{6 …
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