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Results tagged with riemann-zeta-function
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user 81776
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.
5
votes
Accepted
Proof of the sum of the reciprocal non trivial zeros cubed
This comes directly from the Hadamard product given in the Wolfram page you
refer to by taking logarithmic derivatives and identifying powers of $s$.
However, following Harold Stark, the classical for …
- 13.1k
8
votes
Accepted
On a certain integral representation for Hurwitz zeta functions
All these are special cases of the Abel--Plana formula: under simple regularity
and growth condition which are easily given, for any $a$ with $\Re(a)>-1/2$ we have
$$\sum_{n\ge 0}f'(n+a)=-\dfrac{\pi}{ …
- 13.1k
3
votes
Accepted
Can this quantity be expressed as $x\cdot \zeta(k)+y, x,y \in \mathbb{Q}$?
Yes, and yes.
1) We have
$$|l(a)|_k^k=\sum_{d\mid a}\left(\prod_{p\mid d}\left(1-\dfrac{1}{p^k}\right)\right)\left(\zeta(k)-\sum_{j=1}^{a/d}\dfrac{1}{j^k}\right)\;,$$
which checks with your special …
- 13.1k
6
votes
Derivative of zeta at positive even integers
Partial answer: in my book Springer Graduate Texts in Math 240 page 142
Exercise 92, I give six equivalent formulas for the very similar sum $\sum_{n\ge2}\zeta'(n)$. I suggest first that you prove the …
- 13.1k
9
votes
Malmsten-like integrals for $\zeta(n) $
Three remarks:
If you change $x$ into $\exp(-t)$ your integrals are linear combinations
of $\int_0^\infty \log(t)\exp(-mt)/(1+\exp(-2t))^n\,dt$.
In turn, these integrals are the derivative with respe …
- 13.1k
16
votes
Accepted
$\zeta(2n+1)$ - Is this formulation helpful?
This follows directly from the Euler-MacLaurin formula applied
to $\sum_{1\le m<N}m^{2k}\log(m)$: the constant term is essentially
$\zeta'(-2k)$, itself directly related via the functional equation to …
- 13.1k
5
votes
Accepted
Do the roots of this equation involving two Euler products all reside on the critical line?
This problem (and more general ones) was solved completely and
unconditionally (i.e., without any RH) by P.R. Taylor
before the second world war, but I don't remember the reference,
it should be easy …
- 13.1k
13
votes
Accepted
A conjectured series expression for the Riemann $\xi$-function and/or Completed L-series. Co...
Your conjecture is true, and follows trivially from the Poisson summation
formula. For instance, use Theorem 9.4.2 in my book joint with K. Belabas "Numerical Algorithms for number theory", GSM 254, A …
- 13.1k
4
votes
Accepted
Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?
I would expect no: the big difference between $\sigma_1$ and $\sigma_k$ for
$k\ge2$ is that in the latter case $\sigma_k(n)/n^k$ is trivially bounded by
$\zeta(k)$, which doesn't leave much room for a …
- 13.1k
4
votes
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationali...
I also want to mention the formula (origin unknown)
$$\dfrac{56\zeta(3)}{3}=\sum_{n\ge1}\dfrac{64^n}{n^3D_nD_{n-1}}\;,$$
where $D_n=W_4(2n)$ is the $n$th Domb number, equivalent to the continued fract …
- 13.1k
1
vote
Asymptotic behavior of weighted sums involving the fractional part function
Not an answer but a conjectural answer for the value of $C(m)$ supported by
extensive numerical evidence:
$$C(m)=\dfrac{1}{m-1}\sum_{2\le k\le m-1}\binom{m-1}{k}\zeta(k)+\sum_{2\le k\le m-2}\dfrac{1}{ …
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