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Results tagged with gamma-function
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user 11919
used only for functions based on gamma, not functions with some obscure relation to gamma
64
votes
Why is the Gamma function shifted from the factorial by 1?
I would argue against the OP's opinion. The definition $\Gamma(z)$ becomes very natural if you write it as $\Gamma(z) = \int_0^\infty t^{z} e^{-t} d^\times t$, where $d^\times t=dt/t$ is the Haar mea …
- 105k
4
votes
Accepted
Is this a valid way of generating the non-trivial zeros of $\zeta(s)$?
Well, both sides of (1) are holomorphic as functions of $\sigma$ in the strip $0<\Re(\sigma)<1$, because the integrand is holomorphic and rapidly decaying as $x\to\infty$ and the right hand side is ho …
- 105k
4
votes
Accepted
Inequality for the modulus of Riemann zeta on horizontal lines and alleged partial result of...
For fixed $t>12$, let us consider for $0\leq\sigma\leq \frac{1}{2}$ the function
$$ G(\sigma):=|\pi^{-(\sigma-it)/2}\Gamma(\sigma+it)|^2 = \pi^{-\sigma}|\Gamma(\sigma+it)|^2. $$
Following the accept …
- 105k