Probability Distributions – Bound to the central moments of a $ chi ^ 2_d $ r.v.

To let $ d geq 1 $, and $ X $ be a chi square r.v. With $ d $ Degrees of freedom. For an integer $ k geq 1 $I would like to get a bond on the $ k $-th central moment of $ X $i.e.
$$
mathbb {E} ((X-d) ^ k) leq C cdot (k!) ^ k tag {1}
$$

for an absolute constant $ C> 0 $, I think I can prove it by combining properties of sub-exponential r.v.s with $ C = 1/4 $, Is there a more direct proof?

(Maybe I've made a mistake, in this case, the best bond that can be achieved in (1).) Also, I would still be glad if $ C $ is replaced by $ C ^ k $if the proof is simple.)

Definition of the $ chi ^ 2 $ distance of probability measures

I have found the following definition $ chi ^ 2 $Removal of two probability measures $ mu, nu $ on a measurable room $ (E, mathcal E) $: $$ chi ^ 2 ( mu, nu): = begin {cases} int left | frac {{ rm d} mu} {{ rm d} nu} -1 right | ^ 2 : { rm d} mu = int left | frac {{ rm d} nu} {{ rm d} mu} right | ^ 2 : { rm d} mu-1 & text {, if} nu ll mu \ infty & text {, otherwise}. end {cases} $$ First and foremost: why is that exactly defined in the case? $ nu ll mu $? In this case, the Radon-Nikodým derivative $ frac {{ rm d} nu} {{ rm d} mu} $ Surely exists, but why? $ frac {{ rm d} mu} {{ rm d} nu} $ And if it exists, how can we achieve equality? $ int left | frac {{ rm d} mu} {{ rm d} nu} -1 right | ^ 2 : { rm d} mu = int left | frac {{ rm d} nu} {{ rm d} mu} right | ^ 2 : { rm d} mu-1 $?

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Reference Request – Does the gauss sum attached to $ chi $ ever belong to $ mathbb {Q} ( chi) $?

To let $ p $ be a prime and $ chi $ Be a primitive Dirichlet character of the conductor $ p $, We let $$ g ( chi) = sum_ {a = 1} ^ {p} { chi (a) e ^ {2i pi a / p}} $$ be the Gauss sum $ chi $, Is that known? $ g ( chi) $ does not belong to $ mathbb {Q} ( chi) $the algebraic extension of $ mathbb {Q} $ generated by the values ​​of $ chi $?

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LIVE | 192.169.139.94:53412 | 0.85 | Ho Chi Minh | Unknown | s3d35.cloudnetwork.vn | Vietnam | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 192.169.202.104:52317 | 4.33 | Arizona | 85260 | ip-192-169-202-104.ip.secureserver.net | United States | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 54.36.203.240:19390 | 0.24 | Washington | 99163 | cust-216-98-238-030.turbonet.com | United States | Blacklist: No | Checked at http://dichvusocks.us
LIVE | 47.254.34.150:26006 | 0.31 | California | 95125 | Alibaba | United States | Blacklist: No | Checked at http://dichvusocks.us
LIVE | 208.113.182.97:24150 | 0.24 | California | 92821 | ps511566.dreamhost.com | United States | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 95.110.228.190:8975 | 0.44 | Unknown | Unknown | | Unknown | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 92.222.212.38:62213 | 0.27 | Washington | 99360 | 360 Communications, LLC | United States | Blacklist: No | Checked at http://dichvusocks.us
LIVE | 174.138.6.73:3127 | 0.79 | Arges | 110180 | static-86-123-54-134.rdsnet.ro | Romania | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 43.224.8.121:6667 | 2.26 | Gujarat | 363001 | Gtpl Broadband Pvt. | India | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 43.224.8.124:6667 | 2.3 | Gujarat | 363001 | Gtpl Broadband Pvt. | India | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 88.255.185.216:9050 | 0.64 | Hatay | 31040 | 88.255.65.123.static.ttnet.com.tr | Turkey | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 192.169.141.90:53412 | 0.94 | Ho Chi Minh | Unknown | s3d35.cloudnetwork.vn | Vietnam | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 192.169.193.183:30955 | 9.02 | Arizona | 85260 | ip-198.12-157-77.ip.secureserver.net | United States | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 14.52.216.9:1081 | 0.77 | Seoul-t. Ukpyolsi | Unknown | Korea Telecom | Korea, Republic of | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 103.216.82.19:6667 | 1.39 | Gujarat | 396450 | Gtpl Dcpl Private Limited | India | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 109.175.6.166:13101 | 0.52 | Unknown | Unknown | | Unknown | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 37.59.8.29:25307 | 4.36 | Unknown | Unknown | ns3099982.ovh.net | France | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 43.224.8.12: 6667 | 1.06 | Gujarat | 363001 | Gtpl Broadband Pvt. | India | Blacklist: Yes | Checked at http://dichvusocks.us

dichvusocks
Reviewed by dichvusocks on
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[DICHVUSOCKS.US] 07h27 AM UPDATE 24 / 24- Good socks
Payment Immediately perfectmoney, bitcoin, wmtransfer, dash, ETH (Please click on buy socks).
Update Tools Client Dichvusocks.us http://dichvusocks.us/tools.php Link Test Socks http://check.dichvusocks.us/
LIVE | 192.169.139.94:53412 | 0.85 | Ho Chi Minh | Unknown | s3d35.cloudnetwork.vn | Vietnam | Blacklist: Yes | Checked at http://dichvusocks.us
LIVE | 192.169.202.104:52317 | 4.33 | Arizona | 85260 | ip-192-169-202-104.ip.secureserver.net | United States | Blacklist: Yes | Reviewed on

Rating: 5

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