Bounding the derivative of H t - Revision history

Teorth at 15:08, 30 March 2018

2018-03-30T15:08:30Z

← Older revision		Revision as of 08:08, 30 March 2018
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	where <math>S_{\sigma,t}(N)</math> was defined in [[Estimating a sum]]. Similarly		where <math>S_{\sigma,t}(N)</math> was defined in [[Estimating a sum]]. Similarly
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) S_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) S_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>

			[[Category:Polymath15]]

Teorth at 07:12, 9 March 2018

2018-03-09T07:12:52Z

← Older revision		Revision as of 00:12, 9 March 2018
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	:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>		:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>
	We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that		We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) S_{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>
	~~(An unfortunate notational conflict here; the~~ <math>F_{\sigma,t}</math> ~~on the RHS is not the same function as that on the LHS, it is the one~~ in [[Estimating a sum]]. ~~Will need to change the notation.)~~		where <math>S_{\sigma,t}(N)</math> was defined in [[Estimating a sum]]. Similarly
	Similarly		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) S_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>

Teorth at 06:57, 9 March 2018

2018-03-09T06:57:40Z

← Older revision		Revision as of 23:57, 8 March 2018
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	:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>		:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>
	We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that		We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>
			(An unfortunate notational conflict here; the <math>F_{\sigma,t}</math> on the RHS is not the same function as that on the LHS, it is the one in [[Estimating a sum]]. Will need to change the notation.)
	Similarly		Similarly
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>

Teorth at 06:55, 9 March 2018

2018-03-09T06:55:51Z

← Older revision		Revision as of 23:55, 8 March 2018
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	We can bound		We can bound
	:<math> \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) \leq \frac{1}{2(T-3.08)} ( tu^2 + \frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3} )</math>		:<math> \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) \leq \frac{1}{2(T-3.08)} ( tu^2 + \frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3} )</math>
			and also write
			:<math>\exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) ) \|H_{0,n}(s)\| = \exp( \frac{t}{4} \mathrm{Re}(\alpha_1(s)^2) ) \|H_{0,1}(s)\| \frac{b_n}{n^{\mathrm{Re} s_A}}</math>
			:<math>= 8 \|\lambda\| \|B^{eff}_0\| \frac{b_n^2}{n^{\sigma + \frac{t}{2} \mathrm{Re} \alpha_1(s)}}</math>
	to obtain		to obtain
	:<math> \|F'_{t,n}(s)\| \leq \~~frac~~{~~\exp(~~ \frac{t}{4} \mathrm{Re}~~(\alpha_n(s)^2) )~~}~~{n^\sigma} \|H_{0,1~~}~~(s)\|~~ \frac{\exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) }{\sqrt{\pi}} \int_{-\infty}^\infty (\|\alpha_n(s)\| + \|\frac{2u}{\sqrt{t}}\|) \exp( -(1-\frac{t}{2(T-3.08)}) u^2 )\ du.</math>		:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \frac{\exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) }{\sqrt{\pi}} \int_{-\infty}^\infty (\|\alpha_n(s)\| + \|\frac{2u}{\sqrt{t}}\|) \exp( -(1-\frac{t}{2(T-3.08)}) u^2 )\ du.</math>
	Noting that		Noting that
	:<math>\int_{-\infty}^\infty e^{-a u^2}\ du = a^{-1/2} \sqrt{\pi}</math>		:<math>\int_{-\infty}^\infty e^{-a u^2}\ du = a^{-1/2} \sqrt{\pi}</math>
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	for any <math>a>0</math>,		for any <math>a>0</math>,
	we thus have		we thus have
	:<math> \|F'_{t,n}(s)\| \leq \~~frac~~{~~\exp(~~ \frac{t}{4} \mathrm{Re}~~(\alpha_n(s)^2) )~~}~~{n^\sigma} \|H_{0,1~~}~~(s)\|~~ \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>		:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>
	We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that		We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that
	:<math>\sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|H_{~~0,1~~}~~(s)~~\| \exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) \sum_{n=1}^N \~~frac~~{\exp( \frac{t}{4} \~~mathrm~~{Re}(\~~alpha_n~~(s)^2) )}{n^\~~sigma~~}.</math>		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>
			Similarly
			:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>

Teorth at 05:22, 9 March 2018

2018-03-09T05:22:10Z

← Older revision		Revision as of 22:22, 8 March 2018
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	:<math> \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) \leq \frac{1}{2(T-3.08)} ( tu^2 + \frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3} )</math>		:<math> \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) \leq \frac{1}{2(T-3.08)} ( tu^2 + \frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3} )</math>
	to obtain		to obtain
	:<math> \|F'_{t,n}(s)\| \leq \frac{\exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) )}{n^\sigma} \|H_{0,1}(s)\| \frac{\exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) }{\sqrt{\pi}} \int_{-\infty}^\infty} (\|\alpha_n(s)\| + \|\frac{2u}{\sqrt{t}}\|) \exp( -(1-\frac{t}{2(T-3.08)}) u^2 )\ du.</math>		:<math> \|F'_{t,n}(s)\| \leq \frac{\exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) )}{n^\sigma} \|H_{0,1}(s)\| \frac{\exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) }{\sqrt{\pi}} \int_{-\infty}^\infty (\|\alpha_n(s)\| + \|\frac{2u}{\sqrt{t}}\|) \exp( -(1-\frac{t}{2(T-3.08)}) u^2 )\ du.</math>
	Noting that		Noting that
	:<math>\int_{-\infty}^\infty e^{-a u^2}\ du = a^{-1/2} \sqrt{\pi}</math>		:<math>\int_{-\infty}^\infty e^{-a u^2}\ du = a^{-1/2} \sqrt{\pi}</math>

Teorth at 05:14, 9 March 2018

2018-03-09T05:14:31Z

← Older revision		Revision as of 22:14, 8 March 2018
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	We continue using the notation from [[Effective bounds on H_t - second approach]].		We continue using the notation from [[Effective bounds on H_t - second approach]]. We also assume <math>T \geq 10</math> (so <math>x \geq 20</math>).

	Since		Since
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	This identity is true for any <math>\alpha_n</math>; we now set <math>\alpha_n = \alpha_n(s)</math> as in the above wiki page.		This identity is true for any <math>\alpha_n</math>; we now set <math>\alpha_n = \alpha_n(s)</math> as in the above wiki page.
	One can replace <math>\frac{\partial}{\partial s}</math> on the RHS by <math>\frac{1}{\sqrt{t}} \frac{\partial}{\partial u}</math> and integrate by parts to conclude that		One can replace <math>\frac{\partial}{\partial s}</math> on the RHS by <math>\frac{1}{\sqrt{t}} \frac{\partial}{\partial u}</math> and integrate by parts to conclude that
	:<math> F'_{t,n}(s) = \exp( - \frac{t}{4} \alpha_n(s)^2 ) \int_{-\infty}^\infty \exp( - \sqrt{t} \alpha_n(s) u) F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n ) \frac{1}{\sqrt{\pi}} (\alpha_n(s) + 2u) e^{-u^2}\ du.</math>		:<math> F'_{t,n}(s) = \exp( - \frac{t}{4} \alpha_n(s)^2 ) \int_{-\infty}^\infty \exp( - \sqrt{t} \alpha_n(s) u) F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n ) \frac{1}{\sqrt{\pi}} (\alpha_n(s) + \frac{2u}{\sqrt{t}}) e^{-u^2}\ du.</math>
	We have		We have
	:<math> F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n(s)) = H_{0,n}(s) \exp( (\sqrt{t} u + \frac{t}{2} \alpha_n(s)) \alpha_n(s) + O_{\leq}( \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) ) )</math>		:<math> F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n(s)) = H_{0,n}(s) \exp( (\sqrt{t} u + \frac{t}{2} \alpha_n(s)) \alpha_n(s) + O_{\leq}( \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) ) )</math>
	and hence		and hence
	:<math> \|F'_{t,n}(s)\| \leq \exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) ) \|H_{0,n}(s)\| \int_{-\infty}^\infty \exp( \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) ) ) \frac{1}{\sqrt{\pi}} \|\alpha_n(s) + 2u\| e^{-u^2}\ du.</math>		:<math> \|F'_{t,n}(s)\| \leq \exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) ) \|H_{0,n}(s)\| \int_{-\infty}^\infty \exp( \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) ) ) \frac{1}{\sqrt{\pi}} (\|\alpha_n(s)\| + \|\frac{2u}{\sqrt{t}}\|) e^{-u^2}\ du.</math>
			We can bound
			:<math> \frac{1}{4 (T - 3.08)} ( \|\sqrt{t} u + \frac{t}{2} \alpha_n(s)\|^2 + \frac{2}{3} ) \leq \frac{1}{2(T-3.08)} ( tu^2 + \frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3} )</math>
			to obtain
			:<math> \|F'_{t,n}(s)\| \leq \frac{\exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) )}{n^\sigma} \|H_{0,1}(s)\| \frac{\exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) }{\sqrt{\pi}} \int_{-\infty}^\infty} (\|\alpha_n(s)\| + \|\frac{2u}{\sqrt{t}}\|) \exp( -(1-\frac{t}{2(T-3.08)}) u^2 )\ du.</math>
			Noting that
			:<math>\int_{-\infty}^\infty e^{-a u^2}\ du = a^{-1/2} \sqrt{\pi}</math>
			and
			:<math>\int_{-\infty}^\infty e^{-a u^2} \|u\|\ du = a^{-1}</math>
			for any <math>a>0</math>,
			we thus have
			:<math> \|F'_{t,n}(s)\| \leq \frac{\exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) )}{n^\sigma} \|H_{0,1}(s)\| \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>
			We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that
			:<math>\sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|H_{0,1}(s)\| \exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) \sum_{n=1}^N \frac{\exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) )}{n^\sigma}.</math>

Teorth: Created page with "We continue using the notation from Effective bounds on H_t - second approach. Since : $H_t(z) = \frac{1}{8} \xi_t( s)$ with $s := \frac{1+iz}{2}$ ,..."

2018-03-09T04:22:17Z

Created page with "We continue using the notation from Effective bounds on H_t - second approach. Since :<math>H_t(z) = \frac{1}{8} \xi_t( s) </math> with <math>s := \frac{1+iz}{2}</math>,..."

New page

We continue using the notation from [[Effective bounds on H_t - second approach]].

Since
:<math>H_t(z) = \frac{1}{8} \xi_t( s) </math>
with <math>s := \frac{1+iz}{2}</math>, we have
:<math>\frac{d}{dz} H_t(z) = \frac{i}{16} \frac{d}{ds} \xi_t(s).</math>
Next, we have
:<math> \xi_t(s) = \sum_{n=1}^N F_{t,n}(s) + F_{t,n}(1-s) + G_{t,N}(s) + G_{t,N}(1-s)</math>
(using the convention <math>F(\bar{s}) = \bar{F(s)}</math> for <math>s</math> in the lower half-plane). Thus (assuming that we are not at a discontinuity for $latex N$) we have
:<math> \frac{d}{ds} \xi_t(s) = \sum_{n=1}^N F'_{t,n}(s) - F'_{t,n}(1-s) + G'_{t,N}(s) - G'_{t,N}(1-s).</math>
Now we have for any <math>\alpha_n</math> that
:<math> F_{t,n}(s) = \exp( - \frac{t}{4} \alpha_n^2 ) \int_{-\infty}^\infty \exp( - \sqrt{t} \alpha_n u) F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n ) \frac{1}{\sqrt{\pi}} e^{-u^2}\ du,</math>
hence in differentiation under the integral sign (justifiable for instance using the Cauchy integral formula and Fubini's theorem)
:<math> F'_{t,n}(s) = \exp( - \frac{t}{4} \alpha_n^2 ) \int_{-\infty}^\infty \exp( - \sqrt{t} \alpha_n u) \frac{\partial}{\partial s} F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n ) \frac{1}{\sqrt{\pi}} e^{-u^2}\ du.</math>
This identity is true for any <math>\alpha_n</math>; we now set <math>\alpha_n = \alpha_n(s)</math> as in the above wiki page.
One can replace <math>\frac{\partial}{\partial s}</math> on the RHS by <math>\frac{1}{\sqrt{t}} \frac{\partial}{\partial u}</math> and integrate by parts to conclude that
:<math> F'_{t,n}(s) = \exp( - \frac{t}{4} \alpha_n(s)^2 ) \int_{-\infty}^\infty \exp( - \sqrt{t} \alpha_n(s) u) F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n ) \frac{1}{\sqrt{\pi}} (\alpha_n(s) + 2u) e^{-u^2}\ du.</math>
We have
:<math> F_{0,n}( s + \sqrt{t} u + \frac{t}{2} \alpha_n(s)) = H_{0,n}(s) \exp( (\sqrt{t} u + \frac{t}{2} \alpha_n(s)) \alpha_n(s) + O_{\leq}( \frac{1}{4 (T - 3.08)} ( |\sqrt{t} u + \frac{t}{2} \alpha_n(s)|^2 + \frac{2}{3} ) ) )</math>
and hence
:<math> |F'_{t,n}(s)| \leq \exp( \frac{t}{4} \mathrm{Re}(\alpha_n(s)^2) ) |H_{0,n}(s)| \int_{-\infty}^\infty \exp( \frac{1}{4 (T - 3.08)} ( |\sqrt{t} u + \frac{t}{2} \alpha_n(s)|^2 + \frac{2}{3} ) ) ) \frac{1}{\sqrt{\pi}} |\alpha_n(s) + 2u| e^{-u^2}\ du.</math>

← Older revision		Revision as of 00:12, 9 March 2018
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	:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>		:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>
	We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that		We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) S_{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>
	~~(An unfortunate notational conflict here; the~~ <math>F_{\sigma,t}</math> ~~on the RHS is not the same function as that on the LHS, it is the one~~ in [[Estimating a sum]]. ~~Will need to change the notation.)~~		where <math>S_{\sigma,t}(N)</math> was defined in [[Estimating a sum]]. Similarly
	Similarly		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) S_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>

← Older revision		Revision as of 23:57, 8 March 2018
Line 35:		Line 35:
	:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>		:<math> \frac{1}{8} \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\| \frac{b_n}{n^{\mathrm{Re} s_A}} \exp( \frac{1}{2(T-3.08)} (\frac{t^2}{4} \|\alpha_n(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_n(s)\| (1-\frac{t}{2(T-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T-3.08)})^{-1}).</math>
	We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that		We can bound <math>\|\alpha_n(s)\| \leq \|\alpha_1(s)\|</math> and <math>T \geq T_N</math> and conclude that
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(s)\| \leq \|\lambda\| \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{\mathrm{Re} s_A - \frac{t}{2} \log N, t}(N).</math>
			(An unfortunate notational conflict here; the <math>F_{\sigma,t}</math> on the RHS is not the same function as that on the LHS, it is the one in [[Estimating a sum]]. Will need to change the notation.)
	Similarly		Similarly
	:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>		:<math>\frac{1}{8} \sum_{n=1}^N \|F'_{t,n}(1-s)\| \leq \|B^{eff}_0\|\exp( \frac{1}{2(T_N-3.08)} (\frac{t^2}{4} \|\alpha_1(1-s)\|^2 + \frac{1}{3}) ) ( \|\alpha_1(1-s)\| (1-\frac{t}{2(T_N-3.08)})^{-1/2} + \frac{2}{\sqrt{\pi t}} (1-\frac{t}{2(T_N-3.08)})^{-1}) F_{\mathrm{Re} s_B - \frac{t}{2} \log N, t}(N).</math>