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- :<math>H_t(x) = \int_{\bf R} \frac{1}{8} \xi(\frac{1+ix}{2} + i |t|^{1/2} v) \frac{1}{\s in the regime when <math>x</math> is large and <math>t</math> is large and negative. ...22 KB (3,943 words) - 08:56, 10 January 2019
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- :<math>\|xy\| \leq \|x\| + \|y\| \quad (1)</math> :<math> \|x^n \| = |n| \|x\| \quad (2) </math> ...10 KB (1,762 words) - 10:12, 27 November 2018
- ...et <math>I</math> denote the integral <math>I := \int_a^b e^{\phi(x)} \psi(x)\ dx</math>. 1. If <math>\mathrm{Re} \phi(x) < 0</math> for all <math>a \leq x \leq b</math>, then ...20 KB (3,730 words) - 07:07, 30 March 2018
- :<math>H_t(x) = \int_{\bf R} \frac{1}{8} \xi(\frac{1+ix}{2} + i |t|^{1/2} v) \frac{1}{\s in the regime when <math>x</math> is large and <math>t</math> is large and negative. ...22 KB (3,943 words) - 08:56, 10 January 2019
- ...0.4</math>, can one prove that <math>H_t(x+iy) \neq 0</math> for all <math>x \geq 0</math>? ...h>H_t(x+iy)</math> at <math>y=0.4</math>) that one can show that <math>H_t(x+iy) \neq 0</math> for all <math>y \geq 0.4</math> as well. This would give ...22 KB (3,658 words) - 22:26, 1 January 2019
- :<math> T' := \frac{x}{2} + \frac{\pi t}{8} </math> ...ath>E_1+E_2+E_3</math> on <math>H - H^{eff}</math> at some points of <math>x</math> [https://terrytao.wordpress.com/2018/02/12/polymath15-third-thread-c ...28 KB (3,817 words) - 07:09, 30 March 2018
- we thus have for <math>z=x+iy</math> ...^\infty 2\pi^2 n^4 I_{t,\theta}(x-(9-y)i, \pi n^2) - 3\pi n^2 I_{t,\theta}(x-(5-y)i, \pi n^2) + ...12 KB (2,253 words) - 07:10, 30 March 2018
- '''Theorem 1''' Suppose that all zeroes of <math>H_t(x+iy)=0</math> have imaginary part at most <math>\varepsilon</math>. Then <m ...math>y_j>0</math> and no zeroes in the hyperbola <math>\{ x+iy: y > \sqrt{(x-x_j)^2 + y_j^2}</math>. Then <math>\partial_t y_j</math> is negative. ...12 KB (2,472 words) - 08:39, 5 April 2018
- ...eq}(X)</math> to denote any quantity bounded in magnitude by at most <math>X</math>. An equality <math>A=B</math> using this notation means that any qu '''Lemma 1''' Let <math>x > 0</math>. ...24 KB (3,915 words) - 07:07, 30 March 2018
- To compute <math>H_t(x+iy)</math>, we begin with the heat fundamental solution formula :<math> H_t(x+iy) = \frac{1}{8} \int_{-\infty}^\infty \xi( \frac{1-y+ix}{2} + \sqrt{t} v) ...7 KB (1,245 words) - 07:10, 30 March 2018
- ...o-dimensional bounded connected domain D and let u(x,t) (the heat at point x at time t) satisfy the heat equation with Neumann boundary conditions. We t : For sufficiently large t > 0, u(x,t) achieves its maximum on the boundary of D ...61 KB (9,824 words) - 16:22, 21 March 2018
- ...'c-statistics' is to identify 1s and 3s but leave the 2s intact. Let’s use x to denote letters that are either 1 or 3, then the 81 points in <math>[3]^4 ...32, and so can have at most three points in the eight-point set xyz2 (with x,y,z=1,3) rather than four (to get four points one must alternate in a check ...56 KB (9,089 words) - 20:26, 29 September 2015
- ...ment sets in <math>[3]^2</math> which are line-free, which we denote <math>x = D_{2,2}</math>, <math>y=D_{2,1}</math>, <math>z=D_2</math>, and <math>w</ x = 12 22 .. y = 12 .. 32 z = .. 22 32 w = 12 .. 32 ...32 KB (5,360 words) - 20:26, 29 September 2015
- ...group <math>\mathbb{Z}_3</math> defined by <math>c(z) = x \in \mathbb{Z}, x \equiv a - b \bmod{3}</math>. To verify that this is valid it is sufficient ...f order <math>k</math>. I.e. <math>c(x+y) = c(x)+c(y)</math> for all <math>x,y</math> in the ring. ...19 KB (3,233 words) - 12:40, 4 June 2018
- ...math> in the ground set, write <math>\mathcal{A}_x = \{A \in \mathcal{A} : x \in A\}</math>. If <math>\mathcal{A}</math> is union-closed then there is an element <math>x</math> such that <math>\lvert \mathcal{A}_x \rvert \geq \frac{n-1}{\log_2 n ...8 KB (1,166 words) - 01:38, 27 October 2016
- ...io tends to oscillate roughly between 0.5 and 3 for medium values of <math>x</math>: ! style="text-align:left;"| range of <math>x</math> ...21 KB (3,501 words) - 12:52, 15 April 2018
- ...arious regions of the <math>(t,y,x)</math> parameter space where <math>H_t(x+iy)</math> is known to be non-zero. In some cases the parameter :<math> N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor</math> ...10 KB (1,300 words) - 16:44, 10 May 2018
- where <math>\log_+(x)</math> is equal to <math>\log x</math> when <math>x \geq 1</math> and zero otherwise. We can rewrite this as ...{T} + \frac{X_k}{T}</math>. Using the bound <math>\log_+(x+y) \leq \log_+(x) + \log_+(1+y)</math> we have ...30 KB (4,511 words) - 19:30, 23 October 2014
- ...sed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <m ...]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p ...37 KB (4,581 words) - 07:57, 12 July 2015
- ...,B\vee X]</math> are disjoint, and <math>\mathcal{A}':= \bigcup_B [B,B\vee X]</math> decomposes into a fibre bundle over <math>\mathcal{B}</math>. ...</math>, and the transition map <math>\phi_{B,C} : [B,B\vee X]\to [C,C\vee X]</math> for <math>B\leq C</math> in <math>\mathcal{B}</math> is given by <m ...18 KB (3,249 words) - 01:03, 21 March 2016
- :<math>\Gamma(s) = \int_0^\infty x^s e^{-x} \frac{dx}{x} </math> ...can calculate that this expression has magnitude <math>O( x^{6/4} e^{-\pi x/8} )</math>. ...19 KB (3,626 words) - 07:07, 30 March 2018
