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- == Analytic Number Theory == ...udying large-scale properties of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]].5 KB (849 words) - 15:14, 18 May 2021
- ...s turns out to be not the case. There are many pathological functions of a real variable that cannot occur in complex variables. Here are a few spectacular ...y]]! This is certainly not true of a real function, even a [[real analytic function]].2 KB (271 words) - 21:06, 12 April 2022
- ...eal part]] <math>1/2</math>. From the [[functional equation]] for the zeta function, it is easy to see that <math>\zeta(s)=0</math> when <math>s=-2,-4,-6,\ldot ...c{1}{2}</math>. Let <math>M(n)=\sum_{i=1}^n \mu(i)</math> be the [[Mertens function]]. It is easy to show that if <math>M(n)\le\sqrt{n}</math> for sufficiently2 KB (425 words) - 02:18, 29 June 2024
- The '''Riemann zeta function''' is a function very important in about the roots of the zeta function.9 KB (1,547 words) - 02:04, 13 January 2021
- A '''holomorphic function''' <math>f: \mathbb{C} \to \mathbb{C}</math> is a differentiable [[complex number|complex]] [[function]]. That is, just9 KB (1,537 words) - 20:04, 26 July 2017
- ...field of [[number theory]], where an alternate definition is often used: a function from the [[positive integer]]s to the [[complex number]]s is said to be ''m ...mbers]] by <math>f(x) = x^2</math> is a simple example of a multiplicative function.3 KB (450 words) - 11:59, 21 July 2009
- ...th> to the [[order]] of the [[root|zero]] of the associated <math>L</math>-function <math>L(E, s)</math> at <math>s = 1</math>. ...or no answer) whose solutions can be ''verified'' in polynomial time (as a function of the input, often expressed using big-O notation) can also be ''solved''13 KB (1,969 words) - 16:57, 22 February 2024
- ...bounded by <math>C</math>, and <math>f</math> is a complex-differentiable function on for all <math>n>0</math>. In other words, if a function <math>f</math> is4 KB (689 words) - 16:19, 18 January 2024
- ...ting function''', denoted <math>\pi</math>, is a [[function]] defined on [[real number]]s. The quantity <math>\pi(x)</math> is defined as the number of [[ ...er theorem]]. It is also asymptotically equivalent to [[Chebyshev's theta function]]. It was first proved in 1896 by Jacques Hadamard and by Charles de la Val1 KB (238 words) - 12:45, 13 August 2015
- '''Chebyshev's theta function''', denoted <math>\vartheta</math> or sometimes <math>\theta</math>, is a function of use in [[analytic number theory]].2 KB (264 words) - 12:28, 1 April 2014
- ...is the point-slope form which states <math>y=m \cdot x + b</math> for some real numbers b,x,y. Also, if two lines are perpendicular, then product of the sl ...y perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dime4 KB (591 words) - 00:23, 25 January 2016
- real t = .385, s = 3.5*t-1; ...math>, respectively. Now let the vertex of the equilateral triangle on the real axis be <math>a</math> and let the vertex of the equilateral triangle on th22 KB (3,622 words) - 21:06, 10 October 2024
- ...|real]] or [[Complex numbers| complex]] [[Root (polynomial)|roots]]. For a function <math>f(x)</math>, the approximations are defined [[Recursion|recursively]] ...math> and seek a root of <math>f(x)</math>. Since all nonconstant [[Linear function|linear functions]] have exactly one root, as long as <math>f'(x_i) \neq 0</13 KB (2,298 words) - 22:34, 28 May 2023
- Find the largest possible real part of <cmath>(75+117i)z+\frac{96+144i}{z}</cmath>where <math>z</math> is ==Solution 2b (Simple Analytic Geometry)==15 KB (2,348 words) - 02:18, 16 December 2024
- 2. Let <math>x,y</math> and <math>z</math> be real numbers. Show that where p and m are real numbers converge if <math>m>1</math> or <math>m=1</math> but <math>p>1</mat64 KB (10,960 words) - 01:00, 26 December 2024