Riemann zeta function uses. It is expressed by the .

Riemann zeta function uses png, comparing the modulus of the Riemann zeta function evaluated on the critical line (for complex numbers s=1/2+it) to the Riemann-Siegel Z-function, which is one of the most powerful numerical methods for computing the Riemann zeta function near and on the Feb 22, 2024 · Mathematical function, suitable for both symbolic and numerical manipulation. 1 Trivial zeros; 5. Then for Res >1 the Hurwitz zeta function is defined by H. Written as ζ( x ), it was originally defined as the infinite series ζ( x ) = 1 + 2 − x + 3 − x + 4 − x + ⋯. Other functions called zeta functions, but not analogous to the Riemann zeta function. 1 Known upper bounds on \(\mathrm{LV}(\sigma ,\tau )\) 8 Large value theorems for zeta partial sums; 9 Moment growth for the zeta function Feb 28, 2013 · Many useful and interesting properties, identities, and relations for the Riemann zeta function ζ ( s ) and the Hurwitz zeta function ζ ( s , a ) have been developed. The Riemann Zeta Function The Riemann zeta function is de ned by the p-series (p) = X1 n=1 1 n p = 1 + 1 2p + 3p + 1 4 + :::; valid for p>1, (1) which converges for p > 1 by the Integral Test (and diverges for p 1). The series converges for RE(s) > 1, while for s = 1 we have the series X n>0 1 n which is well known not to converge. Bernoulli polynomials 1. 1 16. The chapter will also introduce concepts such as convergence, Barenoulli Mar 13, 2025 · $\ds \forall s \in \C: \map \xi x := \begin{cases} \dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s & : \map \Re s > 0 \\ \map \xi {1 - s} & : \map \Re s \le 0 \end{cases}$ where $\map \zeta s$ is the Riemann zeta function. Zeta functions are a family of special functions defined through a Dirichlet series. More specifically, we employ Riemann’s zeta function to regularize the otherwise infinite results of the so-called Heisenberg–Euler Zeta function and the Mellin transform of the Theta function. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the “Riemann zeta function”), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula ζ(s)− 1 s−1 = X∞ n=1 n An Introduction to Zeta Functions by Pierre Cartier Table of Contents Introd uction 1. (1) This function is sometimes also called the Hardy function or Hardy Z-function (Karatsuba and Voronin 1992, Borwein et al. Artin conjecture; Birch and Swinnerton-Dyer conjecture; Riemann hypothesis and the generalized Riemann hypothesis. There are the Dirichlet L-functions L(s;˜) de ned as follows: q 1 is an integer, ˜:(Z=qZ) ! C a (primitive) character Jul 7, 2013 · The functional equation $$\zeta(s) = 2^s \pi^{s-1} \sin \left(\dfrac{\pi s}2\right) \Gamma(1-s) \zeta(1-s)$$ can be used to obtain the value of the $\zeta$ function for $\operatorname{Re}(s) < 1$, using the value of the zeta function for $\operatorname{Re}(s)>1$. It is one of the most studied, yet most mysterious functions in all of mathematics. Riemann's zeta function 1. The L-function L(s;˜) = P 1 n=1 ˜(n)n s associated with a Dirichlet character ˜modulo qis de ned by L(s;˜) = X1 n=1 ˜(n)n s: We view (s) = P 1 n=1 n s as a special case of this by Riemann function may refer to one of the several functions named after the mathematician Bernhard Riemann, including: Riemann zeta function; Thomae's function, also called the Riemann function; Riemann theta function, Riemann's R, an approximation of the prime-counting function π(x), see Prime-counting function#Exact form. The Riemann zeta function is the complex function de ned by the series (s) := X n 1 n s; for Re(s) > 1, where nvaries over positive integers. 1 The Riemann zeta function De nition 16. It has zeros at the negative even integers (i. py generates a plot, namely Z(t)_plot. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. 7. The completed Riemann zeta function is also known as the Riemann xi function The Riemann zeta function and L-functions 4. 6 Integer Arguments The most important such infinite process is the following fundamental function. The implementation uses three algorithms: the Borwein algorithm for the Riemann zeta function when \(s\) is close to the real line; the Riemann-Siegel formula for the Riemann zeta function when \(s\) is large imaginary, and Euler-Maclaurin summation in all other cases. Aubert, E. I assume that "generalized Riemann zeta function" is "Hurwitz zeta function". The zeta function can be extended to a meromorphic function on the entire complex plane. In this article, three new Mar 5, 2025 · where is the Riemann zeta function and is the gamma function (Gradshteyn and Ryzhik 2000, p. These are analytic objects (complex functions) that are intimately related to the global elds we have been studying. 4. It plays a role in calculating the Casimir effect, a physical force arising from quantized fields between closely placed conductive plates. Bombieri and D. (Image by MIT OpenCourseWare. We will look at the historical context of the function as well as some more modern Zeta Functions and Polylogarithms Zeta: Identities (6 formulas) Functional identities (6 formulas),] Identities (6 formulas) Zeta. Some special values of the zeta function 1. The simplest of all L-functions is the Riemann zeta function, a function whose analytic properties were first discovered by Bernhard Riemann in 1859 in his attempts to find a formula for the number of primes smaller than a given number x x x. 3. It is expressed by the Mar 13, 2025 · Keywords: Riemann zeta function, Riemann-Siegel formula, Gaussian quadrature, high-precision algo-rithm 2020 Mathematics Subject Classification: Primary 11M06, Secondary 11Y35 1 Introduction and main results There exist many methods for computing the Riemann zeta function. Recall that the global behavior of π(x), the prime distribution function, is well approximated by Riemann’s smooth function R(x) (discussed in Chapter 2). Dec 8, 2016 · Indeed, the symbol "= = =" is being used a bit loosely in this expression, but there is a true fact which it is trying to express which has everything to do with the zeta function. The complexities of these methods have exponents 1/2, 3/8, and 1/3 respectively. 2) fb(r) = Z R f(x)e−2πirxdx. This section continues the discussion of complex analysis and the things you can learn about the zeta function. Section 24. Wolfram documentation lists two zeta functions: Zeta[s] gives the Riemann zeta function TemplateBox[{s}, Zeta]. In addition he conjectured that asymptotically 100% of the zeros are simple. s;1/D R. at s = −2, −4, −6, ). Using a different method, Gallagher and Mueller proved in 1978 that PCC under RH implies that 100% of the zeros are The white spot at s= 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(s) = 1/2 are its zeros. 1) (1. Low-order differentiation. For w = 0, the function reduces to the Riemann zeta function, named for the 19th-century German mathematician Bernhard Riemann, whose study of its properties The implementation uses three algorithms: the Borwein algorithm for the Riemann zeta function when \(s\) is close to the real line; the Riemann-Siegel formula for the Riemann zeta function when \(s\) is large imaginary, and Euler-Maclaurin summation in all other cases. "' where n runs through all integers, and (1. MR0564081 (58 #27837) the Riemann zeta function By Ghaith Ayesh Hiary Abstract The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Sch onhage’s method, or Heath-Brown’s method. The Riemann Hypothesis For Hilbert Spaces of Entire Functions Riemann's functional equation for the zeta function, discussed in lecture 23. (Academic, 1989) 1–10 R. 3 Graphics §25. The Riemann zeta function is de ned as (s) = X n2N 1 ns: The above expression for the Riemann zeta function has the form of a Dirichlet series, that is, a series of the form \[f(s)=\sum_{n=1}^\infty a_n n^{-s}. Zeta-functions in number theory are functions belonging to a class of analytic functions of a complex variable, comprising Riemann's zeta-function, its generalizations and analogues. 9 Asymptotic Approximations 25. Differentiation. s)=f:JI. Riemann Zeta Function. The Riemann Function Riemann worked to extend this function to the entire complex plane (except where s = 1, where there is a pole). 2 The Riemann Zeta Function ¶ permalink Subsection 24. \] Our investigation of the relationship between the zeta function and the distribution of primes will benefit from a quick overview of the theory of Dirichlet series in general. We begin with a systematic derivation and proof of the basic characteristics of the zeta function. 1 Special Notation 25. The Riemann zeta-function {(. The function is defined by when the real part is greater than 1. We now divert our attention from algebraic number theory for the moment to talk about zeta functions and L-functions. Either of these may be taken as the This project uses Python to create visualizations of key concepts related to the Riemann Hypothesis, with a focus on exploring the Riemann zeta function in 3D. This is a variant of the function originally defined by Riemann in his landmark paper (Riemann 1859), where the above now standard notation follows Landau (Edwards 2001, p. 1999). Γ ⁡ (z): gamma function, ζ ⁡ (s): Riemann zeta function, π: the ratio of the circumference of a circle to its diameter, d x: differential of x, ψ ⁡ (z): psi (or digamma) function, ∫: integral, sin ⁡ z: sine function, m: nonnegative integer, x: real variable and s: complex variable Keywords: improper integral, integral Section 6, The zeta function and complex analysis April 28, 2017, version 1. Selberg class S; Explicit formulae for L-functions; Trace formula Oct 6, 2017 · Riemann actually gave two independent proofs of this, the first using contour integration and the second using an identity of Jacobi. Dirichlet Zeta Functions constructs the zeta functions of Fourier analysis on the Cartesian product of the complex plane and an adic plane. Here we also discuss Riemann hypothesis and the uses of the $$\zeta $$-function in the calculations of functional integrals. Theorem 2. 1 Generating function of the Riemann zeta function for nonnegative even integers; 4. Weil, "Prehistory of the zeta-function", from Number Theory, Trace Formulas and Discrete Groups, K. In mathematics, the Riemann zeta function is an important function in number theory. Riemann’s insight was 16. 7 Integrals §25. Jacobi zeta function; Weierstrass zeta function; Topics related to zeta functions. G. Sep 8, 2018 · 3 Laurent expansion of the Riemann zeta function; 4 Riemann zeta function for nonnegative integers. WolframMathworld: Riemann zeta function. Notes on the Riemann Zeta Function January 25, 2007 1 The Zeta Function 1. We define the zeta function (denoted \(\zeta\)) as the sum of the infinite series 1 day ago · Montgomery in 1973 introduced the Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function. 3 Connection to the Riemann hypothesis; 7 Large value estimates . The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. 16). 2) where p runs through all primes. 1 A fundamental function. A. Goldfeld, eds. Conjecture 1 (even-odd antisymmetry): The distribution of the zeta values for odd gram points is the negative of the distribution of zeta values for the even zeta points. 11 Hurwitz Zeta Function §25. Aug 14, 2010 · The Riemann zeta-function ζ(s) is defined for all complex numbers s ≠ 1. 4 Reflection Formulas These conjectures are most likely related to the symmetry properties of the value distribution of the Riemann . 1 Basic facts We prove some results that will be used in the proof of the Prime Number Theorem (for arithmetic progressions). For the function itself. , from the behavior of an electron in a prescribed external electromagnetic field. The L-function of a Dirichlet character ˜modulo qis de ned by L(s;˜) = X1 n=1 ˜(n)n s: We view (s) = P 1 n=1 n s as the L-function of the principal character modulo 1, Chris Caldwell: basic introduction to the Riemann zeta function. 41; Edwards 2001, p. Zeta Functions and Polylogarithms Zeta: Identities. The chapter will also introduce concepts such as convergence, Barenoulli May 14, 2017 · The function whose rule is $\sum_{n=1}^\infty n^{-s}$ and whose domain is $\Re(s)>1$ has a ton of extensions to larger domains, but there is a special and natural one called the analytic continuation that extends the domain to $\mathbb C\setminus\{1\}. Riemann’s insight was 1. zeta function. 25 Zeta and Related Functions Riemann Zeta Function 25. 13 (1974), 383–436. 2 Nontrivial Zeta Functions. Including derivatives of the function,] Identities (6 formulas) Zeta Mar 5, 2025 · There are a number of functions in various branches of mathematics known as Riemann functions. Wolfram|Alpha can compute values for multiple variants of zeta functions as well as help you explore other functionalities, such as visualization and series expansion. General case. As a spin-off, we also calculate some integrals that are useful in mathematics and physics. ) The Riemann zeta function and L-functions We prove some results that will be used in the proof of the Prime Number Theorem (for arithmetic progressions). In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half. Analytic continuation of the zeta function 1. e. The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series = = = + + +Leonhard Euler considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem. One interesting special value [though hard to prove] is (2) = X1 n=1 1 n 2 = 1 + 22 + 1 3 2 + 4 + 1 5 + :::= ˇ2 6: (2 6 Growth exponents for the Riemann zeta function 6. E. \] It is then defined by analytical continuation to a meromorphic function on the whole \(\mathbb{C}\) by a functional equation . Riemann introduced this function in connection with his study of prime numbers and from this has developed the subject of analytic number theory. Results for aD1Cb >1 follow by observing H. It is easy to verify that this series converges absolutely and locally uniformly on Re(s) >1 (use the integral test on an open I know that the Riemann Zeta Function is defined as: $$\zeta (s)=\sum_{n=1}^\infty \frac {1}{n^s}=\frac {1}{\Gamma (s)} \int _0^{\infty}\frac { x^{s-1}}{e^x-1} dx$$ Which I think would prove useful for solving the Basel Problem without having to use Euler's tricks and just evaluating the integral. s;1Cb/D X1 nD0 1. 1) X n∈Z f(n) = X n∈Z fb(n) , which relates the sum over the integers of a function fand its Fourier transform (2. s;a/D X1 nD0 1. Definition of b(. We also present a double Jan 14, 2016 · [Lev] Norman Levinson, More than one third of zeros of Riemann’s zeta-function are on σ =1 / 2, Advances in Math. These are called the trivial zeros. Theorem Let (s) = X1 n=1 25 Zeta and Related Functions Riemann Zeta Function 25. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. The most famous example is the Riemann zeta function. $ It agrees with $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$ and also is defined outside that some of the different facets of Riemann’s zeta function and introduce some of the math that will assist us. Hurwitz zeta function 1. Also known as. 4. 0 1 Introduction. The Riemann zeta function for \(s\in \mathbb{C}\) with \(\operatorname{Re}(s)>1\) is defined as \[ \zeta(s) =\sum_{n=1}^\infty \dfrac{1}{n^s}. 2 Riemann zeta function for nonnegative odd integers; 5 Zeros. 1 De nition and Analyticity The Riemann zeta function is de ned for Re(s) > 1 as follows: (s) = X n 1 1 ns: The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact: Theorem 1 Suppose that f1;f2;::: is a 25 Zeta and Related Functions Riemann Zeta Function 25. The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler’s identity: [ (s) :=] X1 n=1 n @s= Y pprime 0 X1 c p=1 p c ps 1 A= Y pprime 1 1 p s: (1) We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. In particular, the Riemann Hypothesis is a conjecture about the roots of the zeta function. 6. The L-function of a Dirichlet character ˜modulo qis de ned by L(s;˜) = X1 n=1 ˜(n)n s: We view (s) = P 1 n=1 n s as the L-function of the principal character modulo 1, more precisely, (s 15 The Riemann zeta function and prime number theorem. Euler-MacLaurin summation formula 1. Surprisingly, Riemann zeta functions are noted to have an intricate connection to not only the natural numbers but also the distribution of prime numbers. bers can be expressed and thereby computed. 1 Connection with exponent pairs and dual exponent pairs; 6. 2 Known bounds on \(\mu \) 6. s;b/ 1 bs: In order to determine properties of the Hurwitz zeta function, one strategy is to express it in term of ’known’ zeta functions like the Riemann zeta The surface is a graph of the reciprocal of the absolute value of the Riemann zeta function ζ (s). some of the different facets of Riemann’s zeta function and introduce some of the math that will assist us. Motivation Weil conjectures: The Riemann zeta function The prototypical example of a zeta function is the Riemann zeta function, rst studied by Euler and later studied by Riemann who thought of it as a function on the whole complex plane and in this way was able to use complex analysis. The most important such infinite process is the following fundamental function. s/. The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact: Theorem 1 Suppose that f 1,f 2 the real ray t = 0 or by the same proof we used for the real case. Z function in the complex plane, zoomed out. Zeta[s,a] gives the generalized Riemann zeta function TemplateBox[{s, a}, Zeta2]. A Proof of the Riemann Hypothesis applies to Dirichlet zeta–functions and the Euler zeta function. 1 Definition and Analyticity The Riemann zeta function is defined for Re(s) > 1 as follows: ζ(s) = X n≥1 1 ns. Riemann zeta function. Riemann Zeta Function As a complex valued function of a complex variable, the graph of the Riemann zeta function ζ(s) lives in four dimensional real space. Derivatives at zero. 2 Definition and Expansions This is a modern introduction to the analytic techniques used in the investigation of zeta functions, through the example of the Riemann zeta function. nC1Cb/s D H. The conjectures are as follows. Definition of the Riemann Zeta Function. Zeros of the Riemann zeta function zeta(s) come in two different types. Dirichlet L-series 2. The Riemann zeta function is one of the most interesting objects in mathematics. Mar 14, 2019 · In particular, Riemann’s impact on mathematics and physics alike is demonstrated using methods originating from the theory of numbers and from quantum electrodynamics, i. Riemann’s paper is the Poisson summation formula (2. We rst explore the Riemann zeta function within its de nition and slowly work through what happens when we extend it beyond the original domain. 5 Integral Representations. This function is a generalized form of the Riemann zeta function ζ(s). 1076; Hardy 1999, p. His reasoning to support these two conjectures made free use of the Riemann Hypothesis (RH). One of the simplest is the Euler- basic theorems we will use can be found at the end of these notes. In general, a nontrivial zero of zeta(s) is denoted rho, and the nth nontrivial zero with t>0 is commonly denoted rho_n Mar 5, 2025 · For a real positive t, the Riemann-Siegel Z function is defined by Z(t)=e^(itheta(t))zeta(1/2+it). This relationship (along with the functional equation for the Theta function) will provide insight on the Riemann Zeta function. Notes on the Riemann Zeta Function March 29, 2005 1 The Zeta Function 1. Definition 24. Section 4. The Riemann zeta function is a function very important in number theory. The code Plot_Riemann_Zeta_Critical_Line. The completed zeta function ˆζ has an analytic continuation to 1. Examples include the Riemann P-series, Riemann-Siegel functions, Riemann theta function, Riemann zeta function, xi-function, the function F(x) obtained by Riemann in studying Fourier series, the function R(x,y;xi,eta) appearing in the application of the Riemann method for solving the Goursat problem Feb 27, 2025 · Zeta function, in number theory, an infinite series given by where z and w are complex numbers and the real part of z is greater than zero. ) Zeta Functions. 5 Integral Representations 25. We begin with the progenitor Jun 29, 2022 · $\zeta$-function. Ayoub, "Euler and the zeta Nov 15, 2019 · This section is about the properties of the Riemann $$\zeta $$-function. 5. The Riemann zeta function ζ(s) is the most important member of the significantly large family of zeta functions The analytic continuation of ζ n (s, a) is based on the convergence of an n-ple series. May 20, 2017 · This review paper will focus on the study of a very famous and important function, the Riemann zeta function. The coloring of the complex function-values used above: positive real values are presented in red. Many alternative proofs have been discovered since, with Titchmarsh listing seven (The Theory of the Riemann Zeta Function, 1986, second edition). This paper focuses on the introduction and proof of the fundamental properties of ζ(z), i. Oct 1, 2006 · Cell[BoxData[RowBox[List[RowBox[List["Zeta", "[", "s", "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["s", "-", "1"]]], RowBox[List[UnderoverscriptBox Feb 9, 2018 · A more sophisticated analysis of the zeta function along these lines can be used to prove both the analytic prime number theorem and Dirichlet’s theorem on primes in arithmetic progressions 1 1 In the case of arithmetic progressions, one also needs to examine the closely related Dirichlet L –functions in addition to the zeta function itself. In particular, we will get a functional equation and meromorphic continu-uation for the Riemann Zeta function. s). . The spikes correspond to the zeros on the critical line ½ + iy. May 24, 2024 · Quantum Field Theory: In quantum field theory, the Riemann zeta function is used for regularization of divergent series and integrals. Riemann’s zeta function to regularize the otherwise infinite results of the so-called Heisenberg–Euler Lagrangian. Much of complex analysis was developed to study it. 10 Zeros About the Project. Feb 26, 2025 · Riemann zeta function, function useful in number theory for investigating properties of prime numbers. It gives us a model for how to think about other L-functions. The Riemann Zeta Function, denoted as ( \zeta(s) ), is a complex function defined for complex numbers (s) where the real part of (s) is greater than 1. To get an idea of what the function looks like, we must do something clever. Delving into the Riemann Zeta Function reveals its classical roots and modern applications, creating a path for new discoveries. 2. 1. 1 Riemann zeta function for nonnegative even integers. 3 Graphics 25. It is easy to see that this series The Riemann zeta function and L-functions We prove some results that will be used in the proof of the Prime Number Theorem (for arithmetic progressions). §25. Chapter 2 will start by introducing Euler’s zeta series, which is essentially the predecessor of Riemann’s zeta function. Zeta The Riemann Zeta Function is but the rst of a zoo of zeta and L-functions for which we can ask similar questions. $ It agrees with $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$ and also is defined outside that Z function in the complex plane, plotted with a variant of domain coloring. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen’s method. Section 3. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Definition 1. 5. More specifically, if we define the completed zeta function ˆζ(s) by ζˆ(s) := π−s2 Γ(s 2)ζ(s), then we have the following. nCa/s: Clearly, H. However, as any casual math enthusiast who’s searched "Riemann zeta function" knows, its definition references a certain idea known as “analytic continuation”. Almost nowhere The Riemann zeta function \(\zeta(z)\) as defined through the special Dirichlet series above, can be continued analytically to an analytic function through out the complex plane C except to the point \(z=1\), where the continued function has a pole of order 1. 1. s) has its origin in the identity expressed by the two formulae ~I {(. Derivatives at other points Riemann zeta spiral along the critical line from height 999000 to a million (from red to violet) The Riemann zeta function has no zeros to the right of σ = 1 or (apart from the trivial zeros) to the left of σ = 0 (nor can the zeros lie too close to those lines). The Poisson summation formula is valid for functions f with suitable regularity properties, such as Schwartz functions: smooth functions The generating function for the terms of the harmonic series: $\frac{1}{n}$ is $-\ln(1 - x)$. Riemann zeta function and explores its applications in algebra. Riemann’s zeta function Riemann’s zeta function is defined to be ζ(s) = X n>0 1 ns. So-called "trivial zeros" occur at all negative even integers s=-2, -4, -6, , and "nontrivial zeros" occur at certain values of t satisfying s=sigma+it (1) for s in the "critical strip" 0<sigma<1. Does an ordinary generating function exist for the terms of the zeta function $\zeta(s) = \sum_{n=1}^\ THE FUNCTION \(s) AND THE DIRICHLET SERIES RELATED TO IT 1. May 14, 2017 · The function whose rule is $\sum_{n=1}^\infty n^{-s}$ and whose domain is $\Re(s)>1$ has a ton of extensions to larger domains, but there is a special and natural one called the analytic continuation that extends the domain to $\mathbb C\setminus\{1\}. The Riemann zeta function is the complex function de ned by the series (s) := X n 1 n s; for Re(s) >1, where nvaries over positive integers. Wikipedia: Riemann zeta function. For reasons that will become clear after a while, the more convenient function is ξ(s) = π − s 2Γ s 2 ζ(s) The importance of 2 days ago · Montgomery in 1973 introduced the Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function. (When the series does not converge, but it can be extended to all complex numbers except —see below. ecpkyh jbgi fzjjz mngctm enus ueutg qpuuhy wioxcy iul hitsw fdkc dsqgwimf dhfrba dgajd qvm