Riemann Zeta Function, denoted by ζ(s), is a mathematical function that is defined for complex numbers. It’s named after the German mathematician Bernhard Riemann, who introduced it in 1859.

In this article, we will understand the meaning of the Riemann zeta function, the properties of the Riemann zeta function, the Functional Equation, the Euler product formula, Zeros of the Zeta Function and its applications.

What is Riemann Zeta Function?

Riemann zeta function is a mathematical function that is used in number theory and complex analysis. It is written as ζ(s) and is defined for complex numbers (s). The function or s ∈ C with Re(s) > 1 can be written mathematically as:

[Tex]\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}[/Tex]

where (s) is a complex number with a real part greater than 1.

This formula means adding up the reciprocals of all positive integers raised to the power of (s). The function helps in studying the distribution of prime numbers and has important applications in mathematics. It can be extended to other values of (s) through the analytic continuation process.

Riemann zeta function is central to the famous Riemann Hypothesis, which concerns the locations of the function’s zeros.

Definition of Riemann Zeta Function

Riemann zeta function, ζ(s), is a function involving a complex variable s = σ + it, where σ and t are real numbers. For σ > 1, it is defined by the infinite series:

[Tex]\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}[/Tex]

This series converges for σ > 1. The function can also be represented using the gamma function and an integral. The zeta function can be extended to almost all complex values of s, through a process called analytic continuation, except at s = 1 where it has a simple pole.

Properties of Riemann Zeta Function

Various properties of Riemann zeta functions are:

Reciprocal

Reciprocal of the Riemann zeta function can be expressed using the Möbius function, μ(n):

[Tex]\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}[/Tex]

This formula holds for complex numbers ( s ) with a real part greater than 1. Similar relationships exist for other multiplicative functions.

Riemann Hypothesis

Riemann Hypothesis states that the reciprocal expression remains valid when the real part of (s) is greater than 1/2.

Universality

Zeta function has a property called universality, meaning there are points on the critical strip (where 0 < Re(s) < 1) that can approximate any holomorphic function. This was first proved by Sergei Voronin in 1975.

Estimates of Maximum Values

Let F(T; H) and G(s₀​; Δ) represent the maximum modulus of the zeta function on certain intervals:

• F(T; H) = max_{∣t−T∣ ≤ H}​ ∣​ζ(21​+it)∣

• G(s_0​; Δ) = max_{∣s−s0​∣ ≤ Δ} ​∣ζ(s)∣

These functions help estimate how large the zeta function values can be on short intervals along the critical line or within small neighborhoods in the critical strip.

Argument of Zeta Function

The argument of the zeta function is given by:

[Tex]S(t) = \frac{1}{\pi} \arg \zeta\left(\frac{1}{2} + it\right)[/Tex]

This function S(t) is related to the change in the argument of ζ(s) along specific paths in the complex plane.

Sign Changes

There are theorems about the sign changes of S(t). For example, every interval (T, T + H] with H ≥ T^27/82+ϵ
contains at least:

[Tex]H \frac{\sqrt[3]{\ln T}}{e^{c\sqrt{\ln \ln T}}}[/Tex]

points where S(t) changes sign. Similar results were obtained earlier for other conditions by Atle Selberg.

Functional Equation

Riemann zeta function satisfies the following functional equation:

ζ(s) = 2^sπ^s−1 sin(πs/2​) Γ(1−s) ζ(1−s)

where Γ(s) is Gamma Function

This equation is valid for all complex numbers ????s and relates the values of the zeta function at ????s and 1-s.

NOTE:

• Equation shows a symmetry between ζ(s) and ζ(1-s), connecting their values.
• Sine function, sin(πs/2​), causes ζ(s) to have simple zeros at even negative integers, known as the trivial zeros.
• At even positive integers, the product sin(πs/2​) Γ(1−s) does not cancel out, that indicates non-trivial zeros. The gamma function Γ(1-s) has poles at these points, which cancel the zeros of the sine function.

Euler Product Formula

In 1737, Euler discovered the connection between the zeta function and prime numbers. He proved the following identity:

[Tex]\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \, \text{prime}} \frac{1}{1 – p^{-s}} [/Tex]

• Sum and Product: The left side is the sum over all positive integers raised to the power s. The right side is an infinite product over all prime numbers p.

• Geometric Series: This product formula uses the geometric series and the fundamental theorem of arithmetic.

• Convergence: Both sides converge when the real part of s is greater than 1.

Infinite Primes: When s=1, the harmonic series diverges, implying there are infinitely many primes.

Sum of Reciprocals: The logarithm of the product term [Tex]\frac{p}{p – 1}[/Tex]​ approximates to 1/p​, which proves that the sum of the reciprocals of primes is infinite.

Density of Primes: Combining this with the sieve of Eratosthenes shows that the density of primes among positive integers is zero.

Coprime Probability: The Euler product formula can calculate the probability that s randomly selected integers are coprime. The probability that any single number is divisible by a prime p is 1/p​. Thus, the probability that at least one of s numbers is not divisible by p is [Tex]1 – \frac{1}{p^s}[/Tex]. The probability that s numbers are coprime is:

[Tex]\prod_{p \, \text{prime}} \left(1 – \frac{1}{p^s}\right) = \left(\prod_{p \, \text{prime}} \frac{1}{1 – p^{-s}}\right)^{-1} = \frac{1}{\zeta(s)}[/Tex]

This shows that the Euler product formula links the zeta function to the probability of numbers being coprime.

Zeros of Riemann Zeta Function

Zeros of the Riemann zeta function, denoted as ζ(s), are the values of the complex variable s for which ζ(s) = 0. There are two types of zeros: trivial and non-trivial.

Trivial Zeros

Trivial zeros occur at negative even integers s = -2, -4, -6,…. These are considered “trivial” because their existence is relatively easy to prove using the functional equation of the zeta function.

Non-trivial Zeros

Non-trivial zeros are the complex values of s for which ζ(s) = 0 and have real part between 0 and 1. They are called “non-trivial” because their distribution is less understood and their study is important for understanding prime numbers and related objects in number theory.

Riemann Hypothesis

Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, proposed by Bernhard Riemann in 1859. It concerns the zeros of the Riemann zeta function ζ(s), a complex function defined for complex numbers ‘s’.

Statement of the Riemann Hypothesis

Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function, and denoted as ζ(s). It lies on the critical line Re(s) = 1/2​. In other words, if ζ(s) = 0 and s is not a negative even integer, then Re(s) = 1/2​.

This conjecture was formulated by Bernhard Riemann in 1859. It remains one of the most important and challenging unsolved problems in mathematics.

Also Read: Applications of Conjectures

Special Values of Riemann Zeta Function

Riemann zeta function, denoted as ????(????)ζ(s), takes on special values at certain inputs.

Zeta at Even Positive Integers

For positive even integers 2n, the Riemann zeta function evaluates to:

[Tex]\zeta(2n) = \frac{|B_{2n}| (2\pi)^{2n}}{2(2n)!}[/Tex]​

Here, B_2n​ represents the (2n)^th Bernoulli number. This formula provides a direct way to calculate the zeta function at even positive integers.

Zeta at Odd Positive Integers

For odd positive integers 2n + 1, there isn’t a known simple expression like the one for even integers. However, these values are related to the algebraic K-theory of integers.

Zeta at Negative Integers

For negative integers -n, where ????n is a non-negative integer, the zeta function evaluates to:

ζ(-n)= [Tex]-\frac{B_{n+1}}{n+1}[/Tex]

Here, B_k​ denotes the k^th Bernoulli number. Particularly, ζ(0) and ζ(-1) have notable values:

• ζ(0) = -1/2​: This value can be interpreted as assigning a finite result to the divergent series 1+1+1+⋯.

• ζ(-1) = -1/12​: This value arises from analytic continuation and has applications in contexts like string theory.

Applications of Riemann Zeta Function

Riemann Zeta Function find its uses in various fields and some of them are:

Quantum Field Theory: In quantum field theory, the Riemann zeta function is used for regularization of divergent series and integrals. It plays a role in calculating the Casimir effect, a physical force arising from quantized fields between closely placed conductive plates.

Dynamical Systems Analysis: The zeta function is also valuable in analyzing dynamical systems, providing insights into their behavior and properties.

Musical Tuning: In music theory, the zeta function helps find equal divisions of the octave (EDOs) that closely match the intervals of the harmonic series. This aids in creating musical scales with specific tonal properties.

Infinite Series Representations: The zeta function appears in various infinite series representations of constants. For example, it is involved in the sum of reciprocals of squares, which equals π²/6​, known as the Basel problem. Other series involve polygamma functions and Euler’s constant.

FAQs on Riemann Zeta Function

How is zeta (- 2) zero?

Zeta(-2) is zero because it satisfies the functional equation of the Riemann zeta function, resulting in its value being zero at negative even integers.

What is the value of zeta 1?

The value of zeta(1) is infinite, as the series 1 + 1/2 + 1/3 + ⋯ diverges.

What is Real Part of Riemann Zeta?

Real part of Riemann zeta function varies depending on the input parameter ‘s’. For s with real part greater than 1, the real part of zeta(s) is greater than 1. However, for ????s with real part less than or equal to 1, the real part of zeta(s) is less than or equal to 1.

Who gives Zeta Function?

Riemann zeta function was introduced by the German mathematician Bernhard Riemann in his 1859 paper “On the Number of Primes Less Than a Given Magnitude,” which revolutionized the study of prime numbers.

What is Zeta Function Symbol?

Lowercase Greek letter “zeta” (ζ) is the symbol used to represent the Riemann zeta function in mathematical notation.

What is Use of Zeta function?

Riemann zeta function is used in various areas of mathematics, physics, and music theory. It helps analyze the distribution of prime numbers, regularize divergent series in quantum field theory, study dynamical systems, and determine equal divisions of the octave in musical tuning, among other applications.