Riemann zeta function

The Riemann zeta function, often called the Euler–Riemann zeta function, is a vital mathematical tool. It is a function of a complex variable, which means it uses numbers with both real and imaginary parts. Mathematicians use the Greek letter zeta (ζ) to represent it. This function is a cornerstone of analytic number theory. It also finds important uses in physics, probability theory, and applied statistics.

To understand how the function works, we look at a specific type of summation. For values where the real part of the variable is greater than one, the function is defined as an infinite sum. This sum is a classic example of a Dirichlet series. In this range, the series converges absolutely to an analytic function. However, the series does not work for all numbers. For example, when the variable is one, the series becomes the harmonic series. The harmonic series diverges, meaning its sum grows toward infinity.

Because the simple sum only works for some numbers, mathematicians use a process called analytic continuation. This allows the function to be defined for almost all complex values. Through this process, the zeta function is revealed to be a meromorphic function. This means it is smooth and well-behaved everywhere on the complex plane, except at one specific point. That point is a simple pole at $s=1$, where the function reaches infinity. Riemann also established a functional equation that relates different values of the function. This equation connects the values at $s$ to the values at $1-s$, using the gamma function to bridge them.

History shows how this function evolved through the work of great thinkers. Leonhard Euler first studied the function using real, positive integers during the mid-eighteenth century. In 1737, Euler discovered a profound connection between this function and prime numbers. He proved the Euler product formula, which shows the function is equal to an infinite product involving every prime number. This discovery helped prove that there are infinitely many primes. In 1859, Bernhard Riemann published a landmark paper titled "On the Number of Primes Less Than a Given Magnitude."

Riemann extended Euler's work to complex variables and changed number theory forever.

Riemann's work highlighted the importance of the function's zeros. There are two types of zeros in the zeta function. The first type is called the trivial zeros. These occur at every even negative integer, such as -2, -4, and -6. They are called trivial because their existence is easy to prove using the functional equation. The second type consists of the non-trivial zeros. These are much more mysterious and are located within a specific region called the critical strip. This strip is defined as the area where the real part of the variable is between zero and one.

The most famous mystery in mathematics is the Riemann hypothesis. This is a conjecture about where the non-trivial zeros are located. Riemann suggested that all these non-trivial zeros lie exactly on a single line called the critical line. This line is located where the real part of the complex number is exactly one-half.

While mathematicians have not yet proven this, they have made significant progress. In 1989, mathematician Conrey proved that more than 40% of the non-trivial zeros lie on this line. Later research improved this figure to 41.7%. The location of these zeros is vital because it directly relates to how prime numbers are distributed.

Beyond the mystery of primes, the zeta function has many specific, measurable properties. Euler calculated the values of the function at even positive integers. For example, the value at $s=2$ provides a solution to the famous Basel problem. In 1979, Roger Apéry proved that the value at $s=3$ is an irrational number, which is now known as Apéry's constant. The function also behaves predictably at negative integers, where it produces rational numbers. These values are important in the theory of modular forms. Today, the study of the zeta function continues to connect complex analysis, prime number distribution, and even the fundamental laws of physics.