Zeros of the Riemann Zeta Function
The roots/zeros of the zeta function, when ζ(s)=0, can be divided into two types which have been dubbed the “trivial” and the “non-trivial” zeros of the Riemann zeta function.
Existence of zeros with real part Re(s) < 0
The trivial zeros are the zeros which are easy to find and explain. They are most easily noticable in the following functional form of the zeta function:
This product becomes zero when the sine term becomes zero. It does so at kπ. So, e.g for a negative even integer s = -2n, the zeta function becomes zero. For positive even integers s = 2n however, the zeros are cancelled out by the poles of the gamma function Γ(z). This is easier to see in the original functional form, where if you put in for s = 2n, the first part of the term becomes undefined.
So, the Riemann zeta function has zeros at every negative even integer s = -2n. These are the trivial zeros, and they can be seen in the plot of the function below:
Existence of zeros with real part Re(s) > 1
From Euler’s product formulation of zeta, we can immediately see that zeta ζ(s) cannot be zero in the area with real part of s larger than 1 because a convergent infinite product can only be zero if one of its factors is zero. The proof of the infinity of the primes denies this.
Existence of zeros with real part 0 ≤ Re(s) ≤ 1
We’ve now found the trivial zeros of zeta in the negative half plane when Re(s) < 0 and shown that there cannot be any zeros in the area Re(s) > 1.
The area between these two areas however, called the critical strip, is where much of the focus of analytic number theory has taken place for the last few hundred years.
In the plot above I have graphed the real parts of zeta ζ(s) in red and the imaginary parts in blue. We see the first two trivial zeros in the lower left when the real part of s is -2 and -4. In between 0 and 1, I have highlighted the critical strip and marked off where the real and imaginary parts of zeta ζ(s) intersect. These are the non-trivial zeros of the Riemann zeta function. Going to higher values we see more zeros, and two seemingly random functions which appear to be getting denser as the imaginary part of s gets larger.
The Riemann Xi Function
We’ve defined the Riemann Xi function ξ(s) (the version of the functional equation which has removed the singularities, and so is defined for all values of s) as:
This function satisfies the relationship
Which means that the function is symmetric about the vertical line Re(s) = 1/2 so that ξ(1) = ξ(0), ξ(2) = ξ(-1) and so on. This functional relationship (the symmetry of s and 1-s) combined with the Euler product formula shows that the Riemann xi function ξ(s) can only have zeros in the range 0 ≤ Re(s) ≤ 1. The zeros of the Riemann xi function in other words correspond to the non-trivial zeros of the Riemann Zeta function. In a sense, the critical line R(s) = 1/2 for the Riemann Zeta function ζ(s) corresponds to the real line (Im(s) = 0) for the Riemann xi function ξ(s).
Looking at the two charts above, one should immediately take note of the fact that all the non-trivial zeros of the Riemann Zeta function ζ(s) (the zeros of the Riemann xi function) have real part Re(s) equal to 1/2. Riemann briefly remarked on this phenomenon in his paper, a fleeting comment which would end up as one of his greatest legacies.
The Riemann Hypothesis
The non-trivial zeros of the Riemann zeta function ζ(s) have real part Re(s) = 1/2.
This is the modern formulation of the unproven conjecture made by Riemann in his famous paper. In words, it states that the points at which zeta is zero, ζ(s) = 0, in the critical strip 0 ≤ Re(s) ≤ 1, all have real part Re(s) = 1/2. If true, all non-trivial zeros of Zeta will be of the form ζ(1/2 + it).
An equivalent statement (Riemann’s actual statement) is that all the roots of the Riemann xi function ξ(s) are real.
In the plot below, the line Re(s) = 1/2 is the horizontal axis. The real part Re(s) of zeta ζ(s) is the red graph and the imaginary part Im(s) is the blue graph. The non-trivial zeros are the intersections between the red and blue graph on the horizontal line.
If the Riemann hypothesis turns out to be true, all the non-trivial zeros of the function will apear on this line as intersections between the two graphs.
Reasons to Believe the Hypothesis
There are many reasons to believe the truth of Riemann’s hypothesis about the zeros of the zeta function. Perhaps the most compelling reason for mathematicians is the consequences it would have for the distribution of prime numbers. The numerical verification of the hypothesis to very high values suggests its truth. In fact, the numerical evidence for the hypothesis is far strong enough to be regarded as experimentally verified in other fields such as physics and chemistry. However, the history of mathematics contains several conjectures that had been shown numerically to very high values and still were proven false. Derbyshire (2004) tells the story of the Skewes number, a very very large number that gave an upper bound, proving the falsity of one of Gauss’ conjectures that the logarithmic integral Li(x) is always greater than the prime counting function. It was disproved by Littlewood without an example, and then shown to must fail above Skewes’ very, very large number ten to the power of ten, to the power of ten, to the power of 34, showing that even though Gauss’ idea had been proven to be wrong, an example of exactly where is far beyond the reach of numerical calculation even today. This could also be the case for Riemann’s hypothesis, which has “only” been verified up to ten to the power of twelve non-trivial zeros.
The Riemann Zeta Function and Prime Numbers
Using the truth of the Riemann hypothesis as a starting point, Riemann began studying its consequences. In his paper he writes; “…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation.” His next objective was relating the zeros of the zeta function to the prime numbers.
Recall the prime counting function π(x)which denotes the number of primes up to and including a real number x. Riemann used π(x) to define his own prime counting function, the Riemann prime counting function J(x). It is defined as:
The first thing to notice about this function is that it is not infinite. At some term, the counting function will be zero because there are no primes for x < 2. So, taking J(100) as an example, the function will be made up of seven terms because the eight term will include an eight root of 100, which is approximately equal to 1.778279.., so this prime counting term becomes zero and the sum becomes J(100) = 28.5333….
Like the prime counting function, the Riemann prime counting function J(x) is a step function which increases in value when:
To relate the value of J(x) to how many primes there are up to and including x, we recover the prime counting function π(x)by a process called Möbius inversion (which I will not show here). The resulting expression is
Remembering that the possible values of the Möbius function are
This means that we can now write the prime counting function as a function of the Riemann prime counting function, giving us
This new expression is still a finite sum because J(x) is zero when x< 2 because there are no primes less than 2.
If we now look at our example of J(100), we get the sum
Which we know is the number of primes below 100.
Translating the Euler Product Formula
Riemann next uses the Euler product formula as a starting point and derives a method for analytically evaluating the prime numbers in the infinitesimal language of calculus. Starting with Euler:
By first taking the logarithm of both sides, then rewriting the denominators in the parenthesis, he derives the relationship
Next, using the well known Maclaurin Taylor series, he expands each log term on the right hand side, creating an infinite sum of infinite sums, one for each term in the prime number series.
Looking at one such term, e.g:
This term, and every other term in the calculation, represents part of the area under the J(x) function. Written as an integral:
In other words, using the Euler product formula, Riemann showed that it is possible to represent the discrete prime counting step function as a continuous sum of integrals. Below our example term is shown as part of the area under the Riemann prime counting function graph.
So, each expression in the finite sum that makes up the prime reciprocal series of Euler product formula can be expressed as integrals, making an infinite sum of integrals that correspond to the area under the Riemann prime counting function. For the prime number 3, this infinite product of integrals is:
Collecting all of these infinite sums together into one integral, the integral under the Riemann prime counting function J(x) can be written as simply:
Or, the more popular form
Riemann had with this connected his zeta function ζ(s) with his Riemann prime counting function J(x) in an identity statement equivalent to the Euler product formula, in the language of calculus.
The Error Term
Having obtained his analytic version of the Euler product formula, Riemann next went on to formulate his own prime number theorem. The explicit form he gave was:
This is Riemann’s explicit formula. It is an improvement on the prime number theorem, a more accurate estimate at how many primes exist up to and including a number x. The formula has four terms:
- The first term, or “principle term” is the logarithmic integral Li(x), which is the better estimation of the prime counting function π(x) from the prime number theorem. It is by far the largest term, and like we have seen earlier, an overestimate on how many primes there are up to a given value x.
- The second term, or “periodic term” is the sum of the logarithmic integral of x to the power ρ, summed over ρ, which are the non-trivial zeros of the Riemann zeta function. It is the term that adjusts the overestimate of the principle term.
- The third is the constant -log(2) = -0.6993147…
- The fourth and final term is an integral which is zero for x< 2 because there are no primes smaller than 2. It has its maximum value at 2, when its integral equals approximately 0.1400101….
The two latter terms are infinitesimal in their contributions to the function’s value as x gets large. The main “contributers” for large numbers are the logarithmic integral function and the periodic sum. See their effects in the chart below:
In the chart above, I have approximated the prime counting function π(x) by using the explicit formula for the Riemann prime counting function J(x), and summed over the first 35 non-trivial zeros of the Riemann zeta function ζ(s). We see that the periodic term causes the function to “resonate” and begin to approach the shape of the prime counting function π(x).
Below you can see the same chart, using more non-trivial zeros.
Using Riemann’s explicit function, one can approximate the number of primes up to and including a given number x to a very high accuracy. In fact, Von Koch proved in 1901 that using the zeros of the Riemann hypothesis to error-correct the logarithmic integral function is equivalent to the “best possible” bound for the error term in the prime number theorem.
“..These zeros act like telephone poles, and the special nature of Riemann’s zeta function dictates precisely how the wire — its graph — must be strung between them..” — Dan Rockmore
Epilogue
Since the death of Riemann in 1866 at the modest age of 39, his groundbreaking paper has stood as a landmark in the field of prime- and analytic number theory. To this day Riemann’s hypothesis about the non-trivial zeros of the Riemann zeta function remains unsolved, despite extensive research by great mathematicians for hundreds of years. Numerous new results and conjectures associated with the hypothesis are published each year, in the hope that one day a proof will be tangible.