Back to Riemann, Part 6

In the last entry, I introduced the prime number counting function, pi(x), and one equation used for approximating it. Remember that pi(x) actually does count the number of prime numbers less than or equal to “x”. But, this is a very slow process, and it doesn’t tell us anything about how the primes are distributed across the real axis. The approximation, x/ln(x), at the least, comes close to pi(x) for sufficiently large values of x. And, the relative error, as given by (pi(x) – x/ln(x))/x goes to zero for large x (there’s still an error, but it’s comparatively small given how many prime numbers we’re really talking about).

Now, we can define any kind of counting function that we like, such as the number of even numbers less than or equal to “x”, the number of odd numbers, or the number of cups of coffee I drink each day of the week. And, if we can plot those numbers, we can try to find an approximation for them in order to predict future events. In cases like with the even numbers, the approximation is going to be 100% accurate (T(x) = x/2, truncated), and the error is going to be 0 for all x. For a function Tc() for counting cups of coffee per day, Tc(x)=Number_of_Days will have an error of +/- 2 depending on how tired I am, but the relative error Tc(x)/x will go to zero for large “x” (that is, for a period of 3 weeks or longer), because I normally only drink one cup a day.

The Russian mathematician Pafnuti Chebyshev, in about 1850, introduced a new logarithmic prime counting function he called psi(x). He defined psi(x) to be the number of prime powers less than or equal to x, but he weighted them based on how frequently they occurred within the range.

psi(x) = T2*ln(2) + T3*ln(3) + T5*ln(5) + T7*ln(7) + T11*ln(11)…

In this case, the T components are counting functions for each prime number. Say we want to look at psi(100).

The powers of 2 less than 100 are: 2, 4, 8, 16, 32 and 64, so T2 = 6.
The powers of 3 less than 100 are: 3, 9, 27 and 81, so T3 = 4.
The powers of 5 less than 100 are: 5 and 25, so T5 = 2.
The powers of 7 less than 100 are: 7 and 49, so T7 = 2.
The powers of all other primes up to 97 occur once only.
After that, T=0 for all primes > 100.

psi(100) = 6*ln(2) + 4*ln(3) + 2*ln(5) + 2*ln(7) + 1*ln(11) + 1*ln(13)…
= 94

The idea here is that although it’s not practical to count the prime powers, this approximation goes to “x” for large values of “x”, and the relative error, (psi(x) – x)/x goes to zero. (Ex.: x = 100, and psi(100) = 94. Rel error = (100-94)/100 = 0.06)

And, psi(x) is going to have pi(x) terms. pi(100) = 25, so there will be 25 terms in psi(100), from T2*ln(2) up to T97*ln(97). psi(x) can be approximated by pi(x)*ln(x), and this is then further proof that pi(x) can be approximated by x/ln(x).

The take away here is that psi(x) is based on the prime numbers themselves, while pi(x) is tied specifically to x and ln(x).

Ok, so let’s go back to the zeta function,
z(x) = 1 + 1/2^x + 1/3^x + 1/4^x…

Euler used a modified form of the zeta function, defined over the range -1 < x < 1, as follows:
M(x) = 1 + x + x^2 + x^3 + x^4…

Now, if you multiply both sides by “x”,
x*M(x) = x + x^2 + x^3 + x^4 + x^5…

And then subtract equation 1 from equation 2, and group the M(x) terms,
M(x) – x*M(x) = 1 + x – x + x^2 – x^2 + x^3 – x^3…

Meuler(x) = 1/(1 – x) for -1 < x < 1 If we plug 2 into the original equation, Moriginal(2) = 1 + 2^2 + 3^2 + 4^2… the series goes to infinity. But, using the identity, Meuler(2) = 1/(1 – 2) = -1, the equation converges. We can use a similar approach with the zeta function. Recall that in its original form, z(x) converges only for x > 1 (so its domain is 1 < x < infinity). We can extend this domain as follows by defining eta(), as invented by Gustav Dirichlet (1805-1859):

eta(x) = 1 – 1/2^x + 1/3^x – 1/4^x + 1/5^x…

The minus signs cause the terms to cancel out faster, and the series then grows more slowly for positive x, even when x < 1.

(Coverage of domains by zeta(x) (top line) and eta(x) (bottom line).)

If we return to the geometric series (which I used to show that e^x is directly related to cos(x) and sin(x)), we get another equation, specifically:

ln(x+1) = x – x^2/2 + x^3/3 – x^4/4 + x^5/5…

Plugging in x = 1,

ln(2) = 1 – 1/2 + 1/3 – 1/4 + 1/5…
eta(1) = 1 – 1/2 + 1/3 – 1/4 + 1/5…
So that eta(1) = ln(2) = 0.693…

and, eta(0.5) = 0.6049…

Ok, what we want is the connection between zeta and eta. First, subtract both sides:

z(x) = 1 + 1/2^x + 1/3^x + 1/4^x…
e(x) = 1 – 1/2^x + 1/3^x – 1/4^x + 1/5^x…

z(x) – e(x) = (1-1) + (1/2^x + 1/2^x) + (1/3^x – 1/3^x)…
z(x) – e(x) = 2/2^x + 2/4^x + 2/6^x…

Factor out 2/2^x = 2^(1-x)

z(x) – e(x) = 2^(1-x) * (1 + 1/x^2 + 1/x^3 + 1/x^4…)
z(x) – e(x) = 2^(1-x) * z(x)

e(x) = z(x) * (1 – 2^(1-x)), or
z(x) = e(x) / (1 – 2^(1-x))

What this means is that, while we can’t find zeta(x) for x between 0 and 1, we can find eta(x) for that range. And, using the conversion formula here, we can map eta(x) to zeta(x) for this same range.

Because eta(x) converges for all real numbers x, with the exception of 1, we can define the behavior of the zeta function for 0 < x < 1. This extends the domain of the zeta function beyond its original 1 < x < infinity.

As an example, eta(0.5) was shown to be 0.6049… above. So,
zeta(0.5) = 0.6049 / (1 – sqrt(2)) = -1.46035…

And,
eta(0) = 0.5
zeta(0) = 0.5 / (-1) = -0.5

The next step is to expand the domain for the zeta function to the complex plane.

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