NTIC The Riemann Explicit Formula

Section 25.7 The Riemann Explicit Formula

Now we are finally ready to see Riemann's result, by plugging in the formula (25.6.1) for

$J$ into the Moebius inverted formula for

$\pi$ we saw just before Remark 25.4.4:

$\begin{equation*} \pi(x)=J(x)-\frac{1}{2}J(\sqrt{x})-\frac{1}{3}J(\sqrt[3]{x})-\frac{1}{5}J(\sqrt[5]{x})+\frac{1}{6}J(\sqrt[6]{x})+\cdots \end{equation*}$

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It is true that Riemann did not prove the following formula fully rigorously, and indeed one of the provers of the Prime Number Theorem mentioned taking decades as part of that effort just to prove all the statements Riemann made in this one paper. Nonetheless, it is certainly Riemann's formula for

$\pi(x)\text{,}$ and an amazing one:

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Fact 25.7.1. Riemann explicit formula.

$\begin{equation*} \pi(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}\left[ Li(x^{1/n})-\sum_{\rho}\left(Li(x^{\rho/n})+Li(x^{\bar{\rho}/n})\right)+\int_{x^{1/n}}^\infty\frac{dt}{t(t^2-1)\log(t)}\right] \end{equation*}$

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It is worth making two points about the transition to this formula. First, if you're wondering where the

$\log(2)$ from (25.6.1) went, it went to 0 because

$\sum_{n=1}^\infty \frac{\mu(n)}{n}=0\text{,}$ though this is very hard to prove. (In fact, it is a consequence of the Prime Number Theorem; see Exercise 25.9.5.)

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Secondly, each

$\rho$ is a zero above the real axis, and then

$\bar{\rho}$ is the corresponding one below the real axis. The summation is over every single zero not on the real axis. In particular, these

$\rho$ are conjectured by the Riemann Hypothesis to all have real part equal to

$1/2\text{,}$ which would make things particularly tidy.

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Now let's see this formula in action.

Prime pi, log integral, and a better approximation — 🔗
Figure $\pi(x)\text{,}$ $Li(x)\text{,}$ and something better than $Li$

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This graphic shows just how good it can get. Again, notice the waviness, which allows it to approximate

$\pi(x)$ not just once per “step” of the function, but along the steps. Try it out interactively below (where we make it somewhat less accurate for the sake of computational speed).

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import mpmath
var('y')
L = lcalc.zeros_in_interval(10,50,0.1)
@interact
def _(n=(100,(60,10^3))):
   P = plot(prime_pi,n-50,n, color='black', legend_label=r'$\pi(x)$')
   P += plot(Li,n-50,n, color='green', legend_label='$Li(x)$')
   G = lambda x: sum([mpmath.li(x^(1/j)) * moebius(j)/j for j in [1..3]])
   P += plot(G,n-50,n, color='red', legend_label = r'$\sum_{j=1}^{%s} \frac{\mu(j)}{j} Li(x^{1/j})$'%3)
   F = lambda x: sum([(mpmath.li(x^(1/j))-log(2) + numerical_integral( 1/(y*(y^2-1)*log(y)), x^(1/j),oo)[0] )*moebius(j)/j for j in [1..3]]) - sum([(mpmath.ei(log(x)*((0.5+l[0]*i)/j)) + mpmath.ei(log(x)*((0.5-l[0]*i)/j))).real for l in L for j in [1..3]])
   P += plot(F,n-50,n,color='blue', legend_label='Really good estimate',plot_points=50)
   show(P)

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We can also just check out some numerical values.

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var('y')
L = lcalc.zeros_in_interval(10,300,0.1)
F = lambda x: sum([(mpmath.li(x^(1/j))-log(2) + numerical_integral(1/(y*(y^2-1)*log(y)),x^(1/j),oo)[0] )*moebius(j)/j for j in [1..3]]) - sum([(mpmath.ei(log(x)*((0.5+l[0]*i)/j)) + mpmath.ei(log(x)*((0.5-l[0]*i)/j))).real for l in L for j in [1..3]])
var('y')
L = lcalc.zeros_in_interval(10,300,0.1)
F = lambda x: sum([(mpmath.li(x^(1/j))-log(2) + numerical_integral(1/(y*(y^2-1)*log(y)),x^(1/j),oo)[0] )*moebius(j)/j for j in [1..3]]) - sum([(mpmath.ei(log(x)*((0.5+l[0]*i)/j)) + mpmath.ei(log(x)*((0.5-l[0]*i)/j))).real for l in L for j in [1..3]])
@interact
def _(n=300):
    print(F(n))
    print(prime_pi(n))
    print(Li(n.n()))
    print(Li(n.n()) - 1/2*Li(sqrt(n.n())) - 1/3*Li((n.n())^(1/3)))

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Many wonderful facts would follow from the truth of the Riemann Hypothesis, or from a natural generalization.

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Fact 25.7.3. Consequences of the (generalized) Riemann Hypothesis.

The following follow from the Riemann Hypothesis or a generalization for things like general Dirichlet series.

The Dirichlet series of the Möbius function would be the multiplicative inverse of the zeta function for lots more complex values than just the real ones we proved it for in .
The value (not just average) of $\sigma(n)$ would have the following bound once $n$ is big enough:

$\begin{equation*} \sigma(n)<e^\gamma \log(\log(n)) \end{equation*}$
The biggest gap between consecutive prime numbers could not be too big (to be precise, $O(\sqrt{p}\log(p)$ ).
We would know exactly what it means for a type of prime to win the ‘prime races’ (see Section 22.1).
Artin's conjecture (Conjecture 17.5.3) on primitive roots follows from a generalization as well.

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So can you prove that there are no other zeros other than those on the critical line to contribute to these approximations to

$\pi(x)\text{?}$ If so, welcome to the future of number theory!