Practice: Analytic number theory

Analytic number theory

Warm-upHuman

Compute $\mu(30) + \mu(36) + \mu(7)$ , where $\mu$ is the Möbius function.

Analytic number theory

Warm-upHuman

Which Dirichlet series equals $1/\zeta(s)$ for $\operatorname{Re} s > 1$ ?

Choose one

\sum_n \mu(n)\, n^{-s}

\sum_n (-1)^n n^{-s}

\sum_n \lambda(n)\, n^{-s}

with

\lambda(n) = (-1)^{\Omega(n)}

(Liouville)

\sum_n \Lambda(n)\, n^{-s}

Analytic number theory

CoreHuman

The prime number theorem says $\pi(x) \sim x/\log x$ . Which statement is a correct reading of it?

Choose one

\pi(x) \big/ (x/\log x) \to 1

as

x \to \infty

.

\pi(x) - x/\log x \to 0

as

x \to \infty

.

\pi(x) = \lfloor x/\log x \rfloor

for all large

x

.

\pi(x) - \operatorname{li}(x) \to 0

.

Analytic number theory

CoreHuman

In the proof that $\zeta(1 + it) \ne 0$ (Theorem 7), why does $\zeta(\sigma + 2it)$ stay bounded as $\sigma \to 1^+$ (for fixed $t \ne 0$ )?

Choose one

\zeta

is holomorphic at

1 + 2it

, because its only pole is at

s = 1

and

1 + 2it \ne 1

.Because

|\zeta(\sigma + 2it)| \le \zeta(\sigma)

.Because

\zeta

has no zeros on

\operatorname{Re} s = 1

.Because of the functional equation relating

s

and

1 - s

.

Analytic number theory

CoreHuman

Gauss's identity $\sum_{d \mid n}\varphi(d) = n$ says $\varphi * \mathbf{1} = \operatorname{id}$ . Möbius inversion gives $\varphi(n) = \sum_{d \mid n}\mu(d)\,\frac{n}{d} = n\prod_{p \mid n}\left(1 - \frac1p\right)$ . Compute $\varphi(60)$ .

Analytic number theory

CoreHuman

Every non-trivial zero $\rho$ of $\zeta$ with $0 < \operatorname{Im}\rho \le 3 \cdot 10^{12}$ has been verified to lie on the critical line. What follows?

Choose one

This is strong evidence for the Riemann hypothesis, but not a proof: it says nothing about the infinitely many zeros higher up.The Riemann hypothesis is proved for all practical purposes.By the functional equation, the remaining zeros must also be on the line.The prime number theorem would fail if the verification had found a zero off the line.

Analytic number theory

StretchHuman

Suppose (hypothetically) that $\zeta$ had a zero $\rho = 0.9 + i\gamma$ . Using the explicit formula, what is the best possible error term one could then hope for in $\psi(x) = x + E(x)$ ?

Choose one

E(x)

would be as large as

x^{0.9 - \varepsilon}

for arbitrarily large

x

, so no bound

O(x^{\Theta})

with

\Theta < 0.9

could hold.

E(x) = O(\sqrt x\log^2 x)

still holds.

\psi(x) \sim x

would fail.No effect: a single zero contributes a bounded term.

Analytic number theory

CoreFormal

Which Lean proposition correctly states the Riemann hypothesis (for Mathlib's riemannZeta : ℂ → ℂ, which assigns a junk value at the pole $s = 1$ )?

Choose one

∀ s : ℂ, riemannZeta s = 0 → (¬ ∃ n : ℕ, s = -2 * (n + 1)) → s ≠ 1 → s.re = 1 / 2∀ s : ℂ, riemannZeta s = 0 → s.re = 1 / 2∀ s : ℂ, riemannZeta s = 0 → (¬ ∃ n : ℕ, s = -2 * (n + 1)) → 0 < s.re ∧ s.re < 1∃ s : ℂ, riemannZeta s = 0 ∧ s.re = 1 / 2

Analytic number theory

StretchComputational

Write mobius(n) for $n \ge 1$ : return $0$ if $n$ has a repeated prime factor, and $(-1)^k$ if $n$ is a product of $k$ distinct primes. Then check numerically that $\sum_{d \mid n}\mu(d) = 0$ for $1 < n \le 200$ (Proposition 1).

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