How Explicit is the Explicit Formula?

Illustrations for Madison Talk

William Stein, Jan 31, 2013

1. Birch and Swinnerton-Dyer's Hunch and Sato-Tate

Sato-Tate for 11a (rank 0) for $p<10^9$

Sato-Tate for 5077a (rank 3) for $p<10^9$

2. Chebyshev Bias

Distribution of primes mod 4 for $p < 10^9$

Prime race mod 4: $\sum_{2 < p\leq X} \left(\frac{-1}{p}\right)$ for $p < 10^9$

3. Riemann's Explicit Formula

Oscillatory Term

$\varepsilon(X,T)$

4. Raw, Medium, and Well Done Sums

Raw Data on a Log Scale for 11a (rank 0) for $p<10^9$

Raw Data on a Log Scale for 5077a (rank 3) for $p<10^9$

Medium Data on a Log Scale for 11a (rank 0) for $p<10^9$

Medium Data on a Log Scale for 5077a (rank 3) for $p<10^9$

Well Done Data on a Log Scale for 11a (rank 0) for $p<10^9$

Well Done Data on a Log Scale for 5077a (rank 3) for $p<10^9$

5. Sarnak's Letter - Conjectures

Raw Mean on a Log Scale for 11a (rank 0) for $p<10^9$

Raw Mean on a Log Scale for 5077a (rank 3) for $p<10^9$

Medium Mean on a Log Scale for 11a (rank 0) for $p<10^9$

Medium Mean on a Log Scale for 5077a (rank 3) for $p<10^9$

Well Done Mean on a Log Scale for 11a (rank 0) for $p<10^9$

Well Done Mean on a Log Scale for 5077a (rank 3) for $p<10^9$

6. Minimalist Conjecture?

Raw Mean on a Log Scale for 128b (rank 0, but Sym^{7} extra vanishing) for $p<10^9$

Raw Mean on a Log Scale for 2379b (rank 0, but Sym^{3} extra vanishing) for $p<10^9$

7. Oscillatory Term -- the Bayesian

Plot of Oscillatory Term S(X,T)/log(X) for $X<10^{30}$ (log scale) and with $T$ changing from small to $500$ (animated) for 11a (rank 0)

Plot of Oscillatory Term S(X,T)/log(X) for $X<10^{30}$ (log scale) and with $T$ changing from small to $500$ (animated) for 128b (rank 0, sym pow)

Plot of Oscillatory Term S(X,T)/log(X) for $X<10^{30}$ (log scale) and with $T$ changing from small to $500$ (animated) for 5077a (rank 3)

Plot of Distribution of Values of S(X,10000) for 11a (rank 0)

Plot of Distribution of Values of S(X,10000) for 128b (rank 0)

Plot of Distribution of Values of S(X,10000) for 5077a (rank 3)

8. Convergent Term

Plot of C(X,T) for 11a (rank 0) for $X<10^9$ and $T=10000$