mrly-euler

• The multiplicative structure of a digit design S_F over the digit set F: which designs carry an Euler product over primes, and what stands in its place when none does.
• wall: for every digit set of every base 2 <= base <= 12, whether the indicator of S_F is multiplicative, with the constructed coprime witness verified and an independent minimal witness searched.
• pair: the Dirichlet coefficients of zeta_F(s) M_F(s), and the least n > 1 where the product leaves 1.
• position: the position-product identity sum_(n in D_level) n^(-s) = int_0^1 G_level(t) Z(s,t) dt with G_level(t) = prod_(i<level) sum_(d in F) e(d base^i t) and Z the periodic zeta, evaluated against the direct sum; the Lerch evaluator is checked against mpmath polylog.
• dual: the periodic zeta rebuilt from Hurwitz's formula, DLMF 25.13.3, solved for F(-t, s).
• fibre: the Lerch-Mobius series M(s, a/Q) = sum_(n>=1) mu(n) e(-n a/Q) n^(-s) written as a finite combination of inverse Dirichlet L-functions of modulus dividing Q, checked as a coefficient identity.
• word: the free-monoid zeta 1/(1 - fill base^(-s)) expanded as an Euler product over Lyndon words in the digit alphabet.
• beurling: the free semigroup N_F on the primes lying in S_F, with pi_F(x), N_F(x), the Mertens M_B(x) and its exponent by residue class of log_base x, to x = 10^6.
• Domain: base <= 12 for the wall, minimal witnesses searched to product 4000; coefficients to n = 4000 for the pair; level <= 5 for the position identity; n = 3000 for the fibre identity; u^16 for the word product; x = 10^6 for the Beurling census.

RUN

• uv run python research/lab/py/mrly-euler/euler.py wall
• uv run python research/lab/py/mrly-euler/euler.py pair
• uv run python research/lab/py/mrly-euler/euler.py position
• uv run python research/lab/py/mrly-euler/euler.py dual
• uv run python research/lab/py/mrly-euler/euler.py fibre
• uv run python research/lab/py/mrly-euler/euler.py word
• uv run python research/lab/py/mrly-euler/euler.py beurling
• position runs a few minutes; every other verb is seconds.

WITNESSES

• 8177 digit sets at 2 <= base <= 12: 4083 with 1 outside F, 11 full, 4083 witnessed, split 4072 repunit, 5 odd base, 6 even base; hardest minimal witness base 12, F = {1}, pair (5, 377), product 1885
• least witnesses at small base: base 3 {0,1} (2, 5), base 3 {1,2} (2, 5), base 4 {1,2,3} (2, 9), base 5 {1,2,3,4} (2, 13), base 2 {1} (3, 5)
• zeta_F M_F leaves 1 at n = 4 for base 3 {0,1}, n = 10 for base 3 {1,2}, n = 9 for base 10 missing 9, n = 10 for base 10 missing 0; over 257 sets the least such n is at most 50, and the eight full sets have none below 4000
• position identity error 1.95e-16 and 2.04e-16 at base 10 missing 9, level 3 and level 4, s = 3.3; 1.67e-16 and 1.75e-16 at s = 2.7 + 1.9i; 2.9e-16 at base 3 {0,1}, level 3, 4, 5; Lerch evaluator against polylog below 1.7e-25
• Hurwitz reflection error below 2.1e-30 at s = 3.3, 2.7 + 1.9i and 0.6 + 4.1i
• fibre identity maximum coefficient error 2.6e-12 over eleven pairs (Q, a) including Q = 3, 9, 27, 100, Euler-factor step below 7.4e-16
• Lyndon expansion equals 1/(1 - fill u) through u^16 at fill 2, 3, 4, 9, 10; c_2(level) = 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, A001037
• Beurling census: base 3 {0,2} has the single prime 2 and M_B(x) = 0 for x >= 2; base 3 {0,1} has 525 primes, N_F(920483) = 2198, running max abs(M_B) = 98, exponent 0.3339 against alpha/2 = 0.3155; base 10 missing 9 has 35139 primes, N_F(10^6) = 488864 against x^alpha = 531441, M_B(10^6) = 1860, running max 1866, exponent 0.5452 against alpha/2 = 0.4771, climbing 0.4203, 0.4882, 0.5452 at 10^4, 10^5, 10^6
• the full base-10 control reproduces M(10^4) = -23, M(10^5) = -48, M(10^6) = 212, A084237

SOURCES

• DLMF 25.13 - the periodic zeta F(x,s) at 25.13.1 and Hurwitz's formula at 25.13.3.
• DLMF 25.12 - the polylogarithm expansion at 25.12.12, the evaluator used here.
• Beurling 1937 - generalised prime systems.
• Diamond, Montgomery and Vorhauer 2006 - a Beurling system with a regular integer count whose zeta has infinitely many zeros off any fixed half plane.