← Session 5 — Phase 0: paper deep-read + Bombieri–Lagarias literature
thinking (3261 chars)
Committed. Now proceed to fork 2: 3.2 continuation — prime-side Li coefficients (Bombieri–Lagarias exact formulas) to push past n=116.
The Li coefficients: λ_n = Σ_ρ [1 − (1−1/ρ)^n]. Bombieri–Lagarias give an exact finite formula for λ_n in terms of primes + constants:
λ_n = (n/2)(log...)... The known exact expression (Bombieri–Lagarias 1999):
λ_n = −Σ_{k=1}^{n} ... Let me recall:
λ_n = (n/2)(γ_E − log(4π)·...)? Known: λ_n = Σ_{j=1}^{n} (−1)^... Actually the Bombieri–Lagarias formula:
λ_n = (1/2)(n·log(2π)·...)? Let me derive properly rather than misremember.
ξ(s) = ½s(s−1)π^{−s/2}Γ(s/2)ζ(s). Li coefficients λ_n = Σ_ρ [1−(1−1/ρ)^n] = (1/(n−1)!) (d/ds)^n [s^{n−1} log ξ(s)]|_{s=1}... something like that. Bombieri–Lagarias: λ_n = (1/(n−1)!) d^n/ds^n [s^{n−1} log ξ(s)] at s=1. Expanding ξ into its factors gives λ_n = S_1(n) + S_2(n) + S_3(n):
- from ½s(s−1): elementary: λ contribution: 1 − (1−1/1)^n + 1 − (1−1/0)^n?? careful — the poles at s=0,1 aren't zeros; the formula via log ξ derivative still works with ξ including the ½s(s−1) factor, and since ξ(s) = ξ(1−s) product over zeros... The standard result (BL Thm 2 / equivalent forms in the literature):
λ_n = n·(log(2π)/2)·...
The classical decomposition (e.g., as in Lagarias or Bombieri–Lagarias):
λ_n = S_1(n) + S_2(n) + S_3(n) where
S_1(n) = (n/2)(γ_E + log(4π)·...)? I don't trust memory. Better: compute λ_n via the derivative formula directly with mpmath: λ_n = (1/(n−1)!) (d/ds)^n [s^{n−1} log ξ(s)]_{s=1} — mpmath can do high-order derivatives via diffs, but n up to 10^5 — NO. Need the prime-side exact formula.
The known exact formula (Bombieri–Lagarias 1999, "Complements to Li's criterion"):
λ_n = −Σ_{j=1}^n binom(n,j)·(1/(j−1)!)·... hmm.
Actually there's the well-known arithmetic formula:
λ_n = (n/2)log(2π) − n + 1 − (γ_E/2)n·... no...
Let me look it up: the standard statement (Li's criterion literature):
λ_n = S_1(n) + S_2(n) + S_3(n) + S_4(n), with
S_1(n) = (n/2)(γ_E − 1 − ...
I should just derive it or fetch. Derivation sketch: define λ_n = Σ_ρ [1 − (1−1/ρ)^n]. Using (1−1/ρ)^n = e^{n log(1−1/ρ)}... The BL formula:
λ_n = −Σ_{k=1}^{n} (n choose k) η_{k−1}·(−1)^... where η_k are the Laurent coefficients of log ξ... The η-series: log ξ(s) around s=1: coefficients η_m = (−1)^{m+1}/m! (d/ds)^{m+1}...
Cleanest computable exact formula (from the literature, e.g., used in all Li numerics):
λ_n = (n/2)(log 2π...
OK let me just fetch it — Bombieri–Lagarias exact arithmetic formula for λ_n. It's:
λ_n = 1 − (γ_E + log 4π)n/2·... nope. FETCH.
Actually I recall now with more confidence — the formula (e.g., in Lagarias "Li coefficients for automorphic L-functions", Thm 2.1 for zeta):
λ_n = S_1(n) + S_2(n) + S_3(n)
S_1(n) = (n/2) log(2π) − ... no...
FETCH from web. The MathOverflow link in Notes.txt (equivalent forms of RH) likely has the Li formula. Or Wikipedia "Li's criterion":
λ_n = Σ_{j=1}^{n} (−1)^{j−1}...
Wikipedia Li's criterion: λ_n = (1/(n−1)!) d^n/ds^n [s^{n−1} log ξ(s)]|_{s=1}. And the arithmetic formula:
λ_n = (n/2)(γ_E − log(2√π)) + ... I'll fetch.
Given time (evening), maybe do fork 2 as a scoped sub-stage: fetch formula, implement E-011 (λ_n prime-side to n=10^5), validate against E-008's zero-side values for n ≤ 116. Good plan.
Committed. Moving to **fork 2: prime-side Li coefficients** (3.2 continuation). The goal: Bombieri–Lagarias exact arithmetic formula for λ_n, pushing past the zero-side limit n=116. Let me get the exact formula.
tool: WebFetch ok
input
{
"prompt": "Find the EXACT arithmetic (prime-side) formula for the Li coefficients lambda_n of the Riemann zeta function (Bombieri-Lagarias, \"Complements to Li's criterion for the Riemann hypothesis\", 1999). I need the formula expressing lambda_n = sum_rho [1 - (1 - 1/rho)^n] as a finite sum involving: primes/von Mangoldt, binomial coefficients, the Euler-Mascheroni constant, log(pi), and digamma-related terms. Quote the exact formula with all signs and normalizations, including any S_1(n), S_2(n), S_3(n) decomposition. Also give the definition lambda_n = (1/(n-1)!) (d/ds)^n [s^{n-1} log xi(s)] at s=1 if shown.",
"url": "https://en.wikipedia.org/wiki/Li%27s_criterion"
}
result 2026-08-18 03:47:52Z → 2026-08-18 03:48:17Z
bytes=173982 code=200 code_text='OK' result='The supplied page does **not** include the requested prime-side Bombieri–Lagarias formula: there is no von Mangoldt/binomial/Euler–Mascheroni/log π/digamma decomposition, and no `S_1(n), S_2(n), S_3(n)` split.\n\nIt only gives the zero-side/definition side:\n\n- Definition: “\\lambda _{n}={\\frac {1}{(n-1)!}}\\left.{\\frac {d^{n}}{ds^{n}}}\\left[s^{n-1}\\log \\xi (s)\\right]\\right|_{s=1}”\n- Zero form: “\\lambda _{n}=\\sum _{\\rho }\\left[1-\\left(1-{\\frac {1}{\\rho }}\\right)^{n}\\right]”\n- Sum convention: “\\sum _{\\rho }=\\lim _{N\\to \\infty }\\sum _{|\\operatorname {Im} (\\rho )|\\leq N}”\n- Criterion: RH is equivalent to “\\lambda _{n}>0” for every positive integer n.\n\nIt also notes these are “sometimes defined with a slightly different normalization” and called Keiper–Li/Li coefficients.' duration_ms=25209.1846000003 url='https://en.wikipedia.org/wiki/Li%27s_criterion'
raw content_json (database row)
[{"type":"thinking","thinking":"Committed. Now proceed to fork 2: 3.2 continuation — prime-side Li coefficients (Bombieri–Lagarias exact formulas) to push past n=116.\n\nThe Li coefficients: λ_n = Σ_ρ [1 − (1−1/ρ)^n]. Bombieri–Lagarias give an exact finite formula for λ_n in terms of primes + constants:\n\nλ_n = (n/2)(log...)... The known exact expression (Bombieri–Lagarias 1999):\nλ_n = −Σ_{k=1}^{n} ... Let me recall: \n\nλ_n = (n/2)(γ_E − log(4π)·...)? Known: λ_n = Σ_{j=1}^{n} (−1)^... Actually the Bombieri–Lagarias formula:\nλ_n = (1/2)(n·log(2π)·...)? Let me derive properly rather than misremember.\n\nξ(s) = ½s(s−1)π^{−s/2}Γ(s/2)ζ(s). Li coefficients λ_n = Σ_ρ [1−(1−1/ρ)^n] = (1/(n−1)!) (d/ds)^n [s^{n−1} log ξ(s)]|_{s=1}... something like that. Bombieri–Lagarias: λ_n = (1/(n−1)!) d^n/ds^n [s^{n−1} log ξ(s)] at s=1. Expanding ξ into its factors gives λ_n = S_1(n) + S_2(n) + S_3(n):\n- from ½s(s−1): elementary: λ contribution: 1 − (1−1/1)^n + 1 − (1−1/0)^n?? careful — the poles at s=0,1 aren't zeros; the formula via log ξ derivative still works with ξ including the ½s(s−1) factor, and since ξ(s) = ξ(1−s) product over zeros... The standard result (BL Thm 2 / equivalent forms in the literature):\nλ_n = n·(log(2π)/2)·... \n\nThe classical decomposition (e.g., as in Lagarias or Bombieri–Lagarias):\nλ_n = S_1(n) + S_2(n) + S_3(n) where\nS_1(n) = (n/2)(γ_E + log(4π)·...)? I don't trust memory. Better: compute λ_n via the derivative formula directly with mpmath: λ_n = (1/(n−1)!) (d/ds)^n [s^{n−1} log ξ(s)]_{s=1} — mpmath can do high-order derivatives via diffs, but n up to 10^5 — NO. Need the prime-side exact formula.\n\nThe known exact formula (Bombieri–Lagarias 1999, \"Complements to Li's criterion\"):\nλ_n = −Σ_{j=1}^n binom(n,j)·(1/(j−1)!)·... hmm.\n\nActually there's the well-known arithmetic formula:\nλ_n = (n/2)log(2π) − n + 1 − (γ_E/2)n·... no...\n\nLet me look it up: the standard statement (Li's criterion literature): \nλ_n = S_1(n) + S_2(n) + S_3(n) + S_4(n), with\nS_1(n) = (n/2)(γ_E − 1 − ... \n\nI should just derive it or fetch. Derivation sketch: define λ_n = Σ_ρ [1 − (1−1/ρ)^n]. Using (1−1/ρ)^n = e^{n log(1−1/ρ)}... The BL formula:\nλ_n = −Σ_{k=1}^{n} (n choose k) η_{k−1}·(−1)^... where η_k are the Laurent coefficients of log ξ... The η-series: log ξ(s) around s=1: coefficients η_m = (−1)^{m+1}/m! (d/ds)^{m+1}... \n\nCleanest computable exact formula (from the literature, e.g., used in all Li numerics):\nλ_n = (n/2)(log 2π... \n\nOK let me just fetch it — Bombieri–Lagarias exact arithmetic formula for λ_n. It's:\nλ_n = 1 − (γ_E + log 4π)n/2·... nope. FETCH.\n\nActually I recall now with more confidence — the formula (e.g., in Lagarias \"Li coefficients for automorphic L-functions\", Thm 2.1 for zeta):\nλ_n = S_1(n) + S_2(n) + S_3(n)\nS_1(n) = (n/2) log(2π) − ... no...\n\nFETCH from web. The MathOverflow link in Notes.txt (equivalent forms of RH) likely has the Li formula. Or Wikipedia \"Li's criterion\":\nλ_n = Σ_{j=1}^{n} (−1)^{j−1}... \n\nWikipedia Li's criterion: λ_n = (1/(n−1)!) d^n/ds^n [s^{n−1} log ξ(s)]|_{s=1}. And the arithmetic formula:\nλ_n = (n/2)(γ_E − log(2√π)) + ... I'll fetch.\n\nGiven time (evening), maybe do fork 2 as a scoped sub-stage: fetch formula, implement E-011 (λ_n prime-side to n=10^5), validate against E-008's zero-side values for n ≤ 116. Good plan.","signature":""},{"type":"text","text":"Committed. Moving to **fork 2: prime-side Li coefficients** (3.2 continuation). The goal: Bombieri–Lagarias exact arithmetic formula for λ_n, pushing past the zero-side limit n=116. Let me get the exact formula."},{"type":"tool_use","id":"WebFetch_128","name":"WebFetch","input":{"prompt":"Find the EXACT arithmetic (prime-side) formula for the Li coefficients lambda_n of the Riemann zeta function (Bombieri-Lagarias, \"Complements to Li's criterion for the Riemann hypothesis\", 1999). I need the formula expressing lambda_n = sum_rho [1 - (1 - 1/rho)^n] as a finite sum involving: primes/von Mangoldt, binomial coefficients, the Euler-Mascheroni constant, log(pi), and digamma-related terms. Quote the exact formula with all signs and normalizations, including any S_1(n), S_2(n), S_3(n) decomposition. Also give the definition lambda_n = (1/(n-1)!) (d/ds)^n [s^{n-1} log xi(s)] at s=1 if shown.","url":"https://en.wikipedia.org/wiki/Li%27s_criterion"},"extra_content":null}]