feat(EisensteinSeries): q-expansion coefficients and non-vanishing for E_k#1
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feat(EisensteinSeries): q-expansion coefficients and non-vanishing for E_k#1
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…r E_k
We prove the q-expansion coefficients of the normalised Eisenstein series
E_k for even k ≥ 3: the constant term is 1 and higher coefficients are
-(2k/B_k) * σ_{k-1}(m). As a consequence, E_k ≠ 0.
New results:
- `ArithmeticFunction.sigma_le_pow_succ`: crude bound σ_k(n) ≤ n^{k+1}
- `EisensteinSeries.E_qExpansion_coeff`: all q-expansion coefficients
- `EisensteinSeries.E_qExpansion_coeff_zero`: constant coefficient is 1
- `EisensteinSeries.E_ne_zero`: E_k is non-zero
Also improves the docstring for `ModularFormClass.qExpansion`.
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
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Summary
ArithmeticFunction.sigma_le_pow_succ: crude boundσ_k(n) ≤ n^{k+1}EisensteinSeries.E_qExpansion_coeff: all q-expansion coefficients of normalised Eisenstein series E_k (constant term 1, higher terms-(2k/B_k) * σ_{k-1}(m))EisensteinSeries.E_qExpansion_coeff_zero: constant coefficient is 1EisensteinSeries.E_ne_zero: E_k is non-zero for even k ≥ 3ModularFormClass.qExpansionTest plan
lake env leanon all three modified files compiles with no errors/warnings/sorriesE2/Summable.leanstill builds🤖 Generated with Claude Code