AI 中文总结
本文量化了对数平均两点Chowla定理的熵减证明，给出均匀的仿射形式和窗口下的有效界，并证明完全显式的非pretentiousness水平界，方法包括显式无零区域、Harnack不等式和有限谐振子。
AI 中文摘要
我们量化了对数平均两点Chowla定理的熵减证明，一致地针对仿射形式和平均窗口。对于L_i(n)=a_i n+b_i，其中斜率为正整数且行列式Delta=a_1 b_2-a_2 b_1非零，设h=max(a_1,a_2,|b_1|,|b_2|,|Delta|,2)。对于每个位置上的Liouville或Mobius函数，且0<epsilon<=1/10，当x>=omega>=exp_4(C h epsilon^(-2))时，在区间x/omega<n<=x上以1/n加权的相关和，其绝对值至多为epsilon log omega，其中exp_4表示四次迭代指数。一个平移引理将公共平移与塔式结构分离：对于lambda(n+b)lambda(n+b+k)，其中k非零，当|k|=o(log_4 omega)且|b|=omega^(o(1))时，消去性是一致的，其中log_4是四次迭代对数。同样的阈值给出了四个Liouville符号对的对数等分布。常数C是有效的，但依赖于两个解析输入中未计算的常数。另外，记ell=log log x，我们证明了完全显式的界ell/3-(4/3)log ell-7 <= N_lambda(x) <= ell-log ell+2.53/ell（对于ell>=64），其中N_lambda是Tao假设中的非pretentiousness水平。在GRH下，下界改进为ell-log ell-log 2-2/ell；因此N_lambda(x)=ell-log ell+O(1)。论证使用了显式的无零区域、Harnack不等式、有限谐振子和光滑Euler乘积的比较。这些结果涉及对数平均，而非自然密度的两点Chowla相关性。
英文摘要
We quantify the entropy-decrement proof of the logarithmically averaged two-point Chowla theorem, uniformly in the affine forms and the averaging window. For L_i(n)=a_i n+b_i with positive integral slopes and nonzero determinant Delta=a_1 b_2-a_2 b_1, put h=max(a_1,a_2,|b_1|,|b_2|,|Delta|,2). For either the Liouville or Mobius function in each position and 0<epsilon<=1/10, the correlation sum over x/omega<n<=x, weighted by 1/n, has absolute value at most epsilon log omega whenever x>=omega>=exp_4(C h epsilon^(-2)), where exp_4 denotes four iterated exponentials. A translation lemma separates the common translation from the tower: for lambda(n+b)lambda(n+b+k), with k nonzero, cancellation is uniform when |k|=o(log_4 omega) and |b|=omega^(o(1)), where log_4 is the fourth iterated logarithm. The same thresholds give logarithmic equidistribution of the four Liouville sign pairs. The constant C is effective but depends on uncomputed constants in two analytic inputs. Separately, writing ell=log log x, we prove fully explicit bounds ell/3-(4/3)log ell-7 <= N_lambda(x) <= ell-log ell+2.53/ell for ell>=64, where N_lambda is the non-pretentiousness level in Tao's hypothesis. Under GRH the lower bound improves to ell-log ell-log 2-2/ell; hence N_lambda(x)=ell-log ell+O(1). The arguments use an explicit zero-free region, Harnack's inequality, a finite resonator, and a comparison of smoothed Euler products. These results concern logarithmic averages, not natural-density two-point Chowla correlations.
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