Pólya's 1919 conjecture (L(n) ≤ 0 for all n ≥ 2) was disproven in 1958, but the failure pattern is interesting.

  I ran L(n) to 100 million and found:                                                                                     
                                                                                                                           
  - Failures occur only at odd composite numbers                                                                           
  - Even variant: L(n) ≤ 0 holds for all even n (zero exceptions)                                                          
  - Prime variant: L(n) ≤ 0 holds for all primes (5.76M tested)                                                            
                                                                                                                           
  The failures cluster in regions with large prime gaps. Found a predictive formula:                                       
                                                                                                                           
  E[Σλ] ≈ -0.39 + 0.083g                                                                                                   
  where g is gap size. Larger gaps push L(n) positive.                                                                     
                                                                                                                           
  Code and data in the Zenodo link. Curious if these restricted variants are known in the literature - couldn't find       
  references.