{"slug":"riemann-hypothesis","name":"The Riemann Hypothesis","statement":"Every nontrivial zero of the Riemann zeta function has real part 1/2.","status":"open","node_count":81,"edge_count":83,"paper_edge_count":63,"composed_count":4,"novel_count":1,"sufficient_count":6,"last_run_at":"2026-08-12T13:30:19.216264+00:00","hubs":[{"node_key":"generalized-riemann-hypothesis","name":"Generalized Riemann Hypothesis","statement":"Every nontrivial zero of every Dirichlet L-function has real part 1/2.","status":"open","origin":"seed","degree":17},{"node_key":"density-hypothesis","name":"The Density Hypothesis","statement":"N(sigma, T) = O(T^(2 - 2 sigma + epsilon)) for the zero-counting function.","status":"open","origin":"seed","degree":3},{"node_key":"lindelof-hypothesis","name":"The Lindelof Hypothesis","statement":"zeta(1/2 + it) = O(t^epsilon) for every epsilon > 0.","status":"open","origin":"seed","degree":3},{"node_key":"grand-riemann-hypothesis","name":"Grand Riemann Hypothesis","statement":"Every nontrivial zero of every automorphic L-function lies on the critical line.","status":"open","origin":"seed","degree":2},{"node_key":"quasi-riemann-hypothesis","name":"The quasi-Riemann hypothesis","statement":"zeta has no zeros with real part greater than theta for some theta < 1.","status":"open","origin":"seed","degree":2},{"node_key":"hilbert-polya","name":"A Hilbert-Polya spectral realization","statement":"The nontrivial zeros are the spectrum of a self-adjoint operator.","status":"open","origin":"seed","degree":1},{"node_key":"lagarias-criterion","name":"Lagarias' criterion","statement":"sigma(n) <= H_n + exp(H_n) log H_n for every n >= 1.","status":"open","origin":"seed","degree":1},{"node_key":"li-criterion","name":"Li's criterion","statement":"The Li coefficients lambda_n are nonnegative for every n >= 1.","status":"open","origin":"seed","degree":1},{"node_key":"pair-correlation","name":"Montgomery's pair correlation conjecture","statement":"The pair correlation of zeta zeros follows the GUE form factor.","status":"open","origin":"seed","degree":1},{"node_key":"robin-inequality","name":"Robin's inequality","statement":"sigma(n) < e^gamma n log log n for every n > 5040.","status":"open","origin":"seed","degree":1},{"node_key":"mobius-sum-bound","name":"Square-root cancellation in the Mertens function","statement":"M(x) = O(x^(1/2 + epsilon)) for every epsilon > 0.","status":"open","origin":"seed","degree":1},{"node_key":"de-bruijn-newman","name":"The de Bruijn-Newman constant","statement":"The de Bruijn-Newman constant Lambda satisfies Lambda <= 0.","status":"open","origin":"seed","degree":1},{"node_key":"farey-franel","name":"The Franel-Landau criterion","statement":"The Farey fraction discrepancy sum is O(x^(1/2 + epsilon)).","status":"open","origin":"seed","degree":1},{"node_key":"mertens-conjecture","name":"The Mertens Conjecture","statement":"The Mertens function satisfies |M(x)| < sqrt(x) for all x > 1.","status":"refuted","origin":"seed","degree":1},{"node_key":"nyman-beurling","name":"The Nyman-Beurling criterion","statement":"The Nyman-Beurling space of dilations of the fractional part function is dense in L^2(0,1).","status":"open","origin":"seed","degree":1},{"node_key":"prime-counting-error","name":"von Koch prime counting error term","statement":"pi(x) = li(x) + O(sqrt(x) log x).","status":"open","origin":"seed","degree":1},{"node_key":"weil-positivity","name":"Weil positivity","statement":"The Weil distribution in the explicit formula is positive definite on the relevant test function space.","status":"open","origin":"seed","degree":1}],"last_run":{"started_at":"2026-08-12T13:30:19.216264+00:00","finished_at":"2026-08-12T13:30:19.257845+00:00","claims_scanned":0,"edges_added":0,"compositions":4,"novel_found":1}}