A084973 - OEIS

A084973

The maximum departure from the x axis, rounded to the nearest integer, in each cycle of the zeta function for increasingly larger negative values.

%I #3 Mar 30 2012 17:30:55

%S 0,0,0,0,0,0,-1,4,-34,374,-4988,78674,-1449689,30854707,-751125115,

%T 20736542367,-644361764772,22387174696660,-864494448030320,

%U 36906142650945649,-1733457688501062507,89187472319797248472

%N The maximum departure from the x axis, rounded to the nearest integer, in each cycle of the zeta function for increasingly larger negative values.

%C "The zeta function is zero at every negative even number (the trivial zeros) and the successive peaks and troughs now ... get rapidly more and more dramatic as you head west (negative). The last trough I show, which occurs at s = -49.587622654 [6410765611566721701427687663932953145937293907205304283197148592994576700093701122213865946359936710563061421]..., has a depth of about 305,507,128,402,512,981,000,000 (305507128402512978943383.678283221037793184376280971034994413486029678612346873189963110344084662196600996131417814311). You see the difficulty of graphing the zeta function all in one piece." - Derbyshire

%D John Derbyshire, Prime Obsession, Bernhard Riemann And The Greatest Unsolved Primblem In Mathematics, Joseph Henry Press, Washington, D.C., 2003, page 143.

%e a(9) = 34 because between -19 and -21, (at -19.403133257176569932332310530627...), =~ -33.80830359565166465388882152774755514487136542215568...),

%t Table[ Round[ 1/FindMinimum[ 1/Abs[ Zeta[s]], {s, -2t - 1 + {-0.9, +0.9}}, AccuracyGoal -> 50, WorkingPrecision -> 60] [[1]]], {t, 1, 30}]

%K sign

%O -1,8

%A _Robert G. Wilson v_, May 23 2003