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.

0, 0, 0, 0, 0, 0, -1, 4, -34, 374, -4988, 78674, -1449689, 30854707, -751125115, 20736542367, -644361764772, 22387174696660, -864494448030320, 36906142650945649, -1733457688501062507, 89187472319797248472

Offset

-1,8

Comments

"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

References

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

Links

Table of n, a(n) for n=-1..20.

Example

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

Mathematica

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

Crossrefs

Sequence in context: A199752 A264607 A307941 * A234313 A367869 A369191

Adjacent sequences: A084970 A084971 A084972 * A084974 A084975 A084976

Keyword

sign

Author

Robert G. Wilson v, May 23 2003

Status

approved