Riemann zeta function: resources
"The zeta function is probably the most challenging
and mysterious object of modern mathematics, in spite of its utter
M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics
"We may – paraphrasing the famous sentence of George Orwell – say that
'all mathematics is beautiful, yet some is more beautiful than the other'.
But the most beautiful in all mathematics is the zeta function. There is no
doubt about it."
"...a variety of evidence suggests that underlying Riemann's zeta
function is some unknown classical, mechanical system whose trajectories
are chaotic and without [time-reversal] symmetry, with the property that,
when quantised, its allowed energies are the
Riemann zeros. These connections between the
seemingly disparate worlds of quantum mechanics and number theory are
M. Berry, "Quantum
Physics on the Edge of Chaos" (New Scientist, 19 November 1987)
basic introduction to the Riemann zeta function
Wikipedia: Riemann zeta function
WolframMathworld: Riemann zeta function
A. Weil, "Prehistory of the zeta-function", from Number Theory, Trace Formulas
and Discrete Groups, K.E. Aubert, E. Bombieri and D. Goldfeld, eds. (Academic,
R.G. Ayoub, "Euler and the zeta function", American Mathematical Monthly 81 (1974) 1067–1086
K. Devlin, "How Euler discovered the zeta function"
(elementary historical introduction to the function which Riemann later extended to the complex plane)
Z. Rudnick, "Number theoretic
background" (covers all the number theory necessary for a basic understanding of the
Riemann Zeta Function, which is covered in its final section)
E.C. Titchmarsh, "The zeros of the Riemann zeta-function",
Proc. Royal Soc. London 151 (1935) 234–255
Riemann's original eight-page paper introducing his zeta function
(English translation PDF)
Critical Strip Explorer v0.67, a wonderful applet produced by Raymond Manzoni
for this site – explore the behaviour of the Riemann zeta function in
and around the critical strip in a highly visual, interactive way. The
resulting images are quite astonishing!
more applets for exploring the behaviour of the Riemann zeta function can be found on Glen Pugh's homepage, as well as here.
the functional equation of the zeta function and related issues
Noam D. Elkies' Analytic Number Theory lecture notes (Harvard University, Spring 1998)
summary of Ilan Vardi's excellent "Introduction to analytic number theory"
J. Borwein, D. Bradley and R. Crandall, "Computational strategies
for the Riemann zeta function", J. Comp. App. Math. 121 (2000) 247–296
J. Arias-de-Reyna, "X-Ray of Riemann zeta-function" (preprint, revised 09/2003)
Wadim Zudilin's bibliography of literature treating specific values of
Riemann's Zeta Function (Academic Press, 1974)
The Riemann Zeta-Function: The Theory of the Riemann
Zeta-Function with Applications (Wiley, 1985)
An Introduction to the Theory of the Riemann Zeta-Function,
(Cambridge University Press, 1988)
A.A. Karatsuba, S.M. Voronin, N. Koblitz,
The Riemann Zeta-function (de Gruyter, 1992)
The Theory of the Riemann Zeta-Function, 2nd edition
– revised by D. Heath-Brown (Oxford University Press, 1986).
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