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- PNAS - New progress on the zeta function: From old conjectures to a major breakthrough
- CORE - Zeta Functions of Graph Coverings
- Dartmouth Digital Commons - The Zeta Function of a Hypergraph (PDF)
- University of Exeter IT Services - How Euler discovered the zeta function
- University of Kentucky, College of Arts and Sciences - Department of Mathematics - Euler and the Zeta Function (PDF)
- The University of Chicago - Department of Mathematics - The Zeta Function and the Riemann Hypothesis (PDF)
- International Journal of Advanced Research in Science, Communication and Technology - Statistical Study of Euler�s Zeta Function by using Riemann Hypothesis
zeta function
zeta function, innumber theory, aninfinite series given by
wherez andw arecomplex numbers and the real part ofz is greater than zero. Forw = 0, thefunction reduces to theRiemann zeta function, named for the 19th-century German mathematicianBernhard Riemann, whose study of its properties led him to formulate theRiemann hypothesis.
The zeta function has a pole, or isolatedsingularity, atz = 1, where theinfinite series diverges toinfinity. (A function such as this, which only has isolated singularities, is known asmeromorphic.) Forz = 1 andw = 0, the zeta function reduces to the harmonic series, or sum of theharmonic sequence (1,1/2,1/3,1/4,…), which has been studied since at least the 6th centurybce, when Greek philosopher and mathematicianPythagoras and his followers sought to explain through numbers the nature of the universe and the theory of musicalharmony.