DATE: Tuesday, December 8 (2009), at 3:00pm
PLACE: LIT 305
SPEAKER: Rob Vary
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TITLE:
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A motivated account of Selberg's identity for the elementary proof of the Prime Number
Theorem
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ABSTRACT:
The prime number theorem asserts that the
number of primes not exceeding x is asymptotically equal
to x/logx. This striking result was proved independently
by Hadamard and de la Vallee Poussin in 1896. But an
elementary proof was not discovered until 1949, when
P. Erdos and A. Selberg, using an identity previously
proved by Selberg in an elementary way, independently
succeeded in giving elementary proofs. These proofs
by Selberg and Erdos, although elementary (in the sense
that no complex variable theory is used), are not simple.
In his 1969 paper, Norman Levinson gives a self contained
and motivated account of an elementary proof of the prime
number theorem by the use of the Iseki-Tatuzawa identity
to obtain Selberg's identity using simplifications due to
Wright and Levinson. In this talk I will give an overview
of Levinson's paper and discuss the proof of Selberg's
identity. Subsequently (in spring 2010) I plan to continue
and complete the discussion of the elementary proof of the
prime number theorem.
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