Small Gaps Between Primes

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Small Gaps Between Primes

• Daniel Goldston
• San Jose State University

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Are there always primes much closer than the
average ?
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More Modest Goal Prove the existence of
infinitely many consecutive primes whose
distance apart is much smaller than the average
distance i. e.
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HISTORY
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More History
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G-P-Y Conditional Results
Our method uses information on primes in
arithmetic progressions.
EXAMPLE. If you divide the natural numbers up
modulo 3 you get three residue classes
We expect and can prove
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The Prime Number Theorem for Arithmetic
Progressions
where PNT for a. p. \medskip \bf The
Bombieri-Vinogradov Theorem For any \epsilongt0
and Agt1 we have \ \sum_q\le Q
\max_\substacka\\ (a,q)1 \left \pi(xq,a)-
\frac\rm li(x)\phi(q) \right \ll
\fracx(\log x)A \ for Qx1/2/(\log
x)B(A). If this holds for
Qx\vartheta-\epsilon we say the primes have
\textbflevel of distribution \vartheta.
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The Bombieri-Vinogradov Theorem
Level of Distribution of Primes in Progressions
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Prime Tuples

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Weak form of the Prime Tuple Conjecture
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Hardy-Littlewood Prime Tuple Conjecture

• residue classes \pmod p the numbers h\in
  \mathcalH fall into, and extend this
  definition to \nu_d(\mathcalH) for squarefree
  integers d by multiplicativity. If
  \mathfrakS(\mathcalH)\neq 0 then
  \mathcalH is called
• \emphAdmissible

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Approximating Prime Tuples The Old Way
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Results from using Old Approximation
Green and Tao used the asymptotic formula for
in their work on arithmetic progressions of
primes.
Ironically, the new approximation does not work
for Green-Tao.
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Approximating Prime Tuples The New Way
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New Approximation
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What makes everything work
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Sketch of Proof of Small Gaps
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Why do primes get news coverage more easily than
other more important mathematics?
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