Difference between revisions of "Chebyshev theta function"
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'''Chebyshev's theta function''', denoted <math>\vartheta</math> or sometimes | '''Chebyshev's theta function''', denoted <math>\vartheta</math> or sometimes | ||
<math>\theta</math>, is a function of use in [[analytic number theory]]. | <math>\theta</math>, is a function of use in [[analytic number theory]]. | ||
− | It is defined | + | It is defined thus, for real <math>x</math>: |
<cmath> \vartheta(x) = \sum_{p \le x} \log x , </cmath> | <cmath> \vartheta(x) = \sum_{p \le x} \log x , </cmath> | ||
where the sum ranges over all [[prime number | primes]] less than | where the sum ranges over all [[prime number | primes]] less than |
Revision as of 15:17, 29 March 2009
Chebyshev's theta function, denoted or sometimes
, is a function of use in analytic number theory.
It is defined thus, for real
:
where the sum ranges over all primes less than
.
Estimates of the function
The function is asymptotically equivalent to
(the prime counting function) and
. This result
is the Prime Number Theorem, and all known proofs are rather
involved.
However, we can obtain a simpler bound on .
Theorem (Chebyshev). If , then
.
Proof. We induct on . For our base
cases, we note that for
, we have
.
Now suppose that . Let
. Then
so
by inductive hypothesis. Therefore
as desired.