Difference between revisions of "Chebyshev theta function"
Missionary (talk | contribs) |
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so | so | ||
<cmath> x \log 2 \ge \sum_{\lfloor n/2 \rfloor < p \le n} \log p | <cmath> x \log 2 \ge \sum_{\lfloor n/2 \rfloor < p \le n} \log p | ||
− | = \vartheta | + | = \vartheta(x) - \vartheta(\lfloor n/2 \rfloor) |
− | \ge \vartheta | + | \ge \vartheta(x) - 2\lfloor n/2 \rfloor \log 2 \ge \vartheta(x) - x \log 2 , </cmath> |
by inductive hypothesis. Therefore | by inductive hypothesis. Therefore | ||
<cmath> 2x \log 2 \ge \vartheta(x), </cmath> | <cmath> 2x \log 2 \ge \vartheta(x), </cmath> |
Revision as of 19:58, 18 September 2012
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.