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Difference between revisions of "Chebyshev theta function"

(definition and a few properties)
 
m (→‎Estimates of the 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, thus, for real <math>x</math>:
+
It is defined thus, for real <math>x</math>:
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<cmath> \vartheta(x) = \sum_{p \le x} \log x , </cmath>
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<cmath> \vartheta(x) = \sum_{p \le x} \log p , </cmath>
 
where the sum ranges over all [[prime number | primes]] less than
 
where the sum ranges over all [[prime number | primes]] less than
 
<math>x</math>.
 
<math>x</math>.
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The function <math>\vartheta(x)</math> is [[asymptotically equivalent]] to
 
The function <math>\vartheta(x)</math> is [[asymptotically equivalent]] to
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<math>\pi(x)</math> (the [[prime counting function]]) and <math>x</math>.  This result
+
<math>\pi(x)\ln x</math> (<math>\pi(x)</math> is the [[prime counting function]]) and <math>x</math>.  This result
 
is the [[Prime Number Theorem]], and all known proofs are rather
 
is the [[Prime Number Theorem]], and all known proofs are rather
 
involved.
 
involved.
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'''Theorem (Chebyshev).''' If <math>x \ge 0</math>, then <math>\vartheta(x) \le
 
'''Theorem (Chebyshev).''' If <math>x \ge 0</math>, then <math>\vartheta(x) \le
−
2x</math>.
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2x \log 2</math>.
  
 
''Proof.''  We induct on <math>\lfloor x \rfloor</math>.  For our base
 
''Proof.''  We induct on <math>\lfloor x \rfloor</math>.  For our base
 
cases, we note that for <math>0 \le x < 2</math>, we have <math>\vartheta(x) =
 
cases, we note that for <math>0 \le x < 2</math>, we have <math>\vartheta(x) =
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0 \le x</math>.
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0 \le 2x \log 2</math>.
  
 
Now suppose that <math>x \ge 2</math>.  Let <math>n = \lfloor x \rfloor</math>.  Then
 
Now suppose that <math>x \ge 2</math>.  Let <math>n = \lfloor x \rfloor</math>.  Then
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\prod_{\lfloor n/2 \rfloor < p \le n} p , </cmath>
 
\prod_{\lfloor n/2 \rfloor < p \le n} p , </cmath>
 
so
 
so
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<cmath> x \ge x \log 2 \ge \sum_{\lfloor n/2 \rfloor < p \le n} \log p
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<cmath> x \log 2 \ge \sum_{\lfloor n/2 \rfloor < p \le n} \log p
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= \vartheta{x} - \vartheta{\lfloor n/2 \rfloor}
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= \vartheta(x) - \vartheta(\lfloor n/2 \rfloor)
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\ge \vartheta{x} - 2\lfloor n/2 \rfloor \ge \vartheta{x} - x , </cmath>
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\ge \vartheta(x) - 2\lfloor n/2 \rfloor \log 2 \ge \vartheta(x) - x \log 2 , </cmath>
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by inductive hypothesis.  Therefore
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by the inductive hypothesis.  Therefore
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<cmath> 2x \ge \vartheta(x), </cmath>
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<cmath> 2x \log 2 \ge \vartheta(x), </cmath>
 
as desired.  <math>\blacksquare</math>
 
as desired.  <math>\blacksquare</math>
  

Latest revision as of 12:28, 1 April 2014

Chebyshev's theta function, denoted $\vartheta$ or sometimes $\theta$, is a function of use in analytic number theory. It is defined thus, for real $x$: \[\vartheta(x) = \sum_{p \le x} \log p ,\] where the sum ranges over all primes less than $x$.

Estimates of the function

The function $\vartheta(x)$ is asymptotically equivalent to $\pi(x)\ln x$ ($\pi(x)$ is the prime counting function) and $x$. This result is the Prime Number Theorem, and all known proofs are rather involved.

However, we can obtain a simpler bound on $\vartheta(x)$.

Theorem (Chebyshev). If $x \ge 0$, then $\vartheta(x) \le 2x \log 2$.

Proof. We induct on $\lfloor x \rfloor$. For our base cases, we note that for $0 \le x < 2$, we have $\vartheta(x) = 0 \le 2x \log 2$.

Now suppose that $x \ge 2$. Let $n = \lfloor x \rfloor$. Then \[2^x \ge 2^n \ge \binom{n}{\lfloor n/2 \rfloor} \ge \prod_{\lfloor n/2 \rfloor < p \le n} p ,\] so \[x \log 2 \ge \sum_{\lfloor n/2 \rfloor < p \le n} \log p = \vartheta(x) - \vartheta(\lfloor n/2 \rfloor) \ge \vartheta(x) - 2\lfloor n/2 \rfloor \log 2 \ge \vartheta(x) - x \log 2 ,\] by the inductive hypothesis. Therefore \[2x \log 2 \ge \vartheta(x),\] as desired. $\blacksquare$

See also

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