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The continuous solution of the system
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The continuous solution $\omega(u)$ of the [[differential-delay equation]]
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110201.png" /></td> </tr></table>
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(u\,\omega(u))' = \omega(u-1)
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110202.png" /></td> </tr></table>
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for $u > 2$ with initial values
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$$
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\omega(u) = \frac{1}{u}\ ,\ \ (1 \le u \le 2) \ .
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$$
  
 
This function occurs in number theory as the limit
 
This function occurs in number theory as the limit
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$$
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\omega(u) = \lim_{x\rightarrow\infty} \frac{ \Phi(x,x^{1/u}) \log(x^{1/u}) }{ x }
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$$
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where $\Phi(x,y)$ denotes the number of positive integers not exceeding $x$ that are free of prime factors smaller than $y$; see [[#References|[a1]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110203.png" /></td> </tr></table>
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The function $\omega(u)$ is positive-valued and converges to the constant $e^{-\gamma}$ as $u\rightarrow\infty$, where $\gamma$ is the [[Euler constant]]. The difference $\omega(u)-e^{-\gamma}$ behaves asymptotically like a trigonometric function with period $2$ and decaying amplitudes of size $\exp((1+o(1))\log u)$. These and similar results have been exploited in the study of irregularities in the [[distribution of prime numbers]]; see [[#References|[a2]]], [[#References|[a3]]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110204.png" /> denotes the number of positive integers not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110205.png" /> that are free of prime factors smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110206.png" />; see [[#References|[a1]]].
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====References====
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Bukhstab,  "Asymptotic estimates of a general number-theoretic function" ''Mat. Sb.'' , '''44'''  (1937)  pp. 1239–1246  (In Russian)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Friedlander,  A. Granville,  A. Hildebrand,  H. Maier,  "Oscillation theorems for primes in arithmetic progressions and for sifting functions" ''J. Amer. Math. Soc.'' , '''4'''  (1991)  pp. 25–86</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Maier,  "Primes in short intervals" ''Michigan Math. J.'' , '''32'''  (1985)  pp. 221–225</TD></TR>
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</table>
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110207.png" /> is positive-valued and converges to the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110208.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b1110209.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b11102010.png" /> is the [[Euler constant|Euler constant]]. The difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b11102011.png" /> behaves asymptotically like a trigonometric function with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b11102012.png" /> and decaying amplitudes of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111020/b11102013.png" />. These and similar results have been exploited in the study of irregularities in the [[Distribution of prime numbers|distribution of prime numbers]]; see [[#References|[a2]]], [[#References|[a3]]].
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Bukhstab,  "Asymptotic estimates of a general number-theoretic function"  ''Mat. Sb.'' , '''44'''  (1937)  pp. 1239–1246  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Friedlander,  A. Granville,  A. Hildebrand,  H. Maier,  "Oscillation theorems for primes in arithmetic progressions and for sifting functions"  ''J. Amer. Math. Soc.'' , '''4'''  (1991)  pp. 25–86</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Maier,  "Primes in short intervals"  ''Michigan Math. J.'' , '''32'''  (1985)  pp. 221–225</TD></TR></table>
 

Latest revision as of 15:57, 22 September 2017

The continuous solution $\omega(u)$ of the differential-delay equation $$ (u\,\omega(u))' = \omega(u-1) $$ for $u > 2$ with initial values $$ \omega(u) = \frac{1}{u}\ ,\ \ (1 \le u \le 2) \ . $$

This function occurs in number theory as the limit $$ \omega(u) = \lim_{x\rightarrow\infty} \frac{ \Phi(x,x^{1/u}) \log(x^{1/u}) }{ x } $$ where $\Phi(x,y)$ denotes the number of positive integers not exceeding $x$ that are free of prime factors smaller than $y$; see [a1].

The function $\omega(u)$ is positive-valued and converges to the constant $e^{-\gamma}$ as $u\rightarrow\infty$, where $\gamma$ is the Euler constant. The difference $\omega(u)-e^{-\gamma}$ behaves asymptotically like a trigonometric function with period $2$ and decaying amplitudes of size $\exp((1+o(1))\log u)$. These and similar results have been exploited in the study of irregularities in the distribution of prime numbers; see [a2], [a3].

References

[a1] A.A. Bukhstab, "Asymptotic estimates of a general number-theoretic function" Mat. Sb. , 44 (1937) pp. 1239–1246 (In Russian)
[a2] J. Friedlander, A. Granville, A. Hildebrand, H. Maier, "Oscillation theorems for primes in arithmetic progressions and for sifting functions" J. Amer. Math. Soc. , 4 (1991) pp. 25–86
[a3] H. Maier, "Primes in short intervals" Michigan Math. J. , 32 (1985) pp. 221–225
How to Cite This Entry:
Buchstab function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buchstab_function&oldid=19062
This article was adapted from an original article by A. Hildebrand (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article