| PAN CHENGDONG.[J].数学年刊A辑,1980,1(1):149~160 |
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| A NEW MEAN VALUE THEOREM ANDITS APPLICATIONS |
| Received:December 17, 1979 |
| DOI： |
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| Let
\[\pi (x;a,d,l) = \sum\limits_{\scriptstyleap \le x\hfill\atop\scriptstyleap \equiv l(\bmod {\kern 1pt} {\kern 1pt} d)\hfill} 1 and let f(a) be a real function,satisfying the condition
\[\sum\limits_{n \le x} {|f(n)| \ll x{{\log }^{{\lambda _1}}}} x\],\[\sum\limits_{n \le x} {{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{d|n} {|f(d)|} \ll x{{\log }^{{\lambda _2}}}} x\]
then for any A>0,we have
\[\sum\limits_{d < {x^{\frac{1}{2}}}{{\log }^{ - B}}x} {\mathop {\max }\limits_{y < x} } \mathop {\max }\limits_{(l,d) = 1} |\sum\limits_{\scriptstylea \le {x^{1 - \varepsilon }}\hfill\atop
\scriptstyle(a,d) = 1\hfill} {f(a)} (\pi (y;a,d,l) - \frac{{\pi (y;a,1,1)}}{{{\phi _d}}}| \ll \frac{x}{{{{\log }^A}x}}\]
In this paper we use above estimation to prove the following results
1)let \[\Omega = \sum\limits_{\scriptstyle{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({p_{1,2}})\hfill\atop
{\scriptstyle{p_s} \le N/{p_1}{p_2}\hfill\atop
\scriptstyleN - p = {p_1}{p_2}{p_3}\hfill}} 1 \]
where N is a large even integer,and (p1,2) denotes the condition
\[{N^{\frac{1}{{10}}}} < {p_1} \le {N^{\frac{1}{3}}} \le {p_2} \le {(\frac{N}{{{p_1}}})^{\frac{1}{2}}}\]
then we have
\[{\cal Q}(N) = \mathop \prod \limits_{p|N} \frac{{p - 1}}{{p - 2}}\mathop \prod \limits_{p > 2} (1 - \frac{1}{{{{(p - 1)}^2}}})\]
let \[D(N) = \sum\limits_{N = {p_1} + {p_2}} 1 \]
then we obtain
\[7.928{\cal Q}(N)\frac{N}{{{{\log }^2}N}}\]
3)let \[1 \le y \le {x^{1 - s}}(0 < \varepsilon < 1)\],f(a)>0,and satisfying the condition(Δ),then we have
\[\sum\limits_{\scriptstyleap \le x\hfill\atop
\scriptstylea \le y\hfill} {f(a)d(ap - 1) \sim 2x\sum\limits_{d \le {x^{\frac{1}{2}}}} {\frac{1}{{\phi (d)}}} } \sum\limits_{a \le y} {\frac{{f(a)}}{{a\log \frac{x}{a}}}} \]
4)let \[{p_x}\]denote the largest prime factor of
\[\mathop \prod \limits_{0 < p + a < x} (p + a)\]
where a is a given non-zero integer.
Hooley proved \[{p^x} > {x^\theta }\],when \[\theta < \frac{5}{8}\],the key of his prove is the estimation of the summation
\[V(y) = \sum\limits_{\scriptstylep + a = kq\hfill\atop
{\scriptstylep \le x - a\hfill\atop
\scriptstyley < q < ry\hfill}} {\log q} \]
where q denotes primes,and \[{x^{\frac{1}{2}}} < y < {x^{\frac{3}{4}}},1 < r < 2\]
Using the Selberg sieve method ,we can turn the above estimation into the estimation of the follwing sum
\[\sum\limits_{d \le {x^{\frac{1}{2}}}{{\log }^{ - B}}y} {\sum\limits_{k \le \frac{x}{y}} {\sum\limits_{\scriptstylekq \le x\hfill\atop
\scriptstylekq \equiv a(\bmod {\kern 1pt} {\kern 1pt} {\kern 1pt} 4)\hfill} {\log q} } } \] |
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