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On Discrete Mean Values of Dirichlet L -Functions
Czechoslovak Mathematical Journal ( IF 0.5 ) Pub Date : 2021-02-02 , DOI: 10.21136/cmj.2021.0189-20
Ertan Elma

Let χ be a nonprincipal Dirichlet character modulo a prime number p ≽ 3 and let \({\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left( { - 1} \right)} \right)\). Define the mean value

$${{\cal M}_p}\left( { - s,{\cal X}} \right)\,\,: = {2 \over {p - 1}}\,\sum\limits_{\matrix{ {\psi \,\left( {\bmod \,p} \right)} \cr {\psi \left( { - 1} \right) = - 1} \cr } } {L\,\left( {1,\psi } \right)L\left( { - s,{\cal X}\overline \psi } \right)\,\,\,\,\left( {\sigma : = \,\,{\Re _s} > 0} \right).} $$

We give an identity for \({{\cal M}_p}\left( { - s,{\cal X}} \right)\,\,\) which, in particular, shows that

$${{\cal M}_p}\left( { - s,{\cal X}} \right)\,\, = L\left( {1 - s{,_{\cal X}}} \right) + {\mathfrak{a}_{\cal X}}2{p^s}L\left( {1{,_{\cal X}}} \right)\zeta \left( { - s} \right) + o\left( 1 \right)\,\,\,\left( {p \to \infty } \right)$$

for fixed \(0 < \sigma < {1 \over 2}\) and \(\left| {t: = \,{\mathfrak{J}_s}} \right| = o\left( {{p^{\left( {1 - 2\sigma } \right)/\left( {3 + 2\sigma } \right)}}} \right)\).



中文翻译:

Dirichlet L函数的离散均值

令χ为非质数Dirichlet字符,模为质数p 3,令\({\ mathfrak {a} _ {\ cal X}}:= {1 \ over 2} \ left({1 {-_ {\ cal X}} \ left({-1} \ right)} \ right)\)。定义平均值

$$ {{\ cal M} _p} \ left({-s,{\ cal X}} \ right)\,\ :: = {2 \ over {p-1}} \,\ sum \ limits _ {\矩阵{{\ psi \,\ left({\ bmod \,p} \ right)} \ cr {\ psi \ left({-1} \ right)=-1} \ cr}} {L \,\ left ({1,\ psi} \ right)L \ left({-s,{\ cal X} \ overline \ psi} \ right)\,\,\,\,\ left({\ sigma:= \,\ ,{\ Re _s}> 0} \ right)。} $$

我们给出\({{cal M} _p} \ left({-s,{\ cal X}} \ right)\,\,\)的身份,特别是表明

$$ {{\ cal M} _p} \ left({-s,{\ cal X}} \ right)\,\ = L \ left({1-s {,_ {\ cal X}}} \ right)+ {\ mathfrak {a} _ {\ cal X}} 2 {p ^ s} L \ left({1 {,_ {\ cal X}}} \ right)\ zeta \ left({-s} \ right)+ o \ left(1 \ right)\,\,\,\ left({p \ to \ infty} \ right)$$

对于固定的\(0 <\ sigma <{1 \ over 2} \)\(\ left | {t:= \,{\ mathfrak {J} _s}} \ right | = o \ left({{p ^ {\ left({1-2 \ sigma} \ right)/ \ left({3 + 2 \ sigma} \ right)}}}} \ right)\)

更新日期:2021-02-24
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