On discrete mean values of Dirichlet L -functions

Ertan Elma

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 4, page 1035-1048
  • ISSN: 0011-4642

Abstract

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Let χ be a nonprincipal Dirichlet character modulo a prime number p ⩾ 3 and let 𝔞 χ : = 1 2 ( 1 - χ ( - 1 ) ) . Define the mean value ℳ p ( - s , χ ) : = 2 p - 1 ∑ ψ ( mod p ) ψ ( - 1 ) = - 1 L ( 1 , ψ ) L ( - s , χ ψ ¯ ) ( σ : = ℜ s > 0 ) . We give an identity for ℳ p ( - s , χ ) which, in particular, shows that ℳ p ( - s , χ ) = L ( 1 - s , χ ) + 𝔞 χ 2 p s L ( 1 , χ ) ζ ( - s ) + o ( 1 ) ( p → ∞ ) for fixed 0 < σ < 1 2 and | t : = ℑ s | = o ( p ( 1 - 2 σ ) / ( 3 + 2 σ ) ) .

How to cite

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Elma, Ertan. "On discrete mean values of Dirichlet $L$-functions." Czechoslovak Mathematical Journal 71.4 (2021): 1035-1048. <http://eudml.org/doc/297519>.

@article{Elma2021,
abstract = {Let $\chi $ be a nonprincipal Dirichlet character modulo a prime number $p\geqslant 3$ and let $\mathfrak \{a\}_\chi := \tfrac\{1\}\{2\} (1-\chi (-1))$. Define the mean value \[ \mathcal \{M\}\_\{p\}(-s,\chi ) :=\frac\{2\}\{p-1\} \sum \psi \hspace\{10.0pt\}(\@mod \; p) \psi (-1)=-1 L(1,\psi )L(-s,\chi \bar\{\psi \}) \quad (\sigma :=\Re s>0). \] We give an identity for $\mathcal \{M\}_\{p\}(-s,\chi )$ which, in particular, shows that \[ \mathcal \{M\}\_\{p\}(-s,\chi )= L(1-s,\chi )+\mathfrak \{a\}\_\chi 2p^s L(1,\chi )\zeta (-s) +o(1) \quad (p\rightarrow \infty ) \] for fixed $0<\sigma <\frac\{1\}\{2\}$ and $|t:=\Im s|=o (p^\{(1-2\sigma )/(3+2\sigma )\})$.},
author = {Elma, Ertan},
journal = {Czechoslovak Mathematical Journal},
keywords = {Dirichlet $L$-function; mean value; Dirichlet character},
language = {eng},
number = {4},
pages = {1035-1048},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On discrete mean values of Dirichlet $L$-functions},
url = {http://eudml.org/doc/297519},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Elma, Ertan
TI - On discrete mean values of Dirichlet $L$-functions
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 4
SP - 1035
EP - 1048
AB - Let $\chi $ be a nonprincipal Dirichlet character modulo a prime number $p\geqslant 3$ and let $\mathfrak {a}_\chi := \tfrac{1}{2} (1-\chi (-1))$. Define the mean value \[ \mathcal {M}_{p}(-s,\chi ) :=\frac{2}{p-1} \sum \psi \hspace{10.0pt}(\@mod \; p) \psi (-1)=-1 L(1,\psi )L(-s,\chi \bar{\psi }) \quad (\sigma :=\Re s>0). \] We give an identity for $\mathcal {M}_{p}(-s,\chi )$ which, in particular, shows that \[ \mathcal {M}_{p}(-s,\chi )= L(1-s,\chi )+\mathfrak {a}_\chi 2p^s L(1,\chi )\zeta (-s) +o(1) \quad (p\rightarrow \infty ) \] for fixed $0<\sigma <\frac{1}{2}$ and $|t:=\Im s|=o (p^{(1-2\sigma )/(3+2\sigma )})$.
LA - eng
KW - Dirichlet $L$-function; mean value; Dirichlet character
UR - http://eudml.org/doc/297519
ER -

References

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