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Is this function
$\sum _{k\ge 1}\frac{1}{{\left(2k-1\right)}^{s}},\phantom{\rule{thinmathspace}{0ex}}Re\left(s\right)>1$
is well known?
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Dayana Zuniga
Formally,
$\zeta \left(s\right)=\sum _{k=1}^{+\mathrm{\infty }}\frac{1}{{k}^{s}}=\sum _{k=1}^{+\mathrm{\infty }}\frac{1}{{\left(2k\right)}^{s}}+\sum _{k=1}^{+\mathrm{\infty }}\frac{1}{{\left(2k-1\right)}^{s}}=\frac{1}{{2}^{s}}\sum _{k=1}^{+\mathrm{\infty }}\frac{1}{{k}^{s}}+\sum _{k=1}^{+\mathrm{\infty }}\frac{1}{{\left(2k-1\right)}^{s}}$
whence
$\sum _{k=1}^{+\mathrm{\infty }}\frac{1}{{\left(2k-1\right)}^{s}}=\zeta \left(s\right)\left(1-\frac{1}{{2}^{s}}\right).$

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