A computable formula for evaluating the mean square sum of $L$-functions
For Dirichlet characters $\chi$ mod $k$ where $k\geq 3$, we here give a computable formula for evaluating the mean square sums $\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for any positive integer $r\geq 3$. We also give an inductive formula for computing the sum $\sum...
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| creator | Thomas, Neha Elizabeth Namboothiri, K Vishnu |
| description | For Dirichlet characters $\chi$ mod $k$ where $k\geq 3$, we here give a
computable formula for evaluating the mean square sums
$\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for
any positive integer $r\geq 3$. We also give an inductive formula for computing
the sum $\sum\limits_{\substack{1\leq m\leq k \\ (m,
k)=1}}\frac{1}{\left(\sin\left(\frac{\pi m}{k}\right)\right)^{2n}}$ where $n$
is a positive integer in terms of Bernoulli numbers and binomial coefficients. |
| doi_str_mv | 10.48550/arxiv.2307.01889 |
| format | Article |
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computable formula for evaluating the mean square sums
$\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for
any positive integer $r\geq 3$. We also give an inductive formula for computing
the sum $\sum\limits_{\substack{1\leq m\leq k \\ (m,
k)=1}}\frac{1}{\left(\sin\left(\frac{\pi m}{k}\right)\right)^{2n}}$ where $n$
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computable formula for evaluating the mean square sums
$\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for
any positive integer $r\geq 3$. We also give an inductive formula for computing
the sum $\sum\limits_{\substack{1\leq m\leq k \\ (m,
k)=1}}\frac{1}{\left(\sin\left(\frac{\pi m}{k}\right)\right)^{2n}}$ where $n$
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computable formula for evaluating the mean square sums
$\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for
any positive integer $r\geq 3$. We also give an inductive formula for computing
the sum $\sum\limits_{\substack{1\leq m\leq k \\ (m,
k)=1}}\frac{1}{\left(\sin\left(\frac{\pi m}{k}\right)\right)^{2n}}$ where $n$
is a positive integer in terms of Bernoulli numbers and binomial coefficients.</abstract><doi>10.48550/arxiv.2307.01889</doi><oa>free_for_read</oa></addata></record> |
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| title | A computable formula for evaluating the mean square sum of $L$-functions |
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