In his text Multiplicative Number Theory on page 9, Davenport mentions that another means of expanding the L-function is known and then mentions the fact that, $$ \mathcal{F} \sum_{n=1,n \; odd} \frac{1}{n}e_q(mn) = \begin{cases} \frac{1}{4} \pi & if \; 0<m<q/2 \\ -\frac{1}{4} \pi & if \; q/2<m<q \end{cases} $$ Davenport mentions that this result is derived using the logarithmic series. I could not find the same symbol that Davenport used in the text so I used $\mathcal{F}$ since I suspected it was a Fourier Transform. Is my suspicion correct ? How exactly is this result derived ?
2026-03-28 16:14:09.1774714449
A Fact Stated in Davenport's Multiplicative Number Theory
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