# On a certain integral representation for Dirichlet L-functions

It is an ancient result of Jensen that

$$(s-1)\zeta(s)=\frac{\pi}{2} \int_{-\infty}^{\infty} \frac{(1/2+it)^{1-s}}{\cosh^{2}\pi t} \mathrm{d}t$$

where $\zeta$ denotes the Riemann zeta function.

Is there a generalisation of this formula valid for all Dirichlet L-functions?

• It's hard to prove a negative, but my expectation is 'no'. Where could a character appear on the right? An alternative approach would be to try to generalize to Hurwitz zeta functions, and then write the Dirichlet L-function as a linear combination of such. Can you give a citation for Jensen's result? I don't think I've seen it in Titchmarsh. – Stopple Jul 2 '18 at 16:25

As I suspected in my comment above, there is an analog of Jensen's 'ancient' result for the Hurwitz zeta function: For $a>1/2$, $$\zeta(s,a)=\frac{\pi}{2(s-1)}\int_{-\infty}^\infty\frac{(a-1/2+it)^{1-s}}{\cosh^2(\pi t)}\, dt.$$ This is (23) on p. 92 of Series Associated with Zeta and Related Functions, by Srivastava and Choi, Kluwer, 2001. (Unfortunately, without proof, but that's for another MO question.) For $a=1$ this reduces to Jensen.

Now let $d$ odd, and $\chi$ a nontrivial character modulo $d$, and $\epsilon=\pm1=\chi(-1)$.

So, $$L(s,\chi)=d^{-s}\sum_{d/2<j<d}\chi(j)(\zeta(s,j/d)+\epsilon\zeta(s,1-j/d)).$$ Using the identity $$\zeta(s,a)=\zeta(s,a+1)+a^{-s}$$ we get $$L(s,\chi)=d^{-s}\sum_{d/2<j<d}\chi(j)(\zeta(s,j/d)+\epsilon\zeta(s,2-j/d))+\epsilon\sum_{d/2<j<d}\chi(j)(d-j)^{-s}.$$ Now the relevant Hurwitz zeta parameter is $>1/2$, so $$L(s,\chi)=\frac{\pi\cdot d^{-s}}{2(s-1)}\int_{-\infty}^\infty\sum_{d/2<j<d}\chi(j)((j/d-1/2+it)^{1-s}+\epsilon (3/2-j/d+it))^{1-s})\cosh^{-2}(\pi t)\, dt$$ $$+\epsilon\sum_{d/2<j<d}\chi(j)(d-j)^{-s}$$ The singualrity at $s=1$ is removable: by the orthogonality of characters the sum over $j$ wil have a zero at $s=1$.