A reciprocity formula for a cotangent sum. (English) Zbl 1293.30078
Summary: We introduce the cotangent sum
\[ c(h/k) =-\sum_{h=1}^{k-1}\frac{a}{k}\cot\frac{\pi ah}{k} \]
and prove that it satisfies
\[ hc(h/k) + c(k/h) - \frac{1}{\pi k}= g(h/k), \] where \(g\) is a function which is analytic on the complex plane minus the negative real axis. The sum arises in connection with the Nyman-Beurling approach to the Riemann hypothesis.
\[ c(h/k) =-\sum_{h=1}^{k-1}\frac{a}{k}\cot\frac{\pi ah}{k} \]
and prove that it satisfies
\[ hc(h/k) + c(k/h) - \frac{1}{\pi k}= g(h/k), \] where \(g\) is a function which is analytic on the complex plane minus the negative real axis. The sum arises in connection with the Nyman-Beurling approach to the Riemann hypothesis.
MSC:
| 11L03 | Trigonometric and exponential sums (general theory) |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
