Geodesic
Asymptotic behavior of sums of the type $\sum_{k=m}^{n-1}\exp(i\omega\sqrt{nk})$ as $n,m\to\infty$
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 9, Tome 78 (1978), pp. 54-59

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The asymptotic behavior as $n,m\to\infty$ of the sum $$ \sum_{k,l=m}^{n-1}\exp[i\omega\sqrt{n}(\sqrt{k}+\sqrt{l})]\Phi\biggl(1-\frac{|\sqrt{k}-\sqrt{l}|}{\Delta}\biggr), $$ is studied where $\Phi(t)=0$ for $t\leqslant0$ and $t$ for $t>0$. 
@article{ZNSL_1978_78_a3,
     author = {M. V. Buslaeva},
     title = {Asymptotic behavior of sums of the type $\sum_{k=m}^{n-1}\exp(i\omega\sqrt{nk})$ as $n,m\to\infty$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {54--59},
     publisher = {mathdoc},
     volume = {78},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1978_78_a3/}
}
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M. V. Buslaeva. Asymptotic behavior of sums of the type $\sum_{k=m}^{n-1}\exp(i\omega\sqrt{nk})$ as $n,m\to\infty$. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 9, Tome 78 (1978), pp. 54-59. http://geodesic.mathdoc.fr/item/ZNSL_1978_78_a3/