Journal of the Ramanujan Mathematical Society
Volume 41, Issue 3, September 2026 pp. 293–298.
Convolution sums of Legendre symbols arising from modular equations
Authors:
M. V. Yathirajsharma and S. T. P. Srinivas
Author institution:Sarada Vilas College, Mysuru 570 004, India.
Summary:
In this article, we see how modular equations of Ramanujan, specifically of
level 11, result in some non-trivial finite sums involving Legendre symbols.
Particularly, we prove the following elegant identity:
∑{j=1}{{p-1}/{2}} h(j)h(p-2j) = {1}/{2}(p-({p}/{11})),
where p ≠ 2, 11 is a prime and h(n) is defined based on the Legendre
symbol ({k}/{11}) as h(n) = ∑{k|n} ({k}/{11}).
It is quite difficult to explicitly recognize the pattern of h(n). However,
when involved in a sum of the above kind, it simplifies beautifully. Although
we were unable to think of an analytical or pure number theoretic proof, we
show in this article how Ramanujan's theory of modular equations may result
in such non-trivial closed convolution sums involving Legendre symbols.
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