Some Generalizations of Totient Function with Elementary Symmetric Sums
Udvas Acharjee, N. Uday Kiran
TL;DR
This work generalizes Euler’s totient function by replacing the polynomial input with elementary symmetric polynomials, focusing on the second symmetric sum e_2. It develops product-form expressions for φ_{ {e_2} }(n) by counting zeros of e_2 over finite fields 𝔽_p^k using Lidl’s results on quadratic forms, and establishes prime-power formulas for these counts, including degenerate cases at p=2. A key methodological thread is the use of inclusion–exclusion to relate φ_F and φ_J and the translation of counting zeros into multiplicative product forms, thereby linking generalized totients, zeros in finite fields, and restricted linear congruences. The paper also outlines a framework to compute solution counts for restricted linear congruences via φ-functions and demonstrates concrete cases (e.g., φ_{ {1,2} }, φ_{ {1,2,3} }, g_k(b,n)) that illuminate Menon-type identities and Ramanujan-sum connections, highlighting the deep equivalences among these counting problems.
Abstract
We generalize certain totient functions using elementary symmetric polynomials and derive explicit product forms for the totient functions involving the second elementary symmetric sum. This work follows from the work of Toth [The Ramanujan Journal, 2022] where the totient function was generalized using the first and the kth elementary symmetric polynomial. We also provide some observations on the behavior of the totient function with an arbitrary jth elementary symmetric polynomial. We then outline a method for solving a certain the restricted linear congruence problem with a greatest common divisor constraint on a quadratic form, illustrated by a concrete example. Most importantly, we demonstrate the equivalence between obtaining product forms for generalized totient functions, counting zeros of specific polynomials over finite fields, and resolving a broad class of restricted linear congruence problems .