Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums

Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 74 Retrieval Number:100.1/ijam.B121905021025 DOI: 10.54105/ijam.B1219.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. ≥ Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums Shikha Singh, Jagmohan Tanti Abstract: This paper explores the hybrid power mean of three-term exponential sums, incorporating weights derived from general quartic Gauss sums. By employing the theory of Dirichlet characters in conjunction with fundamental properties of classical Gauss sums, we establish several significant results. Furthermore, we determine the corresponding weight function for these three-term exponential sums, with particular applications in coding theory. Keywords: Dirichlet Character, Quartic Gauss Sums, ThreeTerm Exponential Sum. MSC: 11L05. I. INTRODUCTION L et q 3 be an integer and χ a Dirichlet character modulo q. For any positive integer m, a general lthGauss sum G (m, l, χ; q) is defined by 𝐺(𝑚,𝑙,𝑥;𝑞)=∑𝑥(𝑏)𝑒 𝑞 𝑏=1 (𝑚𝑏 1 𝑞) When q is prime, k and t are positive integers, for integers m, n and s, a generalized three-term exponential sum is defined by 𝐶(𝑚,𝑛,𝑠,𝑘,𝑡,𝑥;𝑞)=∑𝑥(𝑎)𝑒 𝑞−1 𝑎=1 (𝑚𝑎 𝑘 +𝑠𝑎 𝑡 +𝑛𝑎 𝑞) where e(g) = e 2iπg . Many researchers have actively contributed to this field. Zhang and Han [1] studied the sixth power mean of the two-term exponential sums, Du and Li [2] studied the fourth power mean of generalized three-term exponential sums and gave the formula: Manuscript received on 22 September 2025 | Revised Manuscript received on 12 October 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. *Correspondence Author(s) Dr. Shikha Singh*, Assistant Professor, Department of Mathematics, Central University of Jharkhand, Ranchi (Jharkhand), India. Email ID: [email protected], ORCID ID: 0009-0008-7936-1752 Prof. Jagmohan Tanti, Department of Mathematics, Central University of Jharkhand, Ranchi (Jharkhand), India. © The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/ Yu and Zhang [6], [5] estimated the trigonometric sums and the properties of the congruence equations. They also studied the sixth power mean value of the generalized threeterm exponential sums and gave an exact computational formula: Huaning LIU and Wanmei LI [7] also gave the identity for any integer p ≥ 3, The primary aim of this article is to investigate the mean value of three-term exponential sums and general quartic Gauss sums by employing estimates of Dirichlet characters and certain properties of Gauss sums, II. PRELIMINARY LEMMAS In this section, we present certain properties of Gauss sums [3, 4] along with a set of lemmas that play a crucial role in establishing the main theorems. Lemma 2.1 For any integer m ≥ 1, we have the formula Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums 75 Retrieval Number:100.1/ijam.B121905021025 DOI: 10.54105/ijam.B1219.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Proof: See Theorem 9.16 of [3]. Lemma 2.2 Let q be an odd prime, k, m, and n be positive integers. Then we have, where χ varies over Dirichlet characters (mod q). Proof. Since q ≡ 3 (mod 4) for any integer t with (t, q) = 1. From the orthogonality property of characters (mod q), Now Lemma 2.3 Let q be an odd prime with q ≡ 3 (mod 4), k, m, n positive integers. Then we have where χ varies over non-principal Dirichlet character (mod q). Proof. As for a non-principal even character χ and an integer k > 0, χ = χ Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 76 Retrieval Number:100.1/ijam.B121905021025 DOI: 10.54105/ijam.B1219.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Σ implies Note that, and Form the equations (2.1), (2.2) and (2.3) we get, Lemma 2.4 Let q be an odd prime and k, m, n positive integers. Then we have Proof. Now, Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums 77 Retrieval Number:100.1/ijam.B121905021025 DOI: 10.54105/ijam.B1219.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. q Lemma 2.5 Let q be an odd prime with q ≡ 3 (mod 4), η be a positive integer with (η, q) = 1. Then, for any non-principal even character χ mod q, we get Proof. Here χ is a non-principal even character mod q, then we can write. Therefore, Similarly, we find the identity for χ If q ≡ 3 (mod 4), then we have −1 = —1. Using this, we get, Combining (2.4) and (2.5), we have Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 78 Retrieval Number:100.1/ijam.B121905021025 DOI: 10.54105/ijam.B1219.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. III. MAIN RESULTS In this section, we give the complete proof of t he main results by using the previous lemmas. Theorem 3.1 Let q be an odd prime with q ≡ 3 (mod 4), k, m, n positive integers. Then we have where χ varies over Dirichlet characters (mod q). Proof. As for χ, an even character mod q, then χ is also an even character mod q, So, Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums 79 Retrieval Number:100.1/ijam.B121905021025 DOI: 10.54105/ijam.B1219.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. [Fig.1: q = 7, 11] [Fig.2: q = 7, 11, 19, 23] [Fig.3: q = 7, 11, 19, 23, 31, 43, 47] These plots represent the dynamics of the hybrid power involving the three-term exponential sum and the general quartic Gauss sum. The results indicate that the function is consistently positive, never zero, and exhibits an increasing trend for varying values of q. Additionally, across different segments of q, the functions' graph displays a clear and striking behaviour along the q—axis. Theorem 3.2 Let q be an odd prime and χ be a Dirichlet character over mod q, then. for any integer η with (η, q) = 1, we have Proof. Since χ is an even character over mod q, then χ is also an even character over mod q. IV. WEIGHT DISTRIBUTION OF CODEWORD C(Q) Let p be prime and q = pr for r ≥ 2. We denote the finite field with q elements by Fq. Any m, s, n ∈ Fq, the threeterm of the exponential sum Theorem 4.1 For codeword ϕ (m, s) ∈ c(q), the ϕ (m, s) is given, Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5 Issue-2, October 2025 80 Retrieval Number:100.1/ijam.B121905021025 DOI: 10.54105/ijam.B1219.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Proof. For all m, s ∈ Fq and all a ∈ F ∗, e (Tr (m, s, a)) is a p-th root of unity and equal to 1 or not according to Tr (m, s, a) = 0 Therefore, Since w (ϕ (m, s)) is the number of non-zero entries of the vector ϕ (m, s) we have, V. CONCLUSION In this paper, we evaluated the hybrid power mean of a three-term exponential sum with weights given by general quartic Gauss sums. Through the use of Dirichlet characters and classical Gauss sum properties, we obtained new identities and explicit evaluations. We also computed the associated weight function, demonstrating its relevance in coding theory, particularly in analyzing code structures. ACKNOWLEDGMENT The authors would like to thank the Central University of Jharkhand for research facilities and support. DECLARATION STATEMENT Some of the references cited are older, noted explicitly as [1], [3], [4], and [6]. However, these works remain significant for the current study, as they are pioneering in their fields. After aggregating input from all authors, I must verify the accuracy of the following information as the article's author. ▪ Conflicts of Interest/ Competing Interests: Based on my understanding, this article has no conflicts of interest. ▪ Funding Support: This article has not been funded by any organizations or agencies. This independence ensures that the research is conducted with objectivity and without any external influence. ▪ Ethical Approval and Consent to Participate: The content of this article does not necessitate ethical approval or consent to participate with supporting documentation. ▪ Data Access Statement and Material Availability: The adequate resources of this article are publicly accessible. ▪ Author’s Contributions: The authorship of this article is contributed equally to all participating individuals. REFERENCES 1. W. Zhang and D . Han, On the sixth power mean of the two-term exponential sums, Journal of Number Theory, 136(2014), 403-413. DOI: https://doi.org/10.1016/j.jnt.2013.10.022, works remain significant, see declaration 2. Xiancun Du, Xiauoxue Li, On the fourth power mean of generalized three-term exponential sums, Journal of Mathematical Research with Applications, 35(1) (2015), 92-96. https://tdl.libra.titech.ac.jp/journaldocs/en/recordID/article.bib03/ZR000000001566 3. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. DOI: https://doi.org/10.1007/978-1-4757-5579-4, works remain significant, see declaration 4. H. Davenport, Multiplicative number theory, Markham, 1967. DOI: https://doi.org/10.1007/978-1-4757-5927-3 works remain significant, see declaration 5. W. Zhang, R. Duan, On the mean square value of Lfunctions with the weight of quadratic Gauss sums, Journal of Number Theory, 179(2017), 77-87. DOI: https://doi.org/10.1016/j.jnt.2017.03.019 6. Y. Yu and W Zhang, On the Sixth Power Mean Value of the Generalized Three-Term Exponential Sums, Abstract and Applied Analysis, 2014(2014). DOI: https://doi.org/10.1155/2014/474726, works remain significant, see declaration 7. H. Liu and W. Li, On the Fourth Power Mean of Generalized Three-Term Exponential Sums, Journal of Mathematical Research with Applications, 37(2017), 169-182. DOI: https://doi.org/10.3770/j.issn:2095-2651.2017.02.005 AUTHOR’S PROFILE Dr. Shikha Singh is an Assistant Professor at Sidhhu Kanhu Murmu University (SKMU). She earned her Ph.D. from Central University of Jharkhand (CUJ), specializing in Algebra and Number Theory. She has published papers in peer-reviewed journals and actively participates in academic conferences. Prof. Jagmohan Tanti is a faculty member at BBAU, where he serves in the Department of Mathematics. He holds extensive experience in Algebra and number theory. He has published numerous articles in reputable national and international peer-reviewed journals, contributing significantly to his field. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of the Lattice Science Publication (LSP)/ journal and/ or the editor(s). The Lattice Science Publication (LSP)/ journal and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.