Research in Mathematics

Advances in Exponential Sums and Power Mean Estimates: Analytic Techniques and Computational Approaches

Exponential sums and mean value estimates stand at the core of modern analytic number theory, serving as indispensable bridges connecting pure mathematics with dynamical systems and computational modelling. Classical problems—ranging from the distribution of prime numbers to the analysis of arithmetic progressions—rely heavily on refined estimates of trigonometric sums and higher-order moments. In recent years, the interplay between traditional analytic techniques (such as the Hardy-Littlewood circle method and Vinogradov’s mean value theorem) and modern computational approaches has opened new avenues for tackling long-standing conjectures.

This Article Collection aims to gather cutting-edge research that advances both the theoretical depth and practical computation of these fundamental tools, fostering dialogue between number theorists, analysts, and mathematical modelers.

The significance of this topic lies in its dual impact on theoretical insight and applied methodology. Sharp bounds for exponential sums directly influence our understanding of equidistribution, Diophantine approximation, and pseudo-randomness—concepts increasingly vital in cryptography, coding theory, and complex systems modelling. Meanwhile, efficient algorithms for computing mean values and sum estimates enable large-scale numerical experiments, bridging the gap between asymptotic theory and verifiable data.

By consolidating contemporary developments, this collection will equip researchers with enhanced tools to address unsolved problems in both pure number theory and interdisciplinary dynamical systems, ultimately contributing to the broader mathematical framework of computational modelling.

This Article Collection welcomes original research and comprehensive review articles focused on, but not limited to, the following subtopics:

Analytic Techniques
• Refinements of classical methods (circle method, sieve theory)
• novel approaches to bounding exponential sums

Mean Value Theorems
• Higher-order moment estimates
• Vinogradov mean-value theorems
• Generalizations to various function fields

Interdisciplinary Applications
• Connections between exponential sum estimates and problems in cryptography
• signal processing Applications
• Ergodic theory

We encourage submissions that present rigorous proofs, innovative computational schemes, or integrative studies combining analytic and modelling perspectives, provided they align with the journal’s scope in mathematical and computer modelling.

Keywords: Exponential Sums, Power Mean Estimates and Computational Approaches, Equidistribution, p-adic Dynamical Systems, Arithmetic Progression

Manuscript Submissions:

Manuscript submission is open until 30th April 2027.

Please carefully review the journal scope and author submission instructions prior to submitting a manuscript as it will be rejected if it does not fall within the scope of the journal.

All manuscripts submitted to this Article Collection will undergo desk assessment and peer-review as part of our standard editorial process. Manuscripts which do not fall within the scope of the journal will be rejected.

We are able to offer a 10% Discount to all authors, and have a limited number of 20% Discount codes only available for early submissions. It should be noted that discount codes must be entered in at the point of submission as they cannot be applied retroactively, nor can these be combined as only the higher valued discount would be applicable.

To submit your papers to this Article Collection, please:

1. Check "Yes" for the question, "Are you submitting your paper for a specific special issue or article collection?"
2. Select the relevant Article Collection from the drop-down menu under the question, "Advances in Exponential Sums and Power Mean Estimates: Analytic Techniques and Computational Approaches"
3. Enter the Discount Code on the Final Page (Services) in the “Apply Discount Code” box on the right-hand side of the screen, otherwise this will not be applied to your submission, and you could be then subject to the journal’s Full APC Fee

Please contact Christopher Montgomery, Commissioning Editor regarding details on obtaining your discount codes, and with any other queries for this Article Collection.

Article Collection Guest Advisors

Prof. Wenpeng Zhang is a second-level professor and doctoral supervisor at Northwest University, who received his Ph.D. from Shandong University in 1988 under the guidance of renowned number theorist Professor Pan Chengdong. He has long specialized in number theory and its applications, publishing over 300 SCI papers and securing eight National Natural Science Foundation projects while achieving significant progress in exponential sums, character sums, and L-function mean value theorems. A recipient of multiple provincial and ministerial science awards, he has been listed among the world’s top 2% scientists from 2022 to 2023.

Prof. Taekyun Kim earned his Ph.D. in Number Theory from Kyushu University, Japan, in 1994. His research focuses on number theory, p-adic analysis, q-series, and degenerate polynomials. A Highly Cited Researcher recognized by Clarivate Analytics, he has published over 800 papers in SCI/SCOPUS-indexed journals. He is a Professor of Mathematics at Kwangwoon University since 2008, the founder of the Jangjeon Mathematical Society, and has served as the Editor-in-Chief of several international mathematics journals.