Get Asia’s leading financial wellness suite for a truly engaged workforce · Financial insecurity causes distraction, absenteeism, and employee-turnover

Refyne - India's first earned wage access platform

Refyne is India&#x27;s first Earned Wage Access platform, on a mission to simplify and help improve personal finances!

Refyne - Crunchbase Company Profile & Funding

They reinforce agile behaviours ... of the company’s lasting culture. . . #agileculture ... We&#x27;re #hiring a new Engineer Intern in Bengaluru, Karnataka. Apply today or share this post with your network. ... Spectrum | Insight #2 The co-founder, Chitresh Sharma&#x27;s address scintillated, with anecdotes from the ‘fail fast, fail often, learn quickly’ entrepreneurship journey. A journey of Refyne encapsulated ...

Refyne India | LinkedIn

Glassdoor gives you an inside look at what it&#x27;s like to work at Refyne, including salaries, reviews, office photos, and more. This is the Refyne company profile. All content is posted anonymously by employees working at Refyne.

Working at Refyne | Glassdoor

Information on valuation, funding, cap tables, investors, and executives for Refyne. Use the PitchBook Platform to explore the full profile.

Refyne Company Profile: Valuation & Investors | PitchBook

Refyne - Online platform for payday loans. Raised a total funding of $106M over 3 rounds from 7 investors.

Refyne - Company Profile - Tracxn

Refyne: Employees financial wellness platform. View company info, jobs, team members, culture, funding and more.

Refyne | AngelList

Refyne has 5 employees across 2 locations and $20.1 m in total funding,. See insights on Refyne including office locations, competitors, revenue, financials, executives, subsidiaries and more at Craft.

Refyne Company Profile - Office Locations, Competitors, Revenue, ...

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Swiggy has partnered with Betterplace and Refyne to enable such loans. There is no limit to the number of loans delivery partners can apply for, provided they maintain a good repayment history. This has enabled delivery partners to take loans up to three times on average during their tenure with the platform, the company ...

Refyne - Products, Competitors, Financials, Employees, Headquarters ...

Get insights about Refyne Company Profile, reviews, salaries, interviews questions, offices, locations, headquarters, employee benefits and more.

Refyne Company Profile & Overview | AmbitionBox

The startup’s plug-and-play software is aimed at employers of all size, and the platform can prove beneficial to blue-collar as well as white-collar workers. More than 100 companies in India are already using Refyne’s platform, serving over 300,000 employees. Some of the clients include ...

Refyne Company Profile, information, investors, valuation & Funding

Company information, business information, directors/partners details and director/partners contact information of REFYNE TECH PRIVATE LIMITED

REFYNE TECH PRIVATE LIMITED - Company, directors and contact details ...

Refyne is India’s first Earned Wage Access (EWA) provider based in Bengaluru. The B2B2C company was founded in 2020 by Chitresh Sharma and Apoorv Kumar. Refyne partners with organisations to allow their employees - full-time and contractual - to access their real-time earned pay anytime before ...

Refyne | Companies | QED Investors

Refyne partners with organizations to extend on-demand Earned Wage Access (EWA) to their employees

Refyne

In the landscape of agile software development, stands as a cornerstone of project success. This process, which involves the prioritization, often estimation, and decomposition of work into manageable pieces, is fundamental for effective sprint planning and efficient project execution. An innovative tool known as the has been created to give a new gamified option to this process. Designed to decrease the complexity and size of backlog items before implementation, these cards are a game-changer in enhancing project predictability. backlog refinement Backlog Refinement Cards Gamification in Backlog Refinement In agile project management, the application of gamification techniques (like ), as seen with Backlog Refinement Cards, has yielded significant benefits. These cards introduce a fun, competitive, and collaborative element to the backlog refinement process, which helps to harness the collective wisdom of the team, ultimately resulting in more reliable and realistic project forecasts. Dependency Poker Improved Forecasts By utilizing Backlog Refinement Cards, team members independently examine and break down tasks into smaller, manageable pieces. This process minimizes bias and undue influence, resulting in balanced and insightful estimates. This approach significantly enhances the predictability and forecastability of the project by providing a granular view of tasks. This improved forecasting aids in superior project planning and resource allocation, offering a distinct advantage in project management. Efficiency and Focus Backlog Refinement Cards facilitate an efficient process to refine and break down work. The game-like structure keeps the process engaging, and the focus on each card ensures the refinement process stays concise and on track. This focused approach leads to efficient use of time and resources. Team Building These cards do more than just support refining tasks; they serve as a team-building exercise, encouraging open communication and mutual respect among team members. The improvement in overall team dynamics is a valuable benefit that extends beyond the project at hand. Visibility and Transparency Backlog Refinement Cards provide visibility into how team members think about work and effort, leading to greater transparency. They allow different perspectives to surface and promote a shared understanding of the project’s scope and potential challenges. This transparency is crucial for effective collaboration and decision-making. Enhanced Understanding and Collaboration Encouraging active participation from all team members, each card prompts consideration of different aspects of the backlog item. This triggers discussions about the item's complexities and potential challenges, enhancing the team's understanding of the task and fostering a collaborative environment. The Backlog Refinement Cards apply the concept of serious gamification, mirroring successful implementations seen in agile methodologies, such as Planning Poker. Each card proposes a distinct approach for backlog refinement, prompting thorough discussions and informed decision-making about task management. The competitive aspect fostered by these cards motivates team members, promoting a culture of continuous learning and improvement. This utilization of gamification ultimately makes the backlog refinement process more enjoyable, engaging, and inspiring, while providing tangible benefits to the project and the team. The Cards Each card in the deck represents a unique perspective on the next backlog refinement step. Here are 10 out of the 30 cards. Job Shadowing Hypothesis Create a Spike Pair Programming Interviewing an Expert Enhances understanding of tasks, challenges, and workflows, which leads to more accurate task decomposition and estimation Stimulates critical thinking by encouraging educated assumptions about a feature's potential impact Allows for uncertainty and risk reduction by exploring ideas or answering questions via spikes Promotes code quality and knowledge sharing, resulting in more effective task allocation and estimation Provides insights and clarifications that increase understanding of backlog items and their interdependencies Attending Meetings Splitting by Happy Paths Splitting by User Roles Splitting by Channels Splitting by Domains Improves communication and shared understanding, leading to more accurate backlog refinement Allows teams to focus on the core functionality first, simplifying the management of the backlog Allows the team to focus on items based on defined User Roles. Enables the efficient management of functionalities across multiple channels Facilitates task management by segregating them into different application areas or services Accessibility and Implementation The are freely accessible to everyone, whether for private use or within organizations and companies. It is, however, required to acknowledge their origin when using these cards. Available in a ready-to-use PNG format, these cards are all set to improve your backlog refinement process. Backlog Refinement Cards In summary, Backlog Refinement Cards provide a novel and engaging method for the backlog refinement process. By decomposing backlog items into smaller units, teams can more accurately predict the time and effort required for each task, enhancing overall project predictability. Integrating gamification principles inspires teams to collaborate and engage in more creative problem-solving, leading to more effective backlog refinement and ultimately, smoother project execution.

Backlog Refinement Cards - Use the Benefits of Gamification for Backlog Refinement

The future of e-commerce lies in refined operation, as all industry players are leveraging data analytics to group their customers and provide tailored services. This is especially the case with Lifease, an e-commerce brand targeting middle-class consumers. In this post, I write about how we carry out refined operations, based on our own Data Management Platform (DMP). Let's start from the basics. As the raw material of our DMP, the data sources of Lifease include: Business logs from all sales ends； Sales data from third-party platforms; Basic data from within the company. These constitute our data assets, from which we derive a number of tags to describe the customers' age, address, preferred products, the device they use, etc. Using these tags as filters, we pick out a group of customers that match certain characteristics (the "Grouping" process). Then we observe the pattern of the target group. What is the DMP used for? From the data user's perspective, they mainly use the DMP for two purposes: tag query and grouping. : Sometimes they find a certain customer (or a group of customers) and check what tags they are attached to. Tag Query : After grouping, they might want to check to see if a certain customer is among a specified group, in order to find grounding for their marketing decisions. They might also pull the grouping result set from the DMP to their own business system for further development. Most of the time, they will start analyzing the shopping mode of the target group directly. Grouping How does the DMP work? Firstly, we, as data platform engineers, define the tags and the rules for grouping. Next, we define the domain-specific language ( ) that describes what we want to do, so we can submit computation tasks to Apache Spark. DSL Then, computing results will be stored in Apache Hive and Apache Doris. Lastly, data users can perform whatever queries they need in Hive and Doris. An abstraction of the DMP can be seen as a four-layered architecture: : All meta-information about the tags is stored in the source data tables; Metadata Management : This layer is supported by Spark, Hive, Doris, and Redis; Computation & Storage : This is like a command center of the DMP. It arranges the tasks throughout the whole platform, such as aggregating data into basic tags, converting data of basic tags into SQL semantics for queries based on the DSL rules, and passing computing results from Spark to Hive and Doris; Scheduling : This is where data users conduct grouping, profile analysis, and check the tags. Service Tags Tags are the most important elements of our DMP, so I'm going to take a whole chapter to introduce them. Every tag in our DMP goes through a five-phased lifecycle: : Data users put forward what kind of tags they need. Demand : Data developers sort out the data and produce the tags. Production : Tags are put into use for grouping. Grouping : Operators launch precision marketing practices on the targeted groups. Marketing : All relevant parties evaluate the utilization rate and effectiveness of tags, and then plan for subsequent improvement. Evaluation Every day, the grouping results will be updated and the outdated data will be cleaned. Both processes are automatic. We have also partially automated the tag production process. Our next priority is to realize 100% automation of it. How Tags are Produced? To derive tags from data, we need to turn our raw data into more organized and structured ones first: : This layer contains the user login history, event tracking logs, transaction data, and the binlogs from various databases. ODS (Operational Data Store) : Data processed by the ODS layer will be sent here to form user login tables, user event tables, and order information sheets. DWD (Data Warehouse Details) : The DM layer stores the aggregated data from DWD. DM (Data Market) Data in DM will be paired by “and”, “or”, and “xor” logic to produce tags that are more comprehensible for data users. Types of Tags Tags can be categorized differently based on different metrics: Where Do We Store Our Data? We made a list of what would look like an ideal data warehouse for us: Support high-performance queries so as to handle massive consumer end traffic; Support SQL for data analysis; Support data update; Be able to store huge amounts of data; Support extension functions to deal with user-defined data structures; Be closely integrated into the big data ecosystem. It was hard to find a single tool that met all these needs, so we tried a mixture of multiple tools. We stored part of our offline and real-time data in Hbase for basic tag queries, most of the offline data in Hive, and double-wrote the rest of our real-time data into Kudu and ES for real-time grouping and data query. The grouping results would be produced by Impala and then cached in Redis. As you can imagine, complicated storage could lead to tricky maintenance. Also, double writing adds to the risk of data inconsistency since one of them might fail. So, in Storage Architecture 2.0, we introduced and Apache Spark. The whole data pipeline was a Y-shaped diagram. Apache Doris We stored offline data in Hive, and the basic tags and real-time data Doris. Then, based on Spark, we conducted federated queries across Hive and Spark. The query results would be stored in Redis. With this new architecture, we compromised a little bit of performance for much lower maintenance costs. P.S. For your reference, here is a summary of the applicable scenarios for the various engines that we've used or investigated. What High-Performance Queries? Some of our data queries demand high performance, such as grouping checks and customer group analysis. Grouping Check Grouping check means to see if certain users are categorized into one or several given groups. It is a two-step check: : perform pre-computations and store the results in Redis; use Lua script for bulk check and thus increase performance. Check-in static group packets : extract data from contexts, API, and Apache Doris for rule-based judgment. Meanwhile, we have tried to increase performance by asynchronous checks, quick short-circuit calculation, query statement optimization, and join table quantity control. Check-in real-time behavior groups Customer Group Analysis Customer group analysis is done to figure out the behavioral path of consumers. It entails joining queries across the group packets and multiple tables. Apache Doris does not support path analysis functions so far, but its computing model is friendly to user-defined function development, so we have built a UDF for this purpose and it works well. Our Gains After Architectural Simplification The newly introduced data warehouse, Apache Doris, can be applied in multiple scenarios, including point queries, batch queries, behavioral path analysis, and grouping. In-pointf queries and small-scale join queries. It delivers a high performance of over 10,000 QPS and an RT99 of less than 50ms. Apart from its strong scalability and easy maintenance, we also benefit from the much simpler tag models brought by the integration of real-time and offline data. Conclusion In this post, I zoomed in on various parts of our DMP, and explained how the data tags, storage and query worked. I believe that a good tag system, faster query speeds, and more targeted consumer operation are the recipe for successful refined operations. So our follow-up efforts will be put into these aspects. Tag system: Build a comprehensive and accurate tag evaluation system; Improve quality and production efficiency of tags; Increase the coverage rate of tags. Storage & computation performance: Gradually move all our workloads from Hive and Spark to Apache Doris. Precision marketing: Develop a rich set of user analysis models; Enhance our user insight system; Enable general profile analysis to empower intelligent decisions. I'm writing this piece to share our practice in the data engineering community and hopefully, collect some valuable suggestions, so if you've got any ideas, meet me in the comment section.

Maximizing E-commerce Potential with Refined Data Analytics and Storage Architecture

There are many approaches to business in 2023, but one thing all leaders agree on is that the future of digital transformation lies in “the great dig” – unearthing the goldmine of data lying just beneath the surface. The trouble is, there is so much data flowing through businesses that it can be hard to separate the wheat from the chaff. How does a business know when it has tapped the right vein of data ore, and more importantly, how can it gather and mobilize that data into actionable business intelligence? How Can Businesses Convert Valuable Data Into Actionable Business Intelligence? Increasingly, the obvious answer is a combination of artificial intelligence (AI) and machine learning (ML).  In other words, . AI models can be deployed to help businesses make predictions, take advantage of market opportunities, measure performance and innovation, and react optimally to external events beyond their control. data science In a recent report, posited that by 2025, AI-driven workflows and seamless interfacing between humans and machines will become standard. Put simply, data-driven decision-making is going to become a fundamental part of day-to-day operations, from menial tasks to broad-sweeping business decisions. McKinsey That’s all good and well, but the sheer volume of data combined with the intricacies of devising and deploying AI models makes this a little too good to be true. The amount of data businesses have access to has reached saturation point, and not all of that data is good or useful. When a business does “strike it rich” by tapping a nice supply of data, employing AI to turn that data into business insight can take weeks if not months, by which time any opportunity to put that data to good use has passed. The current process for setting up AI models to interpret data is long and complex, requiring input from data scientists, developers, analysts, and users. According to , once a use case for an AI model has been identified, it takes two-thirds of organizations more than a month to develop a useable model, and another whole month to deploy it. The amount of time and resources invested is wasted if the opportunities to apply learnings from that model have passed. What results is a revolving door of high-investment development and deployment with minimal gain. Algorithmia’s 2021 report Resource is key here. In order to implement AI modeling effectively, an organization will need advanced in-house knowledge of coding and machine learning, or at least a hefty budget to employ a third party.  Even then, the data efficiency bottleneck remains. Once an expensive team of experts has been assembled, they’ll need time to work on developing a usable model, and once that model is outdated, more time again. So how do businesses get ahead of the curve and speed up the excavation and refinement of their data? The Answer Lies in Generative AI and Natural Language Processing (NLP) can be used to create predictive models based on large datasets. For example, an NLP language model could be trained on customer reviews to predict which products are likely to receive positive or negative feedback. It can also be used to train predictive models in real-time, which is particularly useful when there is limited historical data available, or when new and unexpected events occur that are not accounted for in existing data. A business that sells products online may use generative AI to create new variations of product images that can be used to train models to recognize and classify products more accurately. Or a business that analyzes customer behavior may use generative AI to create new customer profiles that can be used to train a machine learning model to predict customer preferences and behaviors more accurately. Generative AI can also be used to simulate different scenarios and outcomes, allowing businesses to explore the potential impact of different decisions before they’re made. This can help businesses to make more informed decisions, as well as identify potential risks and opportunities. Generative AI The real trick is being able to apply generative AI and NPL model training into databases seamlessly. This is the hump many businesses fail to get over, or if they do get over it, it’s expensive, resource-intensive, and too time-consuming, not to mention the sheer level of expertise required to make it happen. This technology needs to be democratized to the point where virtually all users, with a little bit of training, can use it. There are solutions out there that do just that, allowing users to bring state-of-the-art natural language processing and generative AI models into their databases with just a few lines of basic SQL. What results is a form of real-time in-database machine learning. Instead of AI models existing on the periphery, separate from a business database, the model becomes one with the database and provides real-time insights into datasets that can be actioned in a matter of hours and days instead of weeks and months. If the hunt for high-value data is “the great dig”, then the democratization of generative AI and NPL are the shovels that will help businesses unearth it. Lead image generated with stable diffusion. Prompt: Illustrate a gold mine.

Accelerating Excavation and Refinement of Data Gold Mines

Identifying Axion Conversion in Compact Star Magnetospheres With Radio-Wave Polarization Signatures

Rapid developments in the field of artificial intelligence require handling this technology with cautious. While there might have been initial concerns about the pace of these advancements, and the potential of stronger AI models used for unethical commercial use, we now have the opportunity to shape the future of AI and ensure it’s used ethically and for good, and the way to get there goes through Brussels. The EU AI Act The biggest advancement in AI regulation happened in December 2023. The European Parliament and the Council of the EU reached a political agreement on the AI Act. The AI Act is the first-ever legal framework on AI, which addresses the risks of AI and positions Europe to play a leading role globally. One might think that AI regulation in Europe alone is not sufficient. But it might just be enough thanks to a phenomenon is known as the “Brussels Effect.” The “Brussels Effect” The “Brussels Effect” refers to the tendency of multinational companies to conform to the stricter rules and regulations set by the EU across their entire business, rather than maintaining different standards for the European market and the rest of the world. This is often driven by the desire to avoid the costs and complexities associated with maintaining multiple product lines or processes. The term was coined by Yale Law School professor Anu Bradford in her 2020 book “The Brussels Effect: How the European Union Rules the World.” It highlights the EU’s ability to de facto export its regulations and standards to other markets and countries, given the economic importance and size of the European market. The General Data Protection Regulation (GDPR) is a prime example of the “Brussels Effect” in action. While the GDPR grants European citizens the right to access all the personal data collected about them by companies, its impact extends far beyond the EU’s borders. Due to the complexities of maintaining separate data handling processes, many global tech giants grant the GDPR’s data privacy protections to all their users worldwide, regardless of their location. Side note: You can extract all the information Google collected on you for example here: https://takeout.google.com/ Another example is the case of the USB-C charging mandate. Apple’s decision to adopt this standard globally for iPhones, rather than just for the EU market, is a clear example of the “Brussels Effect” in action. So, if judging by these examples, not all is lost. I would go as far as saying that the regulations suggested in the EU AI Act might set the boundaries and rules for the AI industry worldwide for years to come. A Closer Look at the EU AI Act Let’s go over the ideas in the EU AI Act briefly. The main idea is to categorize AI systems to 4 levels of risk. It bans AI applications considered a clear threat to safety and human rights, such as social scoring by governments or toys using voice assistance that encourages dangerous behaviour. High Risk AI systems are ones that use AI, for example in transportation, robot-assisted surgery or automated examination of visa applications etc. High-risk AI systems used in these critical areas will face strict requirements before market deployment. These include risk assessment, high data quality, human oversight, robustness, and documentation. For AI systems posing limited risks, such as chatbots or AI-generated content, the framework mandates transparency obligations to inform users about their artificial nature. AI systems that are labeled as minimal-risk, include applications like AI-enabled video games and spam filters, can be freely utilized without restrictions. These AI systems represent the vast majority of AI applications currently deployed across the EU. The Road Ahead To conclude, this risk-based framework might very well shape the future of AI, the way companies and users view AI systems and the roles they will function in our lives. Several questions raised from this piece of legislation are: How will the risk levels be decided? Will these risk levels and their definitions change often? How will it impact the the development and adoption of such technologies? As the AI landscape continues to evolve, ongoing dialogue and refinement will be crucial to ensure that the regulatory framework remains effective, ethical, and adaptable to technological advancements. We’ll find out sooner than later if indeed the path of the future of AI goes through Brussels. References https://digital-strategy.ec.europa.eu/en/policies/regulatory-framework-ai?embedable=true https://www.europarl.europa.eu/topics/en/article/20230601STO93804/eu-ai-act-first-regulation-on-artificial-intelligence?embedable=true https://www.brusselseffect.com/?embedable=true Rapid developments in the field of artificial intelligence require handling this technology with cautious. While there might have been initial concerns about the pace of these advancements, and the potential of stronger AI models used for unethical commercial use, we now have the opportunity to shape the future of AI and ensure it’s used ethically and for good, and the way to get there goes through Brussels. The EU AI Act The EU AI Act The biggest advancement in AI regulation happened in December 2023. The European Parliament and the Council of the EU reached a political agreement on the AI Act . The AI Act is the first-ever legal framework on AI, which addresses the risks of AI and positions Europe to play a leading role globally. One might think that AI regulation in Europe alone is not sufficient. But it might just be enough thanks to a phenomenon is known as the “Brussels Effect.” The European Parliament and the Council of the EU reached a political agreement on the AI Act The “Brussels Effect” The “Brussels Effect” The “Brussels Effect” refers to the tendency of multinational companies to conform to the stricter rules and regulations set by the EU across their entire business, rather than maintaining different standards for the European market and the rest of the world. This is often driven by the desire to avoid the costs and complexities associated with maintaining multiple product lines or processes. The term was coined by Yale Law School professor Anu Bradford in her 2020 book “The Brussels Effect: How the European Union Rules the World.” It highlights the EU’s ability to de facto export its regulations and standards to other markets and countries, given the economic importance and size of the European market. The General Data Protection Regulation (GDPR) is a prime example of the “Brussels Effect” in action. While the GDPR grants European citizens the right to access all the personal data collected about them by companies, its impact extends far beyond the EU’s borders. Due to the complexities of maintaining separate data handling processes, many global tech giants grant the GDPR’s data privacy protections to all their users worldwide, regardless of their location. Side note: You can extract all the information Google collected on you for example here: https://takeout.google.com/ Side note: You can extract all the information Google collected on you for example here: https://takeout.google.com/ https://takeout.google.com/ https://takeout.google.com/ Another example is the case of the USB-C charging mandate. Apple’s decision to adopt this standard globally for iPhones, rather than just for the EU market, is a clear example of the “Brussels Effect” in action. So, if judging by these examples, not all is lost. I would go as far as saying that the regulations suggested in the EU AI Act might set the boundaries and rules for the AI industry worldwide for years to come. A Closer Look at the EU AI Act A Closer Look at the EU AI Act Let’s go over the ideas in the EU AI Act briefly. The main idea is to categorize AI systems to 4 levels of risk. It bans AI applications considered a clear threat to safety and human rights, such as social scoring by governments or toys using voice assistance that encourages dangerous behaviour. High Risk AI systems are ones that use AI, for example in transportation, robot-assisted surgery or automated examination of visa applications etc. High-risk AI systems used in these critical areas will face strict requirements before market deployment. These include risk assessment, high data quality, human oversight, robustness, and documentation. For AI systems posing limited risks, such as chatbots or AI-generated content, the framework mandates transparency obligations to inform users about their artificial nature. AI systems that are labeled as minimal-risk, include applications like AI-enabled video games and spam filters, can be freely utilized without restrictions. These AI systems represent the vast majority of AI applications currently deployed across the EU. The Road Ahead The Road Ahead To conclude, this risk-based framework might very well shape the future of AI, the way companies and users view AI systems and the roles they will function in our lives. Several questions raised from this piece of legislation are: How will the risk levels be decided? Will these risk levels and their definitions change often? How will it impact the the development and adoption of such technologies? As the AI landscape continues to evolve, ongoing dialogue and refinement will be crucial to ensure that the regulatory framework remains effective, ethical, and adaptable to technological advancements. We’ll find out sooner than later if indeed the path of the future of AI goes through Brussels. References https://digital-strategy.ec.europa.eu/en/policies/regulatory-framework-ai?embedable=true https://digital-strategy.ec.europa.eu/en/policies/regulatory-framework-ai?embedable=true https://www.europarl.europa.eu/topics/en/article/20230601STO93804/eu-ai-act-first-regulation-on-artificial-intelligence?embedable=true https://www.europarl.europa.eu/topics/en/article/20230601STO93804/eu-ai-act-first-regulation-on-artificial-intelligence?embedable=true https://www.brusselseffect.com/?embedable=true https://www.brusselseffect.com/?embedable=true

Forget Silicon Valley, It’s All About Brussels Now: How the EU AI Act Impacts Global AI Policies

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1
2
3 Proof of Theorem 1
References Keywords and phrases: Riemann’s zeta function, density hypothesis, density theorems. 2020 Mathematics Subject Classification: Primary 11M26, Secondary 11M06. 1 The plausibility of the Riemann Hypothesis (RH) is supported (among others) by results of type (1.1) valid for all σ > 1/2 as T → ∞. (1.1) was first shown by Bohr and Landau in 1914 ([BL1914]). The first estimate of type was shown few years later by Carlson [Car1920] with It was Hoheisel [Hoh1930] who first observed that such estimates lead to arithmetic consequences about the difference of consecutive primes. He proved the first approximation towards the famous conjecture that there exists always a prime between two consecutive squares. This conjecture was characterised by Landau [Lan1913] in his plenary talk at ICM1912 in Cambridge as one of the four main problems of the distribution of primes, besides the Riemann Hypothesis. Hoheisel [Hoh1930] could show (pn denotes the nth prime) In the proof important role was played by Carlson’s density theorem (1.2)– (1.3). Later it was realized that a uniform estimate of the form yields (combined with a slightly better zero-free region than the classical one of de la Vall´ee Poussin) In particular, the best possible estimate A = 2 would almost imply Landau’s conjecture. This is especially remarkable in light of the fact that even assuming the Riemann Hypothesis the best estimate we know is (with C = 2 easily by the Riemann–Von Mangoldt explicit formula, with C = 1 by a deeper argument of Cram´er [Cra1921]). This explains the significance of the Density Hypothesis (DH) which states or, in a slightly weaker form, using the notation (1.2), The Riemann Hypothesis clearly implies the Density Hypothesis. However, Ingham [Ing1937] showed that also the Lindel¨of Hypothesis (LH) (1.10) implies DH. In 1954 Tur´an [Tur1954] used his celebrated power-sum method [Tur1953], [Tur1984] to give a different proof of Ingham’s result that LH implies DH. In the same work he came very close to breaking the DH in the vicinity of the boundary line σ = 1. In the following we will use the notation: His result was with a small constant c1 [Tur1954] or with our new notation It was 16 years later when G. Hal´asz and Tur´an succeeded to break the DH [HT1969], that is, to prove (in the refined form appearing as Theorem 38.2 of Tur´an’s book [Tur1984]) with the notation (1.1)–(1.2), (1.11) The proof was based on (i) Vinogradov’s estimate µ(1 − η) ≪ η 3/2, (ii) Tur´an’s power-sum method; (iii) a simple but ingenious idea of Hal´asz [Hal1968]. Soon after this Bombieri [Bom1971] gave a different proof using ideas from the large sieve coupled with (i) and (iii). Further, Montgomery has shown the DH for σ ≥ 9/10, i.e. η ≤ 1/10 ([Mon1969], [Mon1971]). Using another simple but ingenious idea Huxley [Hux1972] (with a refinement of Montgomery) reached the DH for η ≤ 1/6 and proved that The DH was shown for larger and larger ranges, until improving the result σ > 11/14 of Jutila [Jut1977] Bourgain proved its validity [Bou2000] for Another direction of the research was to give strong density estimates for small values of η (often especially for η → 0). In this direction the most important results were reached (in alphabetical order) by Bourgain, Ford, Heath-Brown, Huxley, Ivic, Jutila and Montgomery. All these results use the large sieve and in some form the mentioned idea of Hal´asz, which needs η ≤ 1/4. This is also the reason that they cannot give improvements of the classical zero-density theorem of Ingham [Ing1940], A(σ) ≤ 3/(2 − σ), which is still the best today in the whole range Recently I gave two alternative variants for proving the DH for small values of η. In [Pin2022] the goal was to reach a possibly simple proof while in [Pin2023] to show the strongest possible result both in specific ranges of η < 1/12 and for η → 0. Both proofs were based on Vinogradov’s method and Hal´asz’s idea. However, in the second [Pin2023] it was important to use recent deep results of Heath-Brown [Hea2017], further of Bourgain, Demeter and Guth [BGD2016]. This paper is available on arxiv under CC 4.0 license. Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu Table of Links 1 2 3 Proof of Theorem 1 References 1 1 2 2 3 Proof of Theorem 1 3 Proof of Theorem 1 References References Keywords and phrases: Riemann’s zeta function, density hypothesis, density theorems. Keywords and phrases: 2020 Mathematics Subject Classification: Primary 11M26, Secondary 11M06. 1 The plausibility of the Riemann Hypothesis (RH) is supported (among others) by results of type (1.1) valid for all σ > 1/2 as T → ∞. (1.1) was first shown by Bohr and Landau in 1914 ([BL1914]). The first estimate of type was shown few years later by Carlson [Car1920] with It was Hoheisel [Hoh1930] who first observed that such estimates lead to arithmetic consequences about the difference of consecutive primes. He proved the first approximation towards the famous conjecture that there exists always a prime between two consecutive squares. This conjecture was characterised by Landau [Lan1913] in his plenary talk at ICM1912 in Cambridge as one of the four main problems of the distribution of primes, besides the Riemann Hypothesis. Hoheisel [Hoh1930] could show (pn denotes the nth prime) In the proof important role was played by Carlson’s density theorem (1.2)– (1.3). Later it was realized that a uniform estimate of the form yields (combined with a slightly better zero-free region than the classical one of de la Vall´ee Poussin) In particular, the best possible estimate A = 2 would almost imply Landau’s conjecture. This is especially remarkable in light of the fact that even assuming the Riemann Hypothesis the best estimate we know is (with C = 2 easily by the Riemann–Von Mangoldt explicit formula, with C = 1 by a deeper argument of Cram´er [Cra1921]). This explains the significance of the Density Hypothesis (DH) which states or, in a slightly weaker form, using the notation (1.2), The Riemann Hypothesis clearly implies the Density Hypothesis. However, Ingham [Ing1937] showed that also the Lindel¨of Hypothesis (LH) (1.10) implies DH. In 1954 Tur´an [Tur1954] used his celebrated power-sum method [Tur1953], [Tur1984] to give a different proof of Ingham’s result that LH implies DH. In the same work he came very close to breaking the DH in the vicinity of the boundary line σ = 1. In the following we will use the notation: His result was with a small constant c1 [Tur1954] or with our new notation It was 16 years later when G. Hal´asz and Tur´an succeeded to break the DH [HT1969], that is, to prove (in the refined form appearing as Theorem 38.2 of Tur´an’s book [Tur1984]) with the notation (1.1)–(1.2), (1.11) The proof was based on (i) Vinogradov’s estimate µ(1 − η) ≪ η 3/2, (ii) Tur´an’s power-sum method; (iii) a simple but ingenious idea of Hal´asz [Hal1968]. Soon after this Bombieri [Bom1971] gave a different proof using ideas from the large sieve coupled with (i) and (iii). Further, Montgomery has shown the DH for σ ≥ 9/10, i.e. η ≤ 1/10 ([Mon1969], [Mon1971]). Using another simple but ingenious idea Huxley [Hux1972] (with a refinement of Montgomery) reached the DH for η ≤ 1/6 and proved that The DH was shown for larger and larger ranges, until improving the result σ > 11/14 of Jutila [Jut1977] Bourgain proved its validity [Bou2000] for Another direction of the research was to give strong density estimates for small values of η (often especially for η → 0). In this direction the most important results were reached (in alphabetical order) by Bourgain, Ford, Heath-Brown, Huxley, Ivic, Jutila and Montgomery. All these results use the large sieve and in some form the mentioned idea of Hal´asz, which needs η ≤ 1/4. This is also the reason that they cannot give improvements of the classical zero-density theorem of Ingham [Ing1940], A(σ) ≤ 3/(2 − σ), which is still the best today in the whole range Recently I gave two alternative variants for proving the DH for small values of η. In [Pin2022] the goal was to reach a possibly simple proof while in [Pin2023] to show the strongest possible result both in specific ranges of η < 1/12 and for η → 0. Both proofs were based on Vinogradov’s method and Hal´asz’s idea. However, in the second [Pin2023] it was important to use recent deep results of Heath-Brown [Hea2017], further of Bourgain, Demeter and Guth [BGD2016].

A Remark on Density Theorems for Riemann’s Zeta-function: 1

When I first heard about Nervos Network's dedication to building a BTC Layer 2, I wasn't surprised. That's because the CKB public blockchain technology not only matches but also goes beyond BTC. It not only maintains BTC's native UTXO features but also allows for more complex programmable extensions. While surpassing BTC may seem challenging for CKB public chain, aiming to be a top-notch BTC Layer 2 seems like a smart choice. Why? Let's delve into my take on CKB. The current BTC Layer 2 market is constrained by the mainnet’s limited validation capabilities, resulting in diverse alternative solutions. The simplicity of BTC's scripting language, combined with its near-zero computation and validation power, has created significant opportunities for market innovation. As of now, aside from limited transaction validation and multisig functionalities within UTXO unlocking conditions, the BTC mainnet is incapable of directly handling more complex transaction logic involving data validation, state changes, etc. It relegates BTC to an asset settlement layer, relying on extending through a powerful public chain for local consensus and computational validation capabilities. This leads to a lack of uniform standards or 'orthodoxy' for BTC Layer 2 solutions, making it challenging to rank them. However, distinctions can be made based on the community's perception between narrow and broad interpretations: Narrowly defined, only solutions like the Lightning Network state channels and the one-time seal approach of RGB qualify as truly 'orthodox' BTC Layer 2s. They effectively utilize BTC's limited script validation capacity without relying on or minimally depending on external local consensus. In a more general sense, any extension chain that gains acceptance for its local consensus, along with a cross-chain bridge solution that ensures the safe migration of assets, can theoretically serve as a BTC Layer 2. This includes contemporary Ethereum EVM chains, Solana with its high-concurrency capabilities, and others. Clearly, the BTC Layer 2 market is divided: on one end are highly-specific solutions like the slowly evolving Lightning Network and RGB, facing substantial challenges; on the other end are broadly defined solutions, where any performance chain capable of securely interacting with the BTC mainnet qualifies as a BTC Layer 2. But is there a 'middle-of-the-road' option? Yes, and it lies in the Nervos Network, which adheres to the UTXO model at its core, enhancing its performance. Notable features include: The CKB Network aligns closely with BTC, sharing the same 'UTXO model and mining consensus mechanism,' unlike the account balance model of mainstream public blockchains like Ethereum. The UTXO model has distinct advantages in transaction privacy, flexible transaction structuring, and parallel processing capabilities to prevent double-spending, which may be one of Satoshi Nakamoto’s most brilliant inventions. This is why post-Ethereum projects like Sui and Aptos have also adopted similar UTXO models. Bitcoin's capacity and block speed may be constrained by its era, but the UTXO model is remarkably forward-thinking. CKB adopts this UTXO model, evolving it into the 'Cell model', which preserves the pure transactional nature of Bitcoin’s UTXO model while also supporting the data states found in account models like Ethereum’s. To simplify: In the Bitcoin UTXO model, the creation and destruction of coins resemble a continuous minting and melting process. The Cell model, however, omits the destruction aspect, focusing on verifying and persistently storing states. Each Cell includes Capacity and Data: Capacity measures the balance in bytes, similar to UTXO, while Data holds any kind of information, including the history of transaction states. Thus, the Cell model can accurately represent balances and manage asset transfers, and also handle a variety of complex smart contract states. In summary, the Cell model is a more persistent and flexible transaction model that greatly broadens the scope of the UTXO model. It is crucial for CKB’s ability to maintain the security of the BTC mainnet while offering a 'speed boost' to slower Bitcoin expansion initiatives like the Lightning Network and RGB. For instance, the recent rollout of RGB++ by CKB illustrates this. In the BTC ecosystem, developing a mature RGB solution involves challenges that are not so much about the one-time sealing process of the BTC mainnet, but rather the communication, coordination, and mutual state maintenance among off-chain client validation nodes, particularly in a decentralized setting. In simpler terms, RGB's theory might seem straightforward, but its practical implementation is hindered by foundational infrastructural limitations and various barriers.
Recognizing this, CKB integrates these off-chain client validation nodes into its on-chain public validation process. This approach significantly accelerates the intended UTXO client extension pathway that RGB aims to achieve. Complex P2P consensus among off-chain nodes is notoriously challenging, fraught with complexity and obstacles such as potential data synchronization delays or inconsistencies, and susceptibility to fraud and attacks. Transposing this process onto the blockchain can simplify these issues. With the increasing discussions around RGB++, let’s also look at CKB's Open Transaction data format, showcasing the chain's forward-thinking features. Open Transaction allows multiple participants to collaboratively build and aggregate different transactions over time. It supports partial construction, amendability, and incremental build-up and aggregation. For example, Alice initiates an Open Transaction to exchange a certain amount of Token A for Token B with Bob. The transaction, once started, remains in an editable state. Bob, upon agreeing to the transaction terms, can then add his Token B and finalize the conditions. It might initially seem abstract. Take cross-chain scenarios as an example: Alice and Bob could independently execute asset trades on various distinct chains, significantly boosting the CKB chain's cross-chain interaction capabilities. In the realm of complex DeFi transactions, where market-driven dynamic adjustments are often necessary, Open Transaction allows contract participants to adapt trading conditions fluidly during the contract's execution. This undoubtedly enhances the ability to manage transactional complexities. From my perspective, Open Transaction mirrors the UTXO transaction unlocking conditions, capable of amalgamating intricate unlocking conditions, multi-party signatures, and complex transactional environments. This represents an evolutionary and valuable innovation that builds on the foundational principles of the BTC main chain. Interestingly, Jan Xie, a core developer from the Ethereum team, chose to embrace the BTC UTXO model for his inaugural major project. Despite the broader application of Ethereum's smart contract model, Jan and his Nervos team decisively chose to expand and refine the BTC UTXO model. This choice reflects a deep respect for Satoshi Nakamoto’s simplistic UTXO transaction model and also subtly lays the groundwork for its transformation into a native BTC Layer 2. In conclusion, I am quite bullish about CKB’s potential as a BTC Layer 2. In the short term, it certainly has the potential to expedite the implementation of projects like the Lightning Network and RGB within UTXO-based chains, offering valuable insights for these 'orthodox' expansion solutions on the BTC mainnet. Looking long-term, the inherent features of CKB's chain and its innovative architectural compatibility may enable it to excel in the complex and standardless arena of BTC Layer 2s. Note: There’s much more to explore regarding the technical nuances and standout features of CKB, which I plan to analyze in greater depth later. It's fascinating to see how BTC Layer 2 not only provides a platform for new chains to rise but also opens up endless possibilities for rejuvenation within established chains. Author: Haotian, Independent Researcher This article is the translation of a Haotian’s tweet. When I first heard about Nervos Network's dedication to building a BTC Layer 2, I wasn't surprised. That's because the CKB public blockchain technology not only matches but also goes beyond BTC . It not only maintains BTC's native UTXO features but also allows for more complex programmable extensions. While surpassing BTC may seem challenging for CKB public chain, aiming to be a top-notch BTC Layer 2 seems like a smart choice. Why? Let's delve into my take on CKB. BTC The current BTC Layer 2 market is constrained by the mainnet’s limited validation capabilities, resulting in diverse alternative solutions. The simplicity of BTC's scripting language, combined with its near-zero computation and validation power, has created significant opportunities for market innovation. As of now, aside from limited transaction validation and multisig functionalities within UTXO unlocking conditions, the BTC mainnet is incapable of directly handling more complex transaction logic involving data validation, state changes, etc. It relegates BTC to an asset settlement layer, relying on extending through a powerful public chain for local consensus and computational validation capabilities. This leads to a lack of uniform standards or 'orthodoxy' for BTC Layer 2 solutions, making it challenging to rank them. However, distinctions can be made based on the community's perception between narrow and broad interpretations: However, distinctions can be made based on the community's perception between narrow and broad interpretations: Narrowly defined, only solutions like the Lightning Network state channels and the one-time seal approach of RGB qualify as truly 'orthodox' BTC Layer 2s. They effectively utilize BTC's limited script validation capacity without relying on or minimally depending on external local consensus. In a more general sense, any extension chain that gains acceptance for its local consensus, along with a cross-chain bridge solution that ensures the safe migration of assets, can theoretically serve as a BTC Layer 2. This includes contemporary Ethereum EVM chains, Solana with its high-concurrency capabilities, and others. Clearly, the BTC Layer 2 market is divided: on one end are highly-specific solutions like the slowly evolving Lightning Network and RGB, facing substantial challenges; on the other end are broadly defined solutions, where any performance chain capable of securely interacting with the BTC mainnet qualifies as a BTC Layer 2. Clearly, the BTC Layer 2 market is divided: But is there a 'middle-of-the-road' option? Yes, and it lies in the Nervos Network, which adheres to the UTXO model at its core, enhancing its performance. Notable features include: The CKB Network aligns closely with BTC, sharing the same 'UTXO model and mining consensus mechanism,' unlike the account balance model of mainstream public blockchains like Ethereum. The UTXO model has distinct advantages in transaction privacy, flexible transaction structuring, and parallel processing capabilities to prevent double-spending, which may be one of Satoshi Nakamoto’s most brilliant inventions. This is why post-Ethereum projects like Sui and Aptos have also adopted similar UTXO models. Bitcoin's capacity and block speed may be constrained by its era, but the UTXO model is remarkably forward-thinking. CKB adopts this UTXO model, evolving it into the 'Cell model', which preserves the pure transactional nature of Bitcoin’s UTXO model while also supporting the data states found in account models like Ethereum’s. To simplify : In the Bitcoin UTXO model, the creation and destruction of coins resemble a continuous minting and melting process. The Cell model, however, omits the destruction aspect, focusing on verifying and persistently storing states. Each Cell includes Capacity and Data: Capacity measures the balance in bytes, similar to UTXO, while Data holds any kind of information, including the history of transaction states. Thus, the Cell model can accurately represent balances and manage asset transfers, and also handle a variety of complex smart contract states. To simplify In summary, the Cell model is a more persistent and flexible transaction model that greatly broadens the scope of the UTXO model. It is crucial for CKB’s ability to maintain the security of the BTC mainnet while offering a 'speed boost' to slower Bitcoin expansion initiatives like the Lightning Network and RGB. For instance, the recent rollout of RGB++ by CKB illustrates this. In the BTC ecosystem, developing a mature RGB solution involves challenges that are not so much about the one-time sealing process of the BTC mainnet, but rather the communication, coordination, and mutual state maintenance among off-chain client validation nodes, particularly in a decentralized setting. In simpler terms, RGB's theory might seem straightforward, but its practical implementation is hindered by foundational infrastructural limitations and various barriers.
Recognizing this, CKB integrates these off-chain client validation nodes into its on-chain public validation process. This approach significantly accelerates the intended UTXO client extension pathway that RGB aims to achieve. Complex P2P consensus among off-chain nodes is notoriously challenging, fraught with complexity and obstacles such as potential data synchronization delays or inconsistencies, and susceptibility to fraud and attacks. Transposing this process onto the blockchain can simplify these issues. With the increasing discussions around RGB++, let’s also look at CKB's Open Transaction data format, showcasing the chain's forward-thinking features. Open Transaction allows multiple participants to collaboratively build and aggregate different transactions over time. It supports partial construction, amendability, and incremental build-up and aggregation. For example, Alice initiates an Open Transaction to exchange a certain amount of Token A for Token B with Bob. The transaction, once started, remains in an editable state. Bob, upon agreeing to the transaction terms, can then add his Token B and finalize the conditions. It might initially seem abstract. Take cross-chain scenarios as an example: Alice and Bob could independently execute asset trades on various distinct chains, significantly boosting the CKB chain's cross-chain interaction capabilities. In the realm of complex DeFi transactions, where market-driven dynamic adjustments are often necessary, Open Transaction allows contract participants to adapt trading conditions fluidly during the contract's execution. This undoubtedly enhances the ability to manage transactional complexities. From my perspective, Open Transaction mirrors the UTXO transaction unlocking conditions, capable of amalgamating intricate unlocking conditions, multi-party signatures, and complex transactional environments. This represents an evolutionary and valuable innovation that builds on the foundational principles of the BTC main chain. Interestingly, Jan Xie , a core developer from the Ethereum team, chose to embrace the BTC UTXO model for his inaugural major project. Despite the broader application of Ethereum's smart contract model, Jan and his Nervos team decisively chose to expand and refine the BTC UTXO model. This choice reflects a deep respect for Satoshi Nakamoto’s simplistic UTXO transaction model and also subtly lays the groundwork for its transformation into a native BTC Layer 2. Jan Xie Jan Xie In conclusion , I am quite bullish about CKB’s potential as a BTC Layer 2. In the short term, it certainly has the potential to expedite the implementation of projects like the Lightning Network and RGB within UTXO-based chains, offering valuable insights for these 'orthodox' expansion solutions on the BTC mainnet. Looking long-term, the inherent features of CKB's chain and its innovative architectural compatibility may enable it to excel in the complex and standardless arena of BTC Layer 2s. In conclusion Note: There’s much more to explore regarding the technical nuances and standout features of CKB, which I plan to analyze in greater depth later. It's fascinating to see how BTC Layer 2 not only provides a platform for new chains to rise but also opens up endless possibilities for rejuvenation within established chains. Author: Haotian, Independent Researcher Author: Haotian , Independent Researcher Author: Haotian Haotian This article is the translation of a Haotian’s tweet. This article is the translation of a Haotian’s tweet. This article is the translation of a Haotian’s tweet . tweet tweet

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