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zety

Do you remember the time when the appeared on the smartphones screens? This game has started the era of casual games with very few in-game actions, and each wrong move means your game is over. The one who lasts alive longest will head the leaderboard. Flappy Bird Today we will see how to write a Python bot that piggybacks on computer vision library OpenCV to beat , an endless runner from . Don’t touch the red Addicting Games Rules of the game The game is pretty simple: green buttons fall, and the player needs to press them before leaving the screen. And of course, don’t touch the red! There’s one crucial feature; if you play in arcade mode, the buttons fall with increasing speed. That makes the game hard for a human player. But it is not an issue for our bot! OpenCV Template Matching The main part of our computer vision bot is the template matching available in the OpenCV library. It is not a neural network approach. It is much simpler and limited. This algorithm is meant for searching for a patch on the target image, e.g., a “green button” on a “gaming screen.” It works the following way: the algorithm takes the template image and then, using a sliding window, tries to find a matching patch on the target image. Using this, we can get the positions and similarity measures for each pixel. In the code application of the template matching looks like this import cv2

template = cv2.imread('template.png')
target = cv2.imread('target.png')
result = cv2.matchTemplate(target, template, cv2.TM_CCOEFF_NORMED)
    _, max_val, _, max_loc = cv2.minMaxLoc(result) As a result, we’ll have the equal to the maximum similarity on the target image and the is the upper left corner of the found match. max_val max_loc This algorithm is faster when it works with smaller target images and smaller patterns. Firstly, I’ve tried to work with whole green buttons, but then I’ve switched to smaller ones that work faster, and with that, I’ve achieved higher scores Taking Screenshots and Interaction The other important parts of the bot are getting screens to analyze and how we’re sending mouse clicks to the game. It is necessary to mention that Addicting Games provide games that you can play using your internet browser, so nothing additional must be installed. There’re two Python packages that help with the tasks above: and , we use them to get the screenshots of a particular part of the screen and to send clicks to the browser window correspondingly. I also use library as it’s very handy to set the “break action” on some key in the case when your mouse is controlled by a bot. The library (and probably )  require rights, so run your Python script as an executable with a proper shebang header. mss pyautogui keyboard keyboard pyautogui sudo Here I provide code snippets on how to get screenshots and send clicks: #!/hdd/anaconda2/envs/games_ai/bin/python

# ^ change above to your python path ^

import keyboard
import mss
import pyautogui

pyautogui.PAUSE = 0.0

print("Press 's' to start")
print("Press 'q' to quit")
keyboard.wait('s')

# setup mss and get the full size of your monitor
sct = mss.mss()
mon = sct.monitors[0]

while True:
    # decide on the part of the screen
    roi = {
        "left": 0, 
        "top": int(mon["height"] * 0.2), 
        "width": int(mon["width"] / 2), 
        "height": int(mon["height"] * 0.23)
    }

    roi_crop = numpy.array(sct.grab(roi))[:,:,:3]
    
    # do something with `roi_crop`

    if keyboard.is_pressed('q'):
        break Here’s also one thing. When you use on Linux, you might face it is possible to overcome with command. pyautogui Xlib.error.DisplayConnectionError xhost + My Algorithm Based on the latter two, I’ve created an algorithm that beats the previous human playing with score of 170 a score of 445. There are two parts to a program. First tries to click the first three-button available on a screen when the game starts. The game field doesn’t move until a player hits the first button, so we can treat a field as static when we do click on the first three. For that purpose, we inspect the three lines of the screen, searching for a small pattern (see the previous figure), and then click on them The first half of the code: #!/hdd/anaconda2/envs/games_ai/bin/python

# if "Xlib.error.DisplayConnectionError" use "xhost +" on linux

import shutil
import os
import keyboard
import mss
import cv2
import numpy
from time import time, sleep
import pyautogui
from random import randint
import math

pyautogui.PAUSE = 0.0

print("Press 's' to start")
print("Press 'q' to quit")
keyboard.wait('s')

try:
    shutil.rmtree("./screenshots")
except FileNotFoundError:
    pass
os.mkdir("./screenshots")

# setup mss and get the full size of your monitor
sct = mss.mss()
mon = sct.monitors[0]

frame_id = 0
# decide where is the region of interest
for idx in range(3,0,-1):
    roi = {
        "left": 0, 
        "top": int(mon["height"] * (idx * 0.2)), 
        "width": int(mon["width"] / 2), 
        "height": int(mon["height"] * 0.23)
    }

    green_button = cv2.imread('green_button.png')
    offset_x = int(green_button.shape[0] / 2)
    offset_y = int(green_button.shape[1] / 2)

    roi_crop = numpy.array(sct.grab(roi))[:,:,:3]
    result = cv2.matchTemplate(roi_crop, green_button, cv2.TM_CCOEFF_NORMED)
    _, max_val, _, max_loc = cv2.minMaxLoc(result)

    print(max_val, max_loc)

    button_center = (max_loc[0] + offset_y, max_loc[1] + offset_x)
    roi_crop = cv2.circle(roi_crop.astype(float), button_center, 20, (255, 0, 0), 2)
    cv2.imwrite(f"./screenshots/{frame_id:03}.jpg", roi_crop)

    abs_x_roi = roi["left"] + button_center[0]
    abs_y_roi = roi["top"] + button_center[1]
    pyautogui.click(x=abs_x_roi, y=abs_y_roi)
    frame_id += 1 In the second part, we press the following 400 buttons; it is implemented as an infinite while loop that captures the screen and clicks on the pixel where it is expected to see a button regarding the current speed. The speed function has been selected as a logarithmic function of the number of iterations. This function provides a pixel offset needed to adjust when time has passed since the pattern had been found. The second half: second_roi = {
    "left": 0, 
    "top": int(mon["height"] * 0.18), 
    "width": int(mon["width"] / 2), 
    "height": int(mon["height"] * 0.06)
}

btn = cv2.imread('center.png')
offset_y = int(btn.shape[0])
offset_x = int(btn.shape[1] / 2)

thresh = 0.9
frame_list = []
btn_cnt = 1
while True:
    frame_id += 1
    second_roi_crop = numpy.array(sct.grab(second_roi))[:,:,:3]
    result = cv2.matchTemplate(second_roi_crop, btn, cv2.TM_CCOEFF_NORMED)
    _, max_val, _, max_loc = cv2.minMaxLoc(result)
    
    # define the speed of the screen
    speed = math.floor(math.log(frame_id)**2.5)
    print(frame_id, max_val, max_loc, speed)
    frame_list.append(max_loc[0])
    if max_val > thresh:
        button_center = (max_loc[0] + offset_x, max_loc[1] + offset_y)
        second_roi_crop = cv2.circle(second_roi_crop.astype(float), button_center, 20, (255, 0, 0), 2)
        cv2.imwrite(f"./screenshots/{frame_id:03}.jpg", second_roi_crop)

        abs_x_sec = second_roi["left"] + button_center[0]
        abs_y_sec = second_roi["top"] + button_center[1] + speed
        pyautogui.click(x=abs_x_sec, y=abs_y_sec)
        btn_cnt += 1

    if keyboard.is_pressed('q'):
        break As you can see, the speed is parameterized, and depending on your PC configuration, you can find better parameters that beat my high score. I encourage you to do that! This is because the code is very dependent on the speed of image processing and it may vary from system to system. Here’s the peek of the one run. How it looks like when the bot is actually running. In order not to be unfounded, here is the leaderboard screenshot. I need to mention that in this particular game, score at all levels of difficulty goes to the leaderboard, so you needn’t play “Hard”. “Easy” level is just fine (btw when you reach 100 pressed button, you can’t tell that it’s easy anymore) The code of the project is available at Github . It would be great to create an extensive library of Hacked Addicting Games and keep all of these algorithms there. So you are invited to create the pull requests! https://github.com/zetyquickly/addicting-games-ai Acknowledgments This video inspired this project: https://www.youtube.com/watch?v=vXqKniVe6P8 Here the author beats the leaderboard of the Kick Ya Chop game, it has similarities with Don’t Touch the Red, but also, there’s a big difference. In Kick Ya Chop, the player decides on the speed of the game. The faster the human/bot clicks faster the tree is falling. In Don’t Touch the Red, the game decides on the speed of upcoming buttons.

How to Make a Gaming Bot that Beats Human Using Python and OpenCV

Cómo hacer un bot de juegos que supere a los humanos usando Python y OpenCV

Newtonian Fractal on . http://usefuljs.net/fractals/ Hey Hackers! I’m Emil Bogomolov and I’m a Research Engineer at the best Russian University, Skoltech. First of all, a huge thank you to the community and staff for nominating me for a few 2021 Noonies awards! HackerNoon I’ve been nominated in the following categories please do check out these award pages and vote: . Hackernoon Contributor of the Year - Computer Vision HackerNoon Contributor of the Year - DEVELOPER-TOOLS As someone in the AI industry, I believe that the most exciting technology of the present is deep learning because it helps to solve problems in various domains, for example even protein folding. Learn more about my thoughts and opinions on AI and my journey in the tech industry via the interview below. 1. What do you do and why do you do it? (tell us your story) I do research and development in the field of Computer Vision (CV). I work a lot with 2D data (images, videos) and 3D data (point clouds, meshes, voxels). Understanding visual data is my main interest, and by understanding, I mean to teach a machine to see things. 2. Tell us more about the things you create / write / manage / build! Apart from my main work where I do research (for example and ) I write articles on the internet regarding technologies applied in CV and teach students machine learning and data analysis. this this The research mostly consists of experimenting with different approaches to apply the deep neural networks (written on PyTorch/TensorFlow) to solve some tasks (for example making a segmentation of a human body on the video or classifying to which type of furniture correspond to this particular 3D scan). 3. How did you end up on your current career path? Do you like it? I graduated with MS in Applied Math and I had had always wanted to use the knowledge I got in the university in my everyday work. This goal is kind of achieved, I do mathematical modeling sometimes. But mostly my work is experimenting with existing tools. 4. What tech are you most excited or passionate about right now and why? I am very passionate about AlphaFold and other protein folding algorithms, I believe that it might give a huge boost to treat an extensive number of diseases. Also, I’m very passionate about Open Insulin this project is not about programming but might also save a lot of lives around the globe. 5. What tech are you most worried about right now and why? I am really worried about neural chips on our laptops and smartphones. For now, I don’t see any other purpose of using them but manufacturer to analyze the data we use/store on our devices. 6. If we gave you 10 million dollars to invest in something today, what would you invest in and why? I’d invest in telemedicine startups that make doctor appointments cheaper for end-user. I’d invest in renewable energy companies, in particular in the research of better solar energy storage. And I’d donate some of this money to Russian education organizations for children to see a world full of opportunities. 7. What are you currently learning? I am learning a lot simultaneously: a little bit of C++ to write efficient code, a little bit of Hydra to manage my experiments, I also study design patterns from the Head First book and Kubernetes book… 8. What’s the best advice you’ve ever given someone? I don’t know how to answer this. It depends on the result that person have regarding my advice :) 9. What’s the best advice you’ve ever received? Work with those who have something new to teach me. About HackerNoon’s 2021 Noonie Awards The annual Noonie Awards celebrate the best and brightest of the tech industry, bringing together all who are making the Internet and the world of tech what it is today. Please be sure to , nominate, and vote for the people and companies who you think are making the biggest impact on the tech industry today. check out our award categories The 2021 Noonies are sponsored by: , , and . Thank you so much to these sponsors who are helping us celebrate the accomplishments of all our nominees. bybit Dottech Domains Avast

Deep Learning, Protein Folding Algorithms, Computer Vision, Math, and AI Research #Noonies2021

Today, we are going to discuss a proposed by researchers from four institutions one of which is ByteDance AI Lab (known for their TikTok App). They give us a new method termed (not confuse with Sparse R-CNN that works with sparse convolutions on 3D computer vision tasks such ) that achieve near state-of-the-art performance in object detection and uses completely sparse and learnable bounding boxes generation. method Sparse R-CNN that Let’s start with a short overview of existing detecting methods. Dense method One of the widely used pipelines is the , which directly predicts the label and location of anchor boxes densely covering spatial positions, scales, and aspect ratios in a single-shot way. For example or . one-stage detector SSD YOLO Let’s consider the YOLO algorithm. Ultimately, it aims to predict a class of an object on the image and the bounding box specifying object location. Each bounding box can be described using four descriptors: center of a bounding box ( ) bx, by width ( ) bw height ( ) bh value is corresponding to a class of an object (such as: car, traffic lights, etc.). c In addition, we have to predict the value, which is the probability that there is an object in the bounding box. pc It is a because it is not searching for interesting regions in the given image that could potentially contain an object. Instead, YOLO is splitting the image into cells, using 19×19 grid. But in general one-stage detector could produce × cells, one per pixel. Each cell is responsible for predicting bounding boxes (for this example is chosen as 5). Therefore, we arrive at a large number of × × bounding boxes for one image. dense method W H k k W H k For each cell in the grid and each bbox mentioned above values are produced ( source ) Dense-To-Sparse Method There are two-stage detectors that piggy-backs on dense proposals generated using RPN like the paper proposed. These detectors have dominated modern object detection for years. Faster R-CNN Using RPN it “ from dense region candidates, and then refines the location of each proposal” and predicts its specific category. obtains a sparse set of foreground proposal boxes Two-stage detector architecture ( source ) Proposals are obtained in a similar way as in one-stage detectors, but instead of predicting the class of object directly, it predicts objectness probability. After that, the second stage predicts classes filtered by objectness and overlap score bounding boxes. Sparse Method Authors of this paper categorize a paradigm of their new Sparse R-CNN as the extension of the existing object detector paradigm which includes from thoroughly dense to dense-to-sparse with a new step which leads to thoroughly sparse. Model architectures comparison ( source ) In the reviewed paper using RPN is avoided and replaced with a small set of proposal boxes (e.g. 100 per image). These boxes are obtained using the learnable part and part of the network. The formal predicts values per proposal and the latter predicts latent representation vector of length 256 of each bbox contents. Learned proposal boxes are used as a reasonable statistic to perform refining steps afterward and learned proposal features used to introduce attention mechanism. This mechanism is very similar to one that is used in the paper. proposal boxes proposal features 4 (x,y,h,w) DETR These manipulations are performed inside the dynamic instance interactive head that we will cover in the following section. Proposed Model Features As the name of the paper implies, this model is end-to-end. The architecture is elegant. It consists of FPN based backbone that acquires features from images, mentioned above learnable and and a dynamic instance interactive head which is the main contribution to neural nets architecture of this very paper. proposal boxes proposal features, Dynamic instance interactive head Given proposal boxes, Sparse R-CNN first utilizes the RoIAlign operation to extract features from the backbone for each region defined with proposal bounding boxes. Each RoI feature “ ”. N is fed into its own exclusive head for object location and classification, where each head is conditioned on specific learnable proposal feature Dynamic module ( source ) Proposal features are used as weights for convolutions, on the image above they are mentioned as “Params.” The RoI feature is processed by this generated convolution to obtain the final feature. In this way, those “ ”. The self-attention module is embedded into the dynamic head to reason about the relations between objects and affects predictions by this convolution. bins with most foreground information make effect on final object location and classification Main Result The authors provide several comparison tables that show the performance of a new method. Sparse R-CNN is compared to RetinaNet, Faster R-CNN, and DETR in two variations with ResNet50 and ResNet100. Model performance ( source ) Here we can see that Sparse R-CNN outperforms RetinaNet and Faster R-CNN in both R50 and R100, but it performs quite similarly to DETR based architectures. According to the authors, the DETR model is in practice a dense-to-sparse model because it utilizes a sparse set of object “ ”. Hence the novelty of the article is arise comparing to DETR. queries, to interact with global(dense) image feature Qualitative analysis ( ) source On that image, you can see the qualitative result of model inference on the COCO Dataset. In the first column learned proposal boxes are shown, they are predicted for any new image. In the next columns, you can see final bboxes that were refined from proposals. They differ depending on the stage in the iterative learning process. Show me the code! In conclusion, I would like to say that now in 2020 we saw a lot of papers that apply transformers to images. Transformers have proved their worth in fields of NLP and now they gradually enter the scene of image processing. This paper shows us that using transformers it is possible to create fast one-stage detectors that perform comparable in terms of quality to the best, for now, two-stage ones. All details on implementation, you can find in the author’s code that based on FAIR’s DETR and detectron2 codebases: https://github.com/PeizeSun/SparseR-CNN References [1] Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks https://arxiv.org/abs/1506.01497 [2] YOLO Algorithm and YOLO Object Detection: An Introduction https://appsilon.com/object-detection-yolo-algorithm/ [3] Sparse R-CNN: End-to-End Object Detection with Learnable Proposals https://arxiv.org/abs/2011.12450 Also published here

Using Sparse R-CNN As A Detection Model

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ZettaBlock, an industry-leading, institutional-grade Web3 data infrastructure platform, is proud to announce a new collaboration with the Stellar Development Foundation, the organization that supports the development and growth of the Stellar network, known for facilitating open and accessible financial transactions. This integration is set to redefine the landscape of blockchain development, offering a streamlined path for developers to create not just any Web3 application, but powerful data-driven applications on Stellar, with ease and efficiency. ZettaBlock and the Stellar Development Foundation are joining forces with a shared goal: to make the blockchain space more accessible and powerful for developers and to open up new possibilities for how financial systems can operate globally. This is about providing the tools and support needed to create applications that are not just scalable and secure, but also truly efficient. “SDF is excited to work with ZettaBlock in giving our community more intelligence tools to build best-in-class dapps,” said Justin Rice, VP of Ecosystem of the Stellar Development Foundation. “We’re committed to making Stellar the highest quality network for developers to build on, and this integration with ZettaBlock furthers this initiative.” By leveraging ZettaBlock’s advanced indexing, real-time data, and analytics capabilities alongside Stellar’s robust network for financial transactions, this integration promises to revolutionize the way developers build and deploy dapps. Developers now have access to a fully managed backend, with rich, actionable on-chain data –  both historical and real-time – but also the ability to seamlessly integrate with private data sources. ZettaBlock empowers developers to swiftly create custom APIs tailored to their specific business needs or dapp logic. With the ability to generate GraphQL endpoints from SQL queries in one click, without the burden of infrastructure overhead, developers can now bring their innovative dapps to the Stellar ecosystem more than 90% faster. "We're thrilled to join forces with Stellar," said Chi Zhang, CEO and Co-Founder of ZettaBlock. "This collaboration mirrors our mutual dedication to unlocking the full potential of blockchain data for everyone. We're excited to see the innovative applications that developers on Stellar will create using our comprehensive data platform." Get Started with ZettaBlock and Stellar: Build powerful, data-driven applications on Stellar using ZettaBlock's advanced tools, or discover how our integration can revolutionize your business. Visit app.zettablock.com to get started, or zettablock.com to learn more. About ZettaBlock ZettaBlock is an institutional-grade, full-stack Web3 data platform for indexing, querying, and analyzing across 12+ blockchains and 500+ datasets. Our platform makes Web3 data accessible and accelerates the development of scalable Web3 applications with real-time data. Swiftly generate custom GraphQL or REST APIs from SQL queries and incorporate your private off-chain data, empowering your applications with unparalleled scalability and flexibility. Follow ZettaBlock on Twitter and LinkedIn. About Stellar Development Foundation The Stellar Development Foundation (SDF) is a non-profit organization that supports the development and growth of Stellar, an open-source network that connects the world’s financial infrastructure. Founded in 2014, the Foundation helps maintain the Stellar codebase, supports the technical and business communities building on the network, and serves as a voice to regulators and institutions. The Foundation seeks to create equitable access to the global financial system, using the Stellar network to unlock the world’s economic potential through blockchain technology. For more information, visit stellar.org/foundation. This story was distributed as a release by Btcwire under HackerNoon’s Business Blogging Program. Learn more about the program here. ZettaBlock, an industry-leading, institutional-grade Web3 data infrastructure platform, is proud to announce a new collaboration with the Stellar Development Foundation, the organization that supports the development and growth of the Stellar network, known for facilitating open and accessible financial transactions. This integration is set to redefine the landscape of blockchain development, offering a streamlined path for developers to create not just any Web3 application, but powerful data-driven applications on Stellar, with ease and efficiency. ZettaBlock and the Stellar Development Foundation are joining forces with a shared goal: to make the blockchain space more accessible and powerful for developers and to open up new possibilities for how financial systems can operate globally. This is about providing the tools and support needed to create applications that are not just scalable and secure, but also truly efficient. “SDF is excited to work with ZettaBlock in giving our community more intelligence tools to build best-in-class dapps,” said Justin Rice, VP of Ecosystem of the Stellar Development Foundation. “We’re committed to making Stellar the highest quality network for developers to build on, and this integration with ZettaBlock furthers this initiative.” “SDF is excited to work with ZettaBlock in giving our community more intelligence tools to build best-in-class dapps,” said Justin Rice, VP of Ecosystem of the Stellar Development Foundation . “We’re committed to making Stellar the highest quality network for developers to build on, and this integration with ZettaBlock furthers this initiative.” Justin Rice, VP of Ecosystem of the Stellar Development Foundation By leveraging ZettaBlock’s advanced indexing, real-time data, and analytics capabilities alongside Stellar’s robust network for financial transactions, this integration promises to revolutionize the way developers build and deploy dapps. Developers now have access to a fully managed backend, with rich, actionable on-chain data –  both historical and real-time – but also the ability to seamlessly integrate with private data sources. ZettaBlock empowers developers to swiftly create custom APIs tailored to their specific business needs or dapp logic. With the ability to generate GraphQL endpoints from SQL queries in one click, without the burden of infrastructure overhead, developers can now bring their innovative dapps to the Stellar ecosystem more than 90% faster. "We're thrilled to join forces with Stellar," said Chi Zhang, CEO and Co-Founder of ZettaBlock. "This collaboration mirrors our mutual dedication to unlocking the full potential of blockchain data for everyone. We're excited to see the innovative applications that developers on Stellar will create using our comprehensive data platform." "We're thrilled to join forces with Stellar," said Chi Zhang, CEO and Co-Founder of ZettaBlock. "This collaboration mirrors our mutual dedication to unlocking the full potential of blockchain data for everyone. We're excited to see the innovative applications that developers on Stellar will create using our comprehensive data platform." Get Started with ZettaBlock and Stellar: Build powerful, data-driven applications on Stellar using ZettaBlock's advanced tools, or discover how our integration can revolutionize your business. Visit app.zettablock.com to get started, or zettablock.com to learn more. app.zettablock.com app.zettablock.com zettablock.com zettablock.com About ZettaBlock ZettaBlock is an institutional-grade, full-stack Web3 data platform for indexing, querying, and analyzing across 12+ blockchains and 500+ datasets. Our platform makes Web3 data accessible and accelerates the development of scalable Web3 applications with real-time data. Swiftly generate custom GraphQL or REST APIs from SQL queries and incorporate your private off-chain data, empowering your applications with unparalleled scalability and flexibility. Follow ZettaBlock on Twitter and LinkedIn . Twitter Twitter LinkedIn LinkedIn About Stellar Development Foundation The Stellar Development Foundation (SDF) is a non-profit organization that supports the development and growth of Stellar, an open-source network that connects the world’s financial infrastructure. Founded in 2014, the Foundation helps maintain the Stellar codebase, supports the technical and business communities building on the network, and serves as a voice to regulators and institutions. The Foundation seeks to create equitable access to the global financial system, using the Stellar network to unlock the world’s economic potential through blockchain technology. For more information, visit stellar.org/foundation . stellar.org/foundation stellar.org/foundation This story was distributed as a release by Btcwire under HackerNoon’s Business Blogging Program. Learn more about the program here. This story was distributed as a release by Btcwire under HackerNoon’s Business Blogging Program. Learn more about the program here . here here

ZettaBlock Integrates With Stellar To Simplify Blockchain Development

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 3 Proof of Theorem 1 Remark. If N ≥ T then the simple Theorem 4.11 of [Tit1951] (provable by simple partial summation) yields Similarly to (4.2) of [Pin2023] we start with Summarizing the above we get 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 3 Proof of Theorem 1 Remark. If N ≥ T then the simple Theorem 4.11 of [Tit1951] (provable by simple partial summation) yields Remark. Similarly to (4.2) of [Pin2023] we start with Summarizing the above we get

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

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 2 The goal of the present work is (i) to give a possibly simple proof for all values of σ > 3/4, i.e. η < 1/4, which (ii) breaks the DH for the non-negligible range σ ≥ 7/8, i.e. η ≤ 1/8, further gives (iii) explicit density theorems for different ranges of all η < 1/4 with B(η) → 0 as η → 0 and (iv) besides the (possibly indispensable) simple idea of Hal´asz uses only classical knowledge of the theory of Riemann’s zeta-function (classical in the sense that known since at least hundred years). This means that we avoid the use of Tur´an’s method, the large sieve, Vinogradov’s method, further mean and large value theorems. However, to break the DH we need the estimate reached by the Weyl–Hardy–Littlewood method [Lit1922] (see also (5.15.1) of the book of Titchmarsh [Tit1951], 2nd edition, or the paper of Landau [Lan1924]): which is proved by Hardy and Littlewood in [Lit1922] for ℓ ≥ 1, but it is also true for ℓ = 0 (µ(0) = 1/2) by the functional equation and µ(1) = 0. Alternatively, the method of Van der Corput would yield results of similar type [Cor1921], [Cor1922]. However, the use of (2.1) seems to be simpler, although Van der Corput’s result may yield even stronger estimates for N(1 − η, T) in many ranges of η. In order to formulate our results we need some more notation. For ℓ ≥ 1 let By the estimate (2.1) of Hardy and Littlewood we have for η < 1/2 ℓ Theorem 1. For ℓ ≥ 1, η < max(1/3, 2 −ℓ) we have This implies a series of explicit density theorems for small values of η and the conditional result of Hal´asz and Tur´an [HT1969] that assuming (LH) we have N(σ, T) ≪ Tε for σ > 3/4, which is for σ > 3/4 a strong improvement of the theorem of Ingham [Ing1937] stating A(σ) = 2 for σ > 1/2 on LH. Corollary 1. The Lindel¨of Hypothesis implies Proof. Choosing ℓ = 2, LH and the definition (2.4)–(2.5) yield uℓ(η) = vℓ(η) = 0 for η < 1/4. Corollary 2. For ℓ ≥ 2, η < 1/2ℓ we have Proof. 4/(ℓ + 2) ≥ 3/(ℓ + 1) for ℓ ≥ 2 with equality for ℓ = 2. Denoting by log2 x the logarithm of base 2 and choosing C large (C = C(η) → ∞ as η → 0, but with C(η) = o(log 1/η)), we obtain for η ∈ [1/2 ℓ(1+o(1)) , 1/2 ℓ+C] from (2.7). This is naturally weaker than a result obtainable by Vinogradov’s method (see [Pin2023] and [Hea2017]) but shows still B(η) → 0 as η → 0 beyond breaking DH in some interval for σ = 1 − η close to the point s = 1. However, Corollary 2 yields also interesting (although already known) consequences form small values of ℓ (ℓ = 2, 3, for example), i.e., for relatively large ranges of η. On the other hand, they are often stronger than earlier results proved in a more complicated way (as 11.30 of Ivic [Ivi1985], for example). This breaks the DH also for η ≤ 1/8, that is, for σ ≥ 7/8, but it is weaker than Corollary 4 for η < 1/8 (sharper for η > 1/8). Remark 1. Corollary 6 is trivial for η ≥ 2/7, i.e. σ ≤ 5/7. Remark 2. It is an interesting feature of Corollary 6 that it gives a non-trivial result even in the range η ∈ [1/4, 2/7] in contrary to most applications of Hal´asz’s method, although it is weaker there than the classical theorem of Carlson [Car1920] or the one of Ingham [Ing1940]. Remark 3. It is also interesting to note that the case ℓ = 1 would yield the inequality of Corollary 6 with any estimate µ(1/2) ≤ 3/16 and any estimate of type µ(1/2) ≤ 1/4 − c2 (c2 > 0) would break the DH for η < c3(c2) with c3(c2) > 0. Finally, similarly to Theorem 1 we can give an alternative proof of the case (12.10) of Theorem 12.1 of Montgomery [Mon1971] which states Remark 4. This represents a slight improvement of the Theorem of Tur´an (cf. (1.12)–(1.13)) since Remark 5. The proof of Theorem 2 does not need for any ℓ the estimate (2.1) of Hardy and Littlewood or the estimate of Van der Corput (apart from the “trivial” case µ(0) = 1/2 following from the functional equation). 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 2 The goal of the present work is (i) to give a possibly simple proof for all values of σ > 3/4, i.e. η < 1/4, which (ii) breaks the DH for the non-negligible range σ ≥ 7/8, i.e. η ≤ 1/8, further gives (iii) explicit density theorems for different ranges of all η < 1/4 with B(η) → 0 as η → 0 and (iv) besides the (possibly indispensable) simple idea of Hal´asz uses only classical knowledge of the theory of Riemann’s zeta-function (classical in the sense that known since at least hundred years). This means that we avoid the use of Tur´an’s method, the large sieve, Vinogradov’s method, further mean and large value theorems. However, to break the DH we need the estimate reached by the Weyl–Hardy–Littlewood method [Lit1922] (see also (5.15.1) of the book of Titchmarsh [Tit1951], 2nd edition, or the paper of Landau [Lan1924]): which is proved by Hardy and Littlewood in [Lit1922] for ℓ ≥ 1, but it is also true for ℓ = 0 (µ(0) = 1/2) by the functional equation and µ(1) = 0. Alternatively, the method of Van der Corput would yield results of similar type [Cor1921], [Cor1922]. However, the use of (2.1) seems to be simpler, although Van der Corput’s result may yield even stronger estimates for N(1 − η, T) in many ranges of η. In order to formulate our results we need some more notation. For ℓ ≥ 1 let By the estimate (2.1) of Hardy and Littlewood we have for η < 1/2 ℓ Theorem 1. For ℓ ≥ 1, η < max(1/3, 2 −ℓ) we have Theorem 1. This implies a series of explicit density theorems for small values of η and the conditional result of Hal´asz and Tur´an [HT1969] that assuming (LH) we have N(σ, T) ≪ Tε for σ > 3/4, which is for σ > 3/4 a strong improvement of the theorem of Ingham [Ing1937] stating A(σ) = 2 for σ > 1/2 on LH. Corollary 1. The Lindel¨of Hypothesis implies Proof. Choosing ℓ = 2, LH and the definition (2.4)–(2.5) yield uℓ(η) = vℓ(η) = 0 for η < 1/4. Corollary 2. For ℓ ≥ 2, η < 1/2ℓ we have Proof. 4/(ℓ + 2) ≥ 3/(ℓ + 1) for ℓ ≥ 2 with equality for ℓ = 2. Denoting by log2 x the logarithm of base 2 and choosing C large (C = C(η) → ∞ as η → 0, but with C(η) = o(log 1/η)), we obtain for η ∈ [1/2 ℓ(1+o(1)) , 1/2 ℓ+C] from (2.7). This is naturally weaker than a result obtainable by Vinogradov’s method (see [Pin2023] and [Hea2017]) but shows still B(η) → 0 as η → 0 beyond breaking DH in some interval for σ = 1 − η close to the point s = 1. However, Corollary 2 yields also interesting (although already known) consequences form small values of ℓ (ℓ = 2, 3, for example), i.e., for relatively large ranges of η. On the other hand, they are often stronger than earlier results proved in a more complicated way (as 11.30 of Ivic [Ivi1985], for example). This breaks the DH also for η ≤ 1/8, that is, for σ ≥ 7/8, but it is weaker than Corollary 4 for η < 1/8 (sharper for η > 1/8). Remark 1. Corollary 6 is trivial for η ≥ 2/7, i.e. σ ≤ 5/7. Remark 2. It is an interesting feature of Corollary 6 that it gives a non-trivial result even in the range η ∈ [1/4, 2/7] in contrary to most applications of Hal´asz’s method, although it is weaker there than the classical theorem of Carlson [Car1920] or the one of Ingham [Ing1940]. Remark 3. It is also interesting to note that the case ℓ = 1 would yield the inequality of Corollary 6 with any estimate µ(1/2) ≤ 3/16 and any estimate of type µ(1/2) ≤ 1/4 − c2 (c2 > 0) would break the DH for η < c3(c2) with c3(c2) > 0. Finally, similarly to Theorem 1 we can give an alternative proof of the case (12.10) of Theorem 12.1 of Montgomery [Mon1971] which states Remark 4. This represents a slight improvement of the Theorem of Tur´an (cf. (1.12)–(1.13)) since Remark 5. The proof of Theorem 2 does not need for any ℓ the estimate (2.1) of Hardy and Littlewood or the estimate of Van der Corput (apart from the “trivial” case µ(0) = 1/2 following from the functional equation).

A Remark on Density Theorems for Riemann’s Zeta-function: 2

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 References [BL1914] Bohr, H., Landau, E., Sur les zeros de la fonction ζ(s) de Riemann, Comptes Rendus de l’Acad. des Sciences (Paris) 158 (1914), 106–110. [Bom1971] Bombieri, Enrico, Density theorems for the zeta function, Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 352–358. Amer. Math. Soc., Providence, R.I., 1971. [Bou1] J. Bourgain, Remarks on Hal´asz–Montgomery type inequalities, in: Geometric aspects of functional analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. 77, Birkh¨auser, Basel, 1995, 25–39. [Bou2000] Bourgain, J., On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function, Internat. Math. Res. Notices (2000), no. 3, 133–146. [BGD2016] Bourgain, J., Demeter, C., Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633-–682. [Car1920] Carlson, F., Uber die Nullstellen der Dirichletschen Reihen und der ¨ Riemannschen ζ-Funktion, Arkiv f¨ur Math. Astron. Fys. 15 (20) (1920), 1–18. [Cor1921] van der Corput, J. G., Zahlentheoretische Absch¨atzungen, Math. Ann. 84 (1921), 53–79. [Cor1922] van der Corput, J. G., Versch¨arfung der Absch¨atzungen beim Teilerproblem, Math. Ann. 87 (1922), 39–65. [Cra1921] Cram´er, H., Some theorems concerning prime numbers, Ark. Mat. Astron. Fys. 15(5 (1921), 1–33. [Hal1968] Hal´asz, G., Uber die Mittelwerte multiplikativer zahlentheoretische ¨ r Funktionen, Acta Math. Hungar. 19 (1968), No. 3-4, 365–404. [HT1969] Hal´asz, G. and P. Tur´an, P., On the distribution of roots of Riemann zeta and allied functions, I, Journal of Number Theory 1 (1969), 121–137. [Hea2017] Heath-Brown, D. R., A new kth derivative estimate for a trigonometric sum via Vinogradov’s integral (in Russian). English version published in Proc. Steklov Inst. Math. 296 (2017), 88–103. [Hoh1930] Hoheisel, G., Primzahlprobleme in der Analysis, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. (1930), 580–588. [Hux1972] Huxley, M. N., On the difference between consecutive primes, Invent. Math. 15 (1972), 164–170. [Ing1937] Ingham, A. E., On the difference between consecutive primes, Quart. J. Math. Oxford Ser. 8 (1937), 255–266. [Ing1940] Ingham, A. E., On the estimation of N(σ, T), Quart. J. Math. Oxford Ser. 11 (1940), 291–292. [Ivi1985] Ivic, A., The Riemann Zeta-function, Wiley, 1985. [Jut1977] Jutila, Matti, Zero-density estimates for L-functions, Acta Arithmetica 32 (1977), no. 1, 55–62. [Lan1913] Landau, E., Gel¨oste und ungel¨oste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. In: Proc. 5th Internat. Congress of Math., Vol. 1, 93–108. Cambridge University Press, Cambridge 1913. [Lan1924] Landau, E., Uber die ¨ ζ-Funktion und die L-funktionen, Math. Z. 20 (1924), 98–104. [Lit1922] Littlewood, J. E., Researches in the theory of the Riemann ζ-function, Proc. London Math. Soc. (2) 20 (1922), Records xxii–xxviii. [Mon1969] Montgomery, H. L., Zeros of L-functions, Invent. Math. 8 (1969), 346– 354. [Mon1971] Montgomery, Hugh L., Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin–New York, 1971. ix+178 pp. [Per1908] Perron, O., Zur Theorie der Dirichletschen L-Reihen, J. Reine Angew. Math. 134 (1908), 95–143. [Pin2022] Pintz, J., On the density theorem of Hal´asz and Tur´an, Acta Math. Hungar. 166 (2022), no. 1, 48–56. [Pin2023] Pintz, J., Density theorems for Riemann’s zeta-function near the line Re s = 1, Acta Arithmetica, to appear (online first version DOI: 10.4064/aa210824-10-5). [Tit1951] Titchmarsh, E. C., The theory of the Riemann zeta-function, Oxford, Clarendon Press (2nd edition), 1951. [Tur1953] Tur´an P´al, Az anal´ızis egy ´uj m´odszer´er˝ol ´es annak egyes alkalmaz´asair´ol (in Hungarian) [On a new method in analysis and on some of its applications], Akad´emiai Kiad´o, Budapest, 1953. 197 pp. [Tur1954] Tur´an P´al, On the roots of the Riemann zeta function (in Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. K¨ozl. 4 (1954), 357–368. [Tur1984] P. Tur´an, On a new method of analysis and its applications, John Wiley & Sons, Inc., New York, 1984. xvi+584 pp. 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 References [BL1914] Bohr, H., Landau, E., Sur les zeros de la fonction ζ(s) de Riemann, Comptes Rendus de l’Acad. des Sciences (Paris) 158 (1914), 106–110. [Bom1971] Bombieri, Enrico, Density theorems for the zeta function, Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 352–358. Amer. Math. Soc., Providence, R.I., 1971. [Bou1] J. Bourgain, Remarks on Hal´asz–Montgomery type inequalities, in: Geometric aspects of functional analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. 77, Birkh¨auser, Basel, 1995, 25–39. [Bou2000] Bourgain, J., On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function, Internat. Math. Res. Notices (2000), no. 3, 133–146. [BGD2016] Bourgain, J., Demeter, C., Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633-–682. [Car1920] Carlson, F., Uber die Nullstellen der Dirichletschen Reihen und der ¨ Riemannschen ζ-Funktion, Arkiv f¨ur Math. Astron. Fys. 15 (20) (1920), 1–18. [Cor1921] van der Corput, J. G., Zahlentheoretische Absch¨atzungen, Math. Ann. 84 (1921), 53–79. [Cor1922] van der Corput, J. G., Versch¨arfung der Absch¨atzungen beim Teilerproblem, Math. Ann. 87 (1922), 39–65. [Cra1921] Cram´er, H., Some theorems concerning prime numbers, Ark. Mat. Astron. Fys. 15(5 (1921), 1–33. [Hal1968] Hal´asz, G., Uber die Mittelwerte multiplikativer zahlentheoretische ¨ r Funktionen, Acta Math. Hungar. 19 (1968), No. 3-4, 365–404. [HT1969] Hal´asz, G. and P. Tur´an, P., On the distribution of roots of Riemann zeta and allied functions, I, Journal of Number Theory 1 (1969), 121–137. [Hea2017] Heath-Brown, D. R., A new kth derivative estimate for a trigonometric sum via Vinogradov’s integral (in Russian). English version published in Proc. Steklov Inst. Math. 296 (2017), 88–103. [Hoh1930] Hoheisel, G., Primzahlprobleme in der Analysis, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. (1930), 580–588. [Hux1972] Huxley, M. N., On the difference between consecutive primes, Invent. Math. 15 (1972), 164–170. [Ing1937] Ingham, A. E., On the difference between consecutive primes, Quart. J. Math. Oxford Ser. 8 (1937), 255–266. [Ing1940] Ingham, A. E., On the estimation of N(σ, T), Quart. J. Math. Oxford Ser. 11 (1940), 291–292. [Ivi1985] Ivic, A., The Riemann Zeta-function, Wiley, 1985. [Jut1977] Jutila, Matti, Zero-density estimates for L-functions, Acta Arithmetica 32 (1977), no. 1, 55–62. [Lan1913] Landau, E., Gel¨oste und ungel¨oste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. In: Proc. 5th Internat. Congress of Math., Vol. 1, 93–108. Cambridge University Press, Cambridge 1913. [Lan1924] Landau, E., Uber die ¨ ζ-Funktion und die L-funktionen, Math. Z. 20 (1924), 98–104. [Lit1922] Littlewood, J. E., Researches in the theory of the Riemann ζ-function, Proc. London Math. Soc. (2) 20 (1922), Records xxii–xxviii. [Mon1969] Montgomery, H. L., Zeros of L-functions, Invent. Math. 8 (1969), 346– 354. [Mon1971] Montgomery, Hugh L., Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin–New York, 1971. ix+178 pp. [Per1908] Perron, O., Zur Theorie der Dirichletschen L-Reihen, J. Reine Angew. Math. 134 (1908), 95–143. [Pin2022] Pintz, J., On the density theorem of Hal´asz and Tur´an, Acta Math. Hungar. 166 (2022), no. 1, 48–56. [Pin2023] Pintz, J., Density theorems for Riemann’s zeta-function near the line Re s = 1, Acta Arithmetica, to appear (online first version DOI: 10.4064/aa210824-10-5). [Tit1951] Titchmarsh, E. C., The theory of the Riemann zeta-function, Oxford, Clarendon Press (2nd edition), 1951. [Tur1953] Tur´an P´al, Az anal´ızis egy ´uj m´odszer´er˝ol ´es annak egyes alkalmaz´asair´ol (in Hungarian) [On a new method in analysis and on some of its applications], Akad´emiai Kiad´o, Budapest, 1953. 197 pp. [Tur1954] Tur´an P´al, On the roots of the Riemann zeta function (in Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. K¨ozl. 4 (1954), 357–368. [Tur1984] P. Tur´an, On a new method of analysis and its applications, John Wiley & Sons, Inc., New York, 1984. xvi+584 pp.

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