Viktor Futó · Stay human
@optimista.bsky.social
Spatially written math & proofs with LaTeX - brainec.com
Alumni: University of Coimbra · Charles University in Prague
Alumni: University of Coimbra · Charles University in Prague
Pinned
Hello maths/physics community at bluesky 👋🏻
My name is Viktor, and having been frustrated by the countless hours I spent studying and deciphering math from long, linear, exhausting paragraphs of text, I am now pioneering spatial writing to make studying and reading much friendlier for our brains.
My name is Viktor, and having been frustrated by the countless hours I spent studying and deciphering math from long, linear, exhausting paragraphs of text, I am now pioneering spatial writing to make studying and reading much friendlier for our brains.
* - for all degrees (HBₖ,HBₖ₊₁,HBₖ₊₂,...)
+2𝑔𝑛 - shift of homology degree coming from:
2 - full twist
𝑔 - number of negative twists in 𝑋
𝑛 - number of free strands
Read the spatial story:
brainec.com/s/oc82wP8HXxMUMTkZ3dlU
Based on van den Berg et al. (2015), Braid Floer Homology.
+2𝑔𝑛 - shift of homology degree coming from:
2 - full twist
𝑔 - number of negative twists in 𝑋
𝑛 - number of free strands
Read the spatial story:
brainec.com/s/oc82wP8HXxMUMTkZ3dlU
Based on van den Berg et al. (2015), Braid Floer Homology.
Corollary 12.3.
Based on: J.B. van den Berg, R. Ghrist, R.C. Vandervorst, and W. Wójcik, Braid Floer homology, J. Differential Equations 259 (2015) 1663–1721
brainec.com
December 8, 2025 at 1:26 PM
* - for all degrees (HBₖ,HBₖ₊₁,HBₖ₊₂,...)
+2𝑔𝑛 - shift of homology degree coming from:
2 - full twist
𝑔 - number of negative twists in 𝑋
𝑛 - number of free strands
Read the spatial story:
brainec.com/s/oc82wP8HXxMUMTkZ3dlU
Based on van den Berg et al. (2015), Braid Floer Homology.
+2𝑔𝑛 - shift of homology degree coming from:
2 - full twist
𝑔 - number of negative twists in 𝑋
𝑛 - number of free strands
Read the spatial story:
brainec.com/s/oc82wP8HXxMUMTkZ3dlU
Based on van den Berg et al. (2015), Braid Floer Homology.
𝑏𝑟𝑎𝑖𝑑 - set of strands
𝑠𝑡𝑟𝑎𝑛𝑑 - 1-dimensional curve (in 𝑆¹×𝐷²)
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 - only positive crossings (strand 𝑖 passes over 𝑖+1)
[𝑋 rel 𝑌] - relative braid class (𝑋 - free strands that twist around 𝑌 - fixed strands)
[𝑋₊ rel 𝑌₊] - positive rel. braid class (𝑋₊ -||- that twist *positively* around 𝑌₊)
𝑠𝑡𝑟𝑎𝑛𝑑 - 1-dimensional curve (in 𝑆¹×𝐷²)
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 - only positive crossings (strand 𝑖 passes over 𝑖+1)
[𝑋 rel 𝑌] - relative braid class (𝑋 - free strands that twist around 𝑌 - fixed strands)
[𝑋₊ rel 𝑌₊] - positive rel. braid class (𝑋₊ -||- that twist *positively* around 𝑌₊)
December 8, 2025 at 1:26 PM
𝑏𝑟𝑎𝑖𝑑 - set of strands
𝑠𝑡𝑟𝑎𝑛𝑑 - 1-dimensional curve (in 𝑆¹×𝐷²)
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 - only positive crossings (strand 𝑖 passes over 𝑖+1)
[𝑋 rel 𝑌] - relative braid class (𝑋 - free strands that twist around 𝑌 - fixed strands)
[𝑋₊ rel 𝑌₊] - positive rel. braid class (𝑋₊ -||- that twist *positively* around 𝑌₊)
𝑠𝑡𝑟𝑎𝑛𝑑 - 1-dimensional curve (in 𝑆¹×𝐷²)
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 - only positive crossings (strand 𝑖 passes over 𝑖+1)
[𝑋 rel 𝑌] - relative braid class (𝑋 - free strands that twist around 𝑌 - fixed strands)
[𝑋₊ rel 𝑌₊] - positive rel. braid class (𝑋₊ -||- that twist *positively* around 𝑌₊)
𝐂𝐨𝐫𝐨𝐥𝐥𝐚𝐫𝐲 𝟏𝟐.𝟑 [𝐀𝐥𝐠𝐞𝐛𝐫𝐚𝐢𝐜 𝐓𝐨𝐩𝐨𝐥𝐨𝐠𝐲/𝐁𝐫𝐚𝐢𝐝-𝐅𝐥𝐨𝐞𝐫 𝐇𝐨𝐦𝐨𝐥𝐨𝐠𝐲]
"Positive braids realize, up to shifts, all possible Braid-Floer homologies"
HB*([𝑋 rel 𝑌]) ≅ HB*+2𝑔𝑛([𝑋₊ rel 𝑌₊])
= "Braid-Floer homology of relative braid class is isomorphic to shifted Braid-Floer homology of positive relative braid class"
"Positive braids realize, up to shifts, all possible Braid-Floer homologies"
HB*([𝑋 rel 𝑌]) ≅ HB*+2𝑔𝑛([𝑋₊ rel 𝑌₊])
= "Braid-Floer homology of relative braid class is isomorphic to shifted Braid-Floer homology of positive relative braid class"
December 8, 2025 at 1:26 PM
𝐂𝐨𝐫𝐨𝐥𝐥𝐚𝐫𝐲 𝟏𝟐.𝟑 [𝐀𝐥𝐠𝐞𝐛𝐫𝐚𝐢𝐜 𝐓𝐨𝐩𝐨𝐥𝐨𝐠𝐲/𝐁𝐫𝐚𝐢𝐝-𝐅𝐥𝐨𝐞𝐫 𝐇𝐨𝐦𝐨𝐥𝐨𝐠𝐲]
"Positive braids realize, up to shifts, all possible Braid-Floer homologies"
HB*([𝑋 rel 𝑌]) ≅ HB*+2𝑔𝑛([𝑋₊ rel 𝑌₊])
= "Braid-Floer homology of relative braid class is isomorphic to shifted Braid-Floer homology of positive relative braid class"
"Positive braids realize, up to shifts, all possible Braid-Floer homologies"
HB*([𝑋 rel 𝑌]) ≅ HB*+2𝑔𝑛([𝑋₊ rel 𝑌₊])
= "Braid-Floer homology of relative braid class is isomorphic to shifted Braid-Floer homology of positive relative braid class"
Reposted by Viktor Futó · Stay human
A spatial story that you write on Brainec can be exported as a single .html file and opened in any web browser.
If there are no images, only math expressions and text, it runs fully offline.
Image embedding for offline use is in development.
If there are no images, only math expressions and text, it runs fully offline.
Image embedding for offline use is in development.
November 17, 2025 at 2:55 PM
A spatial story that you write on Brainec can be exported as a single .html file and opened in any web browser.
If there are no images, only math expressions and text, it runs fully offline.
Image embedding for offline use is in development.
If there are no images, only math expressions and text, it runs fully offline.
Image embedding for offline use is in development.
One recurring theme as I speak with professors is how AI is completely eroding the studying process, verification of assignments, the understanding of concepts.
Linear text is infected with AI, but spatial text forces the author to write and to understand what they are writing.
Linear text is infected with AI, but spatial text forces the author to write and to understand what they are writing.
Adivina qué campus visité ayer!
Guess which campus did I visit yesterday!
Guess which campus did I visit yesterday!
November 6, 2025 at 1:26 PM
One recurring theme as I speak with professors is how AI is completely eroding the studying process, verification of assignments, the understanding of concepts.
Linear text is infected with AI, but spatial text forces the author to write and to understand what they are writing.
Linear text is infected with AI, but spatial text forces the author to write and to understand what they are writing.
Adivina qué campus visité ayer!
Guess which campus did I visit yesterday!
Guess which campus did I visit yesterday!
November 5, 2025 at 12:49 PM
Adivina qué campus visité ayer!
Guess which campus did I visit yesterday!
Guess which campus did I visit yesterday!
A creator just brought their Brainec story to TikTok for the first time 🥲
October 21, 2025 at 1:35 PM
A creator just brought their Brainec story to TikTok for the first time 🥲
PS: Recently been very curious about possible conversion between formalized math theorems/proofs in Lean and their spatial human-readable format optimized for reading and comprehension.
October 20, 2025 at 3:33 PM
PS: Recently been very curious about possible conversion between formalized math theorems/proofs in Lean and their spatial human-readable format optimized for reading and comprehension.
..psychology of learning, transition design. Building a scalable instructional medium and practice of writing high-level material in a way that is aligned with our human cognitive experience. Currently pivoting on mathematics
My pleasure to connect, always 🎩
MSc Thesis: hdl.handle.net/10316/96137
My pleasure to connect, always 🎩
MSc Thesis: hdl.handle.net/10316/96137
hdl.handle.net
October 20, 2025 at 3:33 PM
..psychology of learning, transition design. Building a scalable instructional medium and practice of writing high-level material in a way that is aligned with our human cognitive experience. Currently pivoting on mathematics
My pleasure to connect, always 🎩
MSc Thesis: hdl.handle.net/10316/96137
My pleasure to connect, always 🎩
MSc Thesis: hdl.handle.net/10316/96137
Hello to Australian academics who followed back 👋🏻 Recently got curious about PhD opportunities in 🇦🇺 and simply followed the list curated by Dr. Canonne.
I've been working on this interdisciplinary vision of spatial writing that intertwines interaction design, comp. sci., cognitive science... ↓
I've been working on this interdisciplinary vision of spatial writing that intertwines interaction design, comp. sci., cognitive science... ↓
October 20, 2025 at 3:33 PM
Hello to Australian academics who followed back 👋🏻 Recently got curious about PhD opportunities in 🇦🇺 and simply followed the list curated by Dr. Canonne.
I've been working on this interdisciplinary vision of spatial writing that intertwines interaction design, comp. sci., cognitive science... ↓
I've been working on this interdisciplinary vision of spatial writing that intertwines interaction design, comp. sci., cognitive science... ↓
See how Mertens’ Second Theorem enters the final stage of the proof of Lemma 4.4, a key step toward the density-one Davis–Lelièvre theorem:
𝐋𝐞𝐦𝐦𝐚 𝟒.𝟒. (𝐃𝐞𝐧𝐬𝐢𝐭𝐲-𝐨𝐧𝐞 𝐃𝐚𝐯𝐢𝐬–𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞)
brainec.com/s/PCIEHHEEPE...
Based on: Kontorovich, A., & Zhang, X. (2024). On the local-global conj... arXiv:2409.10682
𝐋𝐞𝐦𝐦𝐚 𝟒.𝟒. (𝐃𝐞𝐧𝐬𝐢𝐭𝐲-𝐨𝐧𝐞 𝐃𝐚𝐯𝐢𝐬–𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞)
brainec.com/s/PCIEHHEEPE...
Based on: Kontorovich, A., & Zhang, X. (2024). On the local-global conj... arXiv:2409.10682
Density-one Lemma 4.4. (Davis–Lelièvre)
[Kontorovich, Alex; and Zhang, Xin (2024). On the Local-Global Conjecture for Combinatorial Period Lengths of Closed Billiards on the Regular Pentagon. arXiv preprint arXiv:2409.10682. See p. 8. Avail...
www.brainec.com
October 17, 2025 at 12:36 PM
See how Mertens’ Second Theorem enters the final stage of the proof of Lemma 4.4, a key step toward the density-one Davis–Lelièvre theorem:
𝐋𝐞𝐦𝐦𝐚 𝟒.𝟒. (𝐃𝐞𝐧𝐬𝐢𝐭𝐲-𝐨𝐧𝐞 𝐃𝐚𝐯𝐢𝐬–𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞)
brainec.com/s/PCIEHHEEPE...
Based on: Kontorovich, A., & Zhang, X. (2024). On the local-global conj... arXiv:2409.10682
𝐋𝐞𝐦𝐦𝐚 𝟒.𝟒. (𝐃𝐞𝐧𝐬𝐢𝐭𝐲-𝐨𝐧𝐞 𝐃𝐚𝐯𝐢𝐬–𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞)
brainec.com/s/PCIEHHEEPE...
Based on: Kontorovich, A., & Zhang, X. (2024). On the local-global conj... arXiv:2409.10682
See how can you write it spatially:
brainec.com/s/ZmkrjF2ml2...
brainec.com/s/ZmkrjF2ml2...
Merten's Second Theorem
Let
brainec.com
October 17, 2025 at 12:36 PM
See how can you write it spatially:
brainec.com/s/ZmkrjF2ml2...
brainec.com/s/ZmkrjF2ml2...
𝐌𝐞𝐫𝐭𝐞𝐧𝐬' 𝐒𝐞𝐜𝐨𝐧𝐝 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
In 1737 Euler showed that Σ 1/𝑝 diverges.
In 1874 Mertens’ Second Theorem estimated how quickly.
The growth is very slow: log log 𝑥.
For 𝑥 ∈ ℝ, 𝑝 prime and 𝑥 ≥ 2:
Σₚ≤𝑥 1/𝑝 = log log 𝑥 + 𝑀 + O(1/log 𝑥)
Meissel-Mertens constant
𝑀 ≈ 0.26149721…
#Math #NumberTheory
In 1737 Euler showed that Σ 1/𝑝 diverges.
In 1874 Mertens’ Second Theorem estimated how quickly.
The growth is very slow: log log 𝑥.
For 𝑥 ∈ ℝ, 𝑝 prime and 𝑥 ≥ 2:
Σₚ≤𝑥 1/𝑝 = log log 𝑥 + 𝑀 + O(1/log 𝑥)
Meissel-Mertens constant
𝑀 ≈ 0.26149721…
#Math #NumberTheory
October 17, 2025 at 12:36 PM
𝐌𝐞𝐫𝐭𝐞𝐧𝐬' 𝐒𝐞𝐜𝐨𝐧𝐝 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
In 1737 Euler showed that Σ 1/𝑝 diverges.
In 1874 Mertens’ Second Theorem estimated how quickly.
The growth is very slow: log log 𝑥.
For 𝑥 ∈ ℝ, 𝑝 prime and 𝑥 ≥ 2:
Σₚ≤𝑥 1/𝑝 = log log 𝑥 + 𝑀 + O(1/log 𝑥)
Meissel-Mertens constant
𝑀 ≈ 0.26149721…
#Math #NumberTheory
In 1737 Euler showed that Σ 1/𝑝 diverges.
In 1874 Mertens’ Second Theorem estimated how quickly.
The growth is very slow: log log 𝑥.
For 𝑥 ∈ ℝ, 𝑝 prime and 𝑥 ≥ 2:
Σₚ≤𝑥 1/𝑝 = log log 𝑥 + 𝑀 + O(1/log 𝑥)
Meissel-Mertens constant
𝑀 ≈ 0.26149721…
#Math #NumberTheory
See how can you write it spatially:
www.brainec.com/s/UlzYMrr8rO...
www.brainec.com/s/UlzYMrr8rO...
Euler's Totient Function
Let
www.brainec.com
October 16, 2025 at 12:20 PM
See how can you write it spatially:
www.brainec.com/s/UlzYMrr8rO...
www.brainec.com/s/UlzYMrr8rO...
𝐄𝐮𝐥𝐞𝐫'𝐬 𝐓𝐨𝐭𝐢𝐞𝐧𝐭 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧
φ(𝑛) = how many positive integers ≤ than 𝑛 are relatively prime to 𝑛
φ(𝑛) = |{ 𝑘 ∈ ℕ : 𝑘 ≤ 𝑛, gcd(𝑘,𝑛)=1 }|
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑝𝑟𝑖𝑚𝑒 ≡ 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 ≡ gcd(𝑎,𝑏)=1
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
φ(9) = 6
{1,2,4,5,7,8}
#math #NumberTheory
φ(𝑛) = how many positive integers ≤ than 𝑛 are relatively prime to 𝑛
φ(𝑛) = |{ 𝑘 ∈ ℕ : 𝑘 ≤ 𝑛, gcd(𝑘,𝑛)=1 }|
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑝𝑟𝑖𝑚𝑒 ≡ 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 ≡ gcd(𝑎,𝑏)=1
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
φ(9) = 6
{1,2,4,5,7,8}
#math #NumberTheory
October 16, 2025 at 12:20 PM
𝐄𝐮𝐥𝐞𝐫'𝐬 𝐓𝐨𝐭𝐢𝐞𝐧𝐭 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧
φ(𝑛) = how many positive integers ≤ than 𝑛 are relatively prime to 𝑛
φ(𝑛) = |{ 𝑘 ∈ ℕ : 𝑘 ≤ 𝑛, gcd(𝑘,𝑛)=1 }|
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑝𝑟𝑖𝑚𝑒 ≡ 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 ≡ gcd(𝑎,𝑏)=1
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
φ(9) = 6
{1,2,4,5,7,8}
#math #NumberTheory
φ(𝑛) = how many positive integers ≤ than 𝑛 are relatively prime to 𝑛
φ(𝑛) = |{ 𝑘 ∈ ℕ : 𝑘 ≤ 𝑛, gcd(𝑘,𝑛)=1 }|
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑝𝑟𝑖𝑚𝑒 ≡ 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 ≡ gcd(𝑎,𝑏)=1
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
φ(9) = 6
{1,2,4,5,7,8}
#math #NumberTheory
See how can you write it spatially:
www.brainec.com/s/2SIKNGSmeF...
www.brainec.com/s/2SIKNGSmeF...
Merten's Third Theorem
Let
www.brainec.com
October 14, 2025 at 12:23 PM
See how can you write it spatially:
www.brainec.com/s/2SIKNGSmeF...
www.brainec.com/s/2SIKNGSmeF...
𝐌𝐞𝐫𝐭𝐞𝐧𝐬' 𝐓𝐡𝐢𝐫𝐝 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
Precursor of the Prime Number Theorem
For 𝑥 ∈ ℝ and 𝑥 ≥ 2:
∏ₚ≤𝑥 (1 − 1/𝑝) = e⁻ᵞ(1+o(1))/ln𝑥
Euler–Mascheroni constant
γ ≈ 0.57721566…
#math #NumberTheory
Precursor of the Prime Number Theorem
For 𝑥 ∈ ℝ and 𝑥 ≥ 2:
∏ₚ≤𝑥 (1 − 1/𝑝) = e⁻ᵞ(1+o(1))/ln𝑥
Euler–Mascheroni constant
γ ≈ 0.57721566…
#math #NumberTheory
October 14, 2025 at 12:23 PM
𝐌𝐞𝐫𝐭𝐞𝐧𝐬' 𝐓𝐡𝐢𝐫𝐝 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
Precursor of the Prime Number Theorem
For 𝑥 ∈ ℝ and 𝑥 ≥ 2:
∏ₚ≤𝑥 (1 − 1/𝑝) = e⁻ᵞ(1+o(1))/ln𝑥
Euler–Mascheroni constant
γ ≈ 0.57721566…
#math #NumberTheory
Precursor of the Prime Number Theorem
For 𝑥 ∈ ℝ and 𝑥 ≥ 2:
∏ₚ≤𝑥 (1 − 1/𝑝) = e⁻ᵞ(1+o(1))/ln𝑥
Euler–Mascheroni constant
γ ≈ 0.57721566…
#math #NumberTheory
See how can you write it spatially: www.brainec.com/s/9jQ7n4IUXU...
Mercator Series
Let
www.brainec.com
October 13, 2025 at 1:09 PM
See how can you write it spatially: www.brainec.com/s/9jQ7n4IUXU...
𝐌𝐞𝐫𝐜𝐚𝐭𝐨𝐫 𝐒𝐞𝐫𝐢𝐞𝐬
Also called = Taylor/Maclaurin expansion of ln(1+𝑥)
If |𝑥|<1 then:
ln(1+𝑥) = Σₙ₌₁ (−1)ⁿ⁺¹ 𝑥ⁿ/𝑛
ln(1+𝑥) = 𝑥 − 𝑥²⁄2 + 𝑥³⁄3 − 𝑥⁴⁄4 + …
#mathematics #RealAnalysis
Also called = Taylor/Maclaurin expansion of ln(1+𝑥)
If |𝑥|<1 then:
ln(1+𝑥) = Σₙ₌₁ (−1)ⁿ⁺¹ 𝑥ⁿ/𝑛
ln(1+𝑥) = 𝑥 − 𝑥²⁄2 + 𝑥³⁄3 − 𝑥⁴⁄4 + …
#mathematics #RealAnalysis
October 13, 2025 at 1:09 PM
𝐌𝐞𝐫𝐜𝐚𝐭𝐨𝐫 𝐒𝐞𝐫𝐢𝐞𝐬
Also called = Taylor/Maclaurin expansion of ln(1+𝑥)
If |𝑥|<1 then:
ln(1+𝑥) = Σₙ₌₁ (−1)ⁿ⁺¹ 𝑥ⁿ/𝑛
ln(1+𝑥) = 𝑥 − 𝑥²⁄2 + 𝑥³⁄3 − 𝑥⁴⁄4 + …
#mathematics #RealAnalysis
Also called = Taylor/Maclaurin expansion of ln(1+𝑥)
If |𝑥|<1 then:
ln(1+𝑥) = Σₙ₌₁ (−1)ⁿ⁺¹ 𝑥ⁿ/𝑛
ln(1+𝑥) = 𝑥 − 𝑥²⁄2 + 𝑥³⁄3 − 𝑥⁴⁄4 + …
#mathematics #RealAnalysis
Read the spatial story: www.brainec.com/s/h16CK8pj8u...
Euler Product for Dirichlet Series
Let
www.brainec.com
October 10, 2025 at 1:04 PM
Read the spatial story: www.brainec.com/s/h16CK8pj8u...
𝐄𝐮𝐥𝐞𝐫 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐟𝐨𝐫 𝐃𝐢𝐫𝐢𝐜𝐡𝐥𝐞𝐭 𝐒𝐞𝐫𝐢𝐞𝐬
Core identity in analytic number theory linking Dirichlet series to primes
If 𝑓 is multiplicative & series converges absolutely at 𝑠:
∑ₙ 𝑓(𝑛)/𝑛ˢ = ∏ₚ ( ∑ₖ 𝑓(𝑝ᵏ)/𝑝ᵏˢ )
“sum over all integers” = “product over primes” (one sum per prime)
#mathematics #NumberTheory
Core identity in analytic number theory linking Dirichlet series to primes
If 𝑓 is multiplicative & series converges absolutely at 𝑠:
∑ₙ 𝑓(𝑛)/𝑛ˢ = ∏ₚ ( ∑ₖ 𝑓(𝑝ᵏ)/𝑝ᵏˢ )
“sum over all integers” = “product over primes” (one sum per prime)
#mathematics #NumberTheory
October 10, 2025 at 1:04 PM
𝐄𝐮𝐥𝐞𝐫 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐟𝐨𝐫 𝐃𝐢𝐫𝐢𝐜𝐡𝐥𝐞𝐭 𝐒𝐞𝐫𝐢𝐞𝐬
Core identity in analytic number theory linking Dirichlet series to primes
If 𝑓 is multiplicative & series converges absolutely at 𝑠:
∑ₙ 𝑓(𝑛)/𝑛ˢ = ∏ₚ ( ∑ₖ 𝑓(𝑝ᵏ)/𝑝ᵏˢ )
“sum over all integers” = “product over primes” (one sum per prime)
#mathematics #NumberTheory
Core identity in analytic number theory linking Dirichlet series to primes
If 𝑓 is multiplicative & series converges absolutely at 𝑠:
∑ₙ 𝑓(𝑛)/𝑛ˢ = ∏ₚ ( ∑ₖ 𝑓(𝑝ᵏ)/𝑝ᵏˢ )
“sum over all integers” = “product over primes” (one sum per prime)
#mathematics #NumberTheory
Used for:
- estimating sums of arithmetic functions
- discrete “integration by parts” for sums
- transforming hard sums into easier integrals
Read the spatial story:
www.brainec.com/s/SXxDYUDZPw...
- estimating sums of arithmetic functions
- discrete “integration by parts” for sums
- transforming hard sums into easier integrals
Read the spatial story:
www.brainec.com/s/SXxDYUDZPw...
Abel's Summation Formula
Let
www.brainec.com
October 9, 2025 at 12:46 PM
Used for:
- estimating sums of arithmetic functions
- discrete “integration by parts” for sums
- transforming hard sums into easier integrals
Read the spatial story:
www.brainec.com/s/SXxDYUDZPw...
- estimating sums of arithmetic functions
- discrete “integration by parts” for sums
- transforming hard sums into easier integrals
Read the spatial story:
www.brainec.com/s/SXxDYUDZPw...
𝐀𝐛𝐞𝐥’𝐬 𝐒𝐮𝐦𝐦𝐚𝐭𝐢𝐨𝐧 𝐅𝐨𝐫𝐦𝐮𝐥𝐚 | #NumberTheory
shows how a weighted sum (by 𝑓) can be expressed using cumulative sum (𝐴(𝑥)) and the way function 𝑓 changes
If 𝑓:[1,𝑁]⟼ℂ continuously differentiable & 𝐴(𝑥) = Σₙ≤𝑥 𝑎ₙ:
Σₙ₌₁ᴺ 𝑎ₙ 𝑓(𝑛) = 𝐴(𝑁) 𝑓(𝑁) − ∫₁ᴺ 𝐴(𝑥) 𝑓′(𝑥) d𝑥
Used for: ↓
shows how a weighted sum (by 𝑓) can be expressed using cumulative sum (𝐴(𝑥)) and the way function 𝑓 changes
If 𝑓:[1,𝑁]⟼ℂ continuously differentiable & 𝐴(𝑥) = Σₙ≤𝑥 𝑎ₙ:
Σₙ₌₁ᴺ 𝑎ₙ 𝑓(𝑛) = 𝐴(𝑁) 𝑓(𝑁) − ∫₁ᴺ 𝐴(𝑥) 𝑓′(𝑥) d𝑥
Used for: ↓
October 9, 2025 at 12:46 PM
𝐀𝐛𝐞𝐥’𝐬 𝐒𝐮𝐦𝐦𝐚𝐭𝐢𝐨𝐧 𝐅𝐨𝐫𝐦𝐮𝐥𝐚 | #NumberTheory
shows how a weighted sum (by 𝑓) can be expressed using cumulative sum (𝐴(𝑥)) and the way function 𝑓 changes
If 𝑓:[1,𝑁]⟼ℂ continuously differentiable & 𝐴(𝑥) = Σₙ≤𝑥 𝑎ₙ:
Σₙ₌₁ᴺ 𝑎ₙ 𝑓(𝑛) = 𝐴(𝑁) 𝑓(𝑁) − ∫₁ᴺ 𝐴(𝑥) 𝑓′(𝑥) d𝑥
Used for: ↓
shows how a weighted sum (by 𝑓) can be expressed using cumulative sum (𝐴(𝑥)) and the way function 𝑓 changes
If 𝑓:[1,𝑁]⟼ℂ continuously differentiable & 𝐴(𝑥) = Σₙ≤𝑥 𝑎ₙ:
Σₙ₌₁ᴺ 𝑎ₙ 𝑓(𝑛) = 𝐴(𝑁) 𝑓(𝑁) − ∫₁ᴺ 𝐴(𝑥) 𝑓′(𝑥) d𝑥
Used for: ↓
See how creating a multiplicative indicator function allows us to prove Lemma 4.4, a key step toward the Density-one Davis–Lelièvre theorem:
www.brainec.com/s/PCIEHHEEPE...
www.brainec.com/s/PCIEHHEEPE...
Density-one Lemma 4.4. (Davis–Lelièvre)
[Kontorovich, Alex; and Zhang, Xin (2024). On the Local-Global Conjecture for Combinatorial Period Lengths of Closed Billiards on the Regular Pentagon. arXiv preprint arXiv:2409.10682. See p. 8. Avail...
www.brainec.com
October 7, 2025 at 12:41 PM
See how creating a multiplicative indicator function allows us to prove Lemma 4.4, a key step toward the Density-one Davis–Lelièvre theorem:
www.brainec.com/s/PCIEHHEEPE...
www.brainec.com/s/PCIEHHEEPE...
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
𝑓(𝑛) = 1 - if 𝑛 squarefree, 𝑓(𝑛) = 0 otherwise
𝑁𝑜𝑡𝑖𝑐𝑒: multiplying a non-squarefree number by any squarefree number always gives a non-squarefree number.
Read the short spatial story: www.brainec.com/s/HO31YvCkft...
𝑓(𝑛) = 1 - if 𝑛 squarefree, 𝑓(𝑛) = 0 otherwise
𝑁𝑜𝑡𝑖𝑐𝑒: multiplying a non-squarefree number by any squarefree number always gives a non-squarefree number.
Read the short spatial story: www.brainec.com/s/HO31YvCkft...
Multiplicative Indicator Function
Let
www.brainec.com
October 7, 2025 at 12:41 PM
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
𝑓(𝑛) = 1 - if 𝑛 squarefree, 𝑓(𝑛) = 0 otherwise
𝑁𝑜𝑡𝑖𝑐𝑒: multiplying a non-squarefree number by any squarefree number always gives a non-squarefree number.
Read the short spatial story: www.brainec.com/s/HO31YvCkft...
𝑓(𝑛) = 1 - if 𝑛 squarefree, 𝑓(𝑛) = 0 otherwise
𝑁𝑜𝑡𝑖𝑐𝑒: multiplying a non-squarefree number by any squarefree number always gives a non-squarefree number.
Read the short spatial story: www.brainec.com/s/HO31YvCkft...