Let's assume the weight is distributed equally throughout the car, AI seems to think you have to be able to deadlift 1000KG, which just cannot be true as in this scenario, part of the weight (it the front of the car) is still on the ground

So I'm 16 (11th to 12th grade) and I am shit terrified for my entrance exams. I have an year to prepare and I have no clue where to start.

So to all the people studying/about to study maths in uni, how did you guys prepare?

Also, realistically, do you think I can make it? I got 96 in 10th maths (finals) but then I dropped to 50% (theory only) in my 11th midterms.

If I spend £454 pounds on some items and then sell them for £80 each there’s 14 in total how much would I profit in total?

Hi, I cannot decided between this books

Hoel "Probability Theory " Achim Klenke "Probability Theory" Lead Better "A first course of probability"

If you have other suggest, please let me know in the comments, thanks again

Does this make any sense? If it is an extraneous solution what would the solution be then?

https://preview.redd.it/qzn60561tdge1.jpg?width=298&format=pjpg&auto=webp&s=e881e88bd9e3a57f0dcdfd5d53582395b098368f

Hi, I recall having a very stupid issue with continuity. Essentially, the title. Is that due to the projectively extended real line? It looks like not.

I read answers stating "it is continuous in its domain"

Ok, so, I have a couple of questions about this.

About first and second species discontinuities: does that mean that if a function is not defined in the discontinuity point, then the function is continuous in its domain?

Say, f(x) defined as follows:

-1 for x<0 1 for x>0

This function, too, is continuous in its domain if I got it right.

About third specie: does it even exist at all then?

Like, f(x) = x*(x+1)/(x+1) for x≠-1 is continuous in its domain, too.

Correct?

I noticed something strange about this code which I sum up here.
First take digitsConstant, a small random semiprime… then use the following pseudocode :

2. Compute : bb=([[digitsConstant^0.5 ]]+1)^2 −digitsConstant
3. Find integers x and y such as (25^2 + x×digitsConstant)÷(y×67) = digitsConstant+bb
4. take z, an unknown variable, then expand ((67z + 25)^2+ x×digitsConstant)÷(y×67) and then take the last Integer part without a z called w. w will always be a perfect square.
5. w=sqrt(w)
6. Find a and b such as a == w (25 + w×b)
7. Solve 0=a^2 ×x^2 +(2a×b-x×digitsConstant)×z+(b^2 -67×y)
8. For each of the 2 possible integer solution, compute z mod digitsConstant.

The fact the result will be a modular square root is expected, but then why if the y computed at step 2 is a perfect square, z mod digitsConstant will always be the same as the integer square root of y and not the other possible modular square ? (that is, the trivial solution).

i’m a final year btech student with a 3.04/4 gpa. I want to pursue maths later down the future but before that i want to beocme a quant trader to earn money. i’m decent at maths but i havent extensively studied it, but recently my interest sparked after watching a grigori perelman documentary. where do i start?

Title itself.

Interesting things in point set topology, metric spaces or anything else in other math areas applying or related to these are welcome.

Hi all,

I am currently training for a numerical reasoning test and was just wondering if anyone has any textbooks they can recommend that could assist in my practice, right now using the website assessment day but if anyone knows of a book that could guide perhaps in better statistical analysis or "understanding of systems" - I'm not really sure how to practice for that part, I would greatly appreciate it. Thanks

I am a Physics undergrad who wants to be a mathematician. I am thinking of doing a Reading project in a pure math topic under a prof, for the sake of knowledge itself and also to build my profile.

But how do I produce proof of doing this project? This is not a part of an official program. I was hoping that I could use this for further projects and grad admission opportunities.

It generated this:

Reasoning:

Okay, so I need to explain how the Riemann Hypothesis relates to the distribution of prime numbers and their deviation from some average. Let me start by recalling what I know about primes and the Riemann Hypothesis.

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They are the building blocks of integers, as every integer can be factored into primes. However, their distribution among the natural numbers is quite irregular. For example, primes become less frequent as numbers get larger, but there isn't a simple formula to predict exactly where the next prime will be.

I remember that mathematicians have tried to find patterns or approximations for the distribution of primes. One famous result is the Prime Number Theorem (PNT), which states that the number of primes less than a given number x, denoted π(x), is approximately x / ln(x). This means that the density of primes around a large number x is roughly 1 / ln(x). So, as x increases, the primes thin out. But this is just an approximation, and the actual count π(x) sometimes deviates from this estimate.

Now, how does the Riemann Hypothesis (RH) come into play here? From what I recall, the RH is about the zeros of the Riemann zeta function, ζ(s). The zeta function is defined for complex numbers s with real part greater than 1 by the series ζ(s) = Σ_{n=1}^∞ 1/n^s. Through analytic continuation, it can be extended to the entire complex plane except for a pole at s=1. The Riemann Hypothesis states that all non-trivial zeros of the zeta function lie on the critical line Re(s) = 1/2.

But how does that relate to prime numbers? There's a connection through the work of Bernhard Riemann, who showed that the distribution of primes is deeply connected to the zeros of the zeta function. The explicit formula for π(x) involves a sum over the zeros of the zeta function. If the RH is true, it would give us much tighter control over the error term in the Prime Number Theorem.

The Prime Number Theorem tells us that π(x) is approximately Li(x) (the logarithmic integral function), and the error term in this approximation is related to the zeros of the zeta function. Specifically, each zero contributes an oscillatory term to the error. The further these zeros are from the critical line Re(s) = 1/2, the larger the deviations in the distribution of primes from the estimate given by Li(x). If all non-trivial zeros lie on the critical line, then the error term is minimized, meaning the primes are distributed as regularly as possible.

So, if the RH holds, the error term in the PNT would be O(x^{1/2 + ε}) for any ε > 0, which is much better than the current best known error term without assuming RH. This means that the primes are distributed in a way that their deviation from the logarithmic integral function is as small as possible, controlled by the square root of x times some logarithmic factors.

Conversely, if there were zeros off the critical line, say with real part greater than 1/2, then the error term would be larger, indicating more significant deviations in the prime distribution. This would mean primes could cluster or spread out more than expected, disrupting the smooth approximation given by Li(x).

Another angle is the explicit formula connecting primes and zeros. The formula involves a sum over primes and a sum over zeros of zeta. Each zero contributes a term that oscillates with a frequency related to its imaginary part. The real part of the zero determines the amplitude of that oscillation. If all zeros are on the line 1/2, then all these oscillations decay as x^{1/2}, leading to the tightest possible control. If there were zeros with larger real parts, those terms would decay more slowly, leading to larger fluctuations.

Also, I think there's a connection with the von Mangoldt function and the Chebyshev function ψ(x), which counts the weighted primes and prime powers. The explicit formula for ψ(x) more directly involves the zeros of zeta. The RH for ψ(x) gives a very precise error term, which in turn influences the error term for π(x).

So, in summary, the Riemann Hypothesis is deeply connected to understanding the deviations of the prime distribution from its average behavior (as approximated by Li(x)). The zeros of the zeta function act like harmonics in the music of the primes, each contributing a note that affects the overall distribution. If RH is true, those harmonics are all in tune, leading to the smoothest possible deviation. If not, there would be dissonant notes causing larger deviations.

Wait, but I need to make sure I'm not conflating different error terms. The PNT error term is related to the zeros, yes. The best unconditional error term is O(x exp(-c sqrt(log x))), which is still pretty good, but under RH it becomes O(x^{1/2} log x). So the difference is between exponential in sqrt(log x) vs polynomial in x^{1/2}. That's a significant improvement.

Also, the connection between primes and zeta zeros is through the explicit formulae, as I mentioned. Riemann's original paper introduced this, and it was later expanded by von Mangoldt and others. The explicit formula shows that each zero contributes a term to the error between π(x) and Li(x). So the more zeros you have off the line, the more these terms can add up constructively or destructively, creating larger errors.

But all known zeros are on the critical line, right? At least the first trillions are. So empirically, it's looking good, but of course, that's not a proof.

Another thought: the distribution of primes is also related to probabilistic models, like the Cramér model, which models primes as random numbers with density 1/log n. The RH would imply that the primes are distributed as regularly as possible, in some sense matching this random model closely. But primes are not truly random, they are deterministic, so the RH is a way of saying their distribution is "psuedorandom" in a precise mathematical sense.

Moreover, the connection to the explicit formula and the error term. Let me recall that the explicit formula looks something like:

ψ(x) = x - Σ_ρ (x^ρ)/ρ - log(2π) - (1/2) log(1 - x^{-2})

where the sum is over the non-trivial zeros ρ of the zeta function. So each zero ρ = β + iγ contributes a term x^ρ / ρ. The real part β of ρ determines the magnitude of x^ρ, which is x^{β}. If all β = 1/2, then each term is x^{1/2} times some oscillatory factor. Summing over all these terms gives the error term. However, the sum over zeros is conditionally convergent, so it's tricky, but under RH, we can bound the sum by x^{1/2} times some logarithmic terms.

If there was a zero with β > 1/2, then the term x^{β} would dominate x^{1/2}, leading to a larger error. Hence, RH is equivalent to the error term being as small as possible.

Therefore, the deviations of the prime counting function π(x) from its approximation Li(x) are controlled by the zeros of ζ(s). The Riemann Hypothesis asserts that these deviations are no worse than O(x^{1/2 + ε}), which is the minimal possible based on the form of the explicit formula.

To wrap up, the Riemann Hypothesis provides a precise statement about the regularity of prime distribution. Its proof would confirm that primes do not deviate too wildly from their expected density, while its disproof would indicate inherent irregularities in the distribution of primes. This is why RH is so central in number theory—it’s the keystone for understanding the fine structure of prime numbers.

Answer:

The Riemann Hypothesis (RH) is a profound conjecture about the zeros of the Riemann zeta function, ζ(s), and it has deep implications for the distribution of prime numbers. Here's a concise explanation of its connection to prime distribution:

1. Prime Number Theorem (PNT): The PNT states that the prime-counting function, π(x) (the number of primes ≤ x), approximates to the logarithmic integral, Li(x), with an error term. This error quantifies how primes deviate from their expected density (~1/ln x). The PNT error term is tied to the zeros of ζ(s).
2. Zeros of the Zeta Function: The Riemann zeta function has both trivial zeros (negative even integers) and non-trivial zeros in the critical strip (0 < Re(s) < 1). The RH posits that all non-trivial zeros lie on the critical line Re(s) = 1/2.
3. Explicit Formula and Error Control: Using analytic tools, the deviation of π(x) from Li(x) is expressed as a sum over the zeros of ζ(s). Each zero ρ = β + iγ contributes an oscillatory term proportional to x^β. If RH holds (β = 1/2 for all zeros), these terms decay like x^{1/2}, minimizing the error. This gives the tightest possible bound: π(x)=Li(x)+O(x1/2log⁡x).π(x)=Li(x)+O(x1/2logx). Without RH, larger β values would imply slower decay and larger deviations.
4. Consequences of RH:
5. Regular Prime Distribution: RH implies primes are distributed as uniformly as possible, with fluctuations no worse than ~√x.Pseudorandom Behavior: Primes would align closely with probabilistic models (e.g., Cramér’s model), behaving "randomly" but deterministically.Sharp Error Terms: Results in number theory (e.g., gaps between primes, prime-counting in intervals) gain precision under RH.
6. Implications of Violations: If RH were false, a zero off the critical line (β > 1/2) would introduce larger oscillations in π(x), revealing unexpected prime clustering or gaps, disrupting the harmony predicted by the PNT.

In summary, the Riemann Hypothesis asserts that the non-trivial zeros of ζ(s) constrain the prime distribution’s deviations to be as small as mathematically possible. Its resolution will either confirm the primes’ ultimate regularity or unveil hidden chaos in their structure.

Mathematics has always been my true calling, but life kept me from pursuing it. I’m 25, from Kerala, and I feel an immense void—almost guilt—for not dedicating myself to it.

Now, I’m determined to change that. I want to pursue an online B.Sc. in Mathematics and eventually become a researcher and teacher. I looked into IGNOU, but I heard it lacks live classes.

If you know any good universities offering structured online math degrees, please share. Your help could bring me closer to the path I was meant to take.

https://preview.redd.it/rter4wdzltfe1.png?width=681&format=png&auto=webp&s=3434abed0f36b37de9cea86704638151ab9c432c

Two numbers with difference 2 and quotient 2.

Is it 1 & 4?

2 / 1 =2

4 - 2 =2

Thanks 🥲