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Results tagged with elementary-number-theory
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user 161898
For questions on introductory topics in number theory, such as divisibility, prime numbers, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields, Pell's equations, and related topics.
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Approximation for the sum of primes
I have attempted to put together an approximation for the sum of primes.
I've used the much simplified $$\operatorname{li}(x)=\frac{x}{\log(x)-1}$$ combined with $$\frac{x}{2}$$ to give:
$$\frac{x^2 …
- 1,379
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votes
1
answer
64
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Trouble understanding part of thereom: Every prime congruent 1 mod (4) can be written as sum...
I've been working through with great difficulty Dudley Underwood's Elementary Number Theory.
I'm having some problem understanding the proof of a thereom regarding the sum of two squares.
I still d …
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votes
1
answer
292
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binomial expansion used in number theory
I am having trouble understanding the following:
now the binomial expansion is defined:
while it not obvious how the left hand side could equal the right hand side, I can see the inequality easily. …
- 1,379
2
votes
1
answer
258
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Step by step derivation of Robin's inequality $\sigma(n) < e^\gamma n \log \log n$
Guy Robin proved that
$$\begin{equation}
\sigma(n) < e^\gamma n \log \log n
\end{equation}$$
is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984).
The paper where he …
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1
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142
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Equation for product of sequence involving 2
Supposing for all even numbers less that or equal to n say 16,
for example the set:
2, 4, 6, 8, 10, 12, 14, 16
2.4.6.8.10.12.14.16 = 10,321,920 = $2^{15}. 3^{2}. 5^{1}. 7^1$
I would like to find …
- 1,379
1
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0
answers
48
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How to approximate the Chebychev $\psi$ function
Using Riemann-Stieltjes integration the following expression is true
$$\theta(n) = \log (t) \pi(t) \Big|_2^n - \int_2^n \frac{\pi(t)}{t}dt. \tag{1}$$
using $Li(x)$ leads to the approximation $\thet …
- 1,379
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0
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53
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Can the prime number theorem be given in terms of d(x) or σ(x)?
Given according to Apostol's Introduction to Analytic Number Theorem p79, these relations satisfy equivalence to the Prime Number Theorem:
$\lim_{x\to \infty}\frac{\pi(x)\ln(x)}{x}=1$
$\lim_{x\to \i …
- 1,379
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Accepted
Step by step derivation of Robin's inequality $\sigma(n) < e^\gamma n \log \log n$
The steps are too numerous to reproduce here but this book$^1$ Chapter 7
covers the theorem plus its consequence. I have not had time to absorb it all but it does go through the background and relati …
- 1,379
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2
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129
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For the divisor function is $d(n^2)$ related to $d(n)$ knowing also n?
The divisor function d(n) is defined as 'the number of positive divisors of n (including 1 and n)' according to Underwood Dudley.
Is the divisor function $d(n^2)$ related to $d(n)$?
for example
d( …
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0
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1
answer
59
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Product of powers of two less than or equal to 2n
Question 1
Is there an equation to calculate or approximate the Product of powers of two less than or equal to in general 2n?
Example 1
Given for example 2n = 14 this would simply be $2.4.8=2.2^2.2^3 …
- 1,379
2
votes
1
answer
222
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Bertrand's Theorem: prime-power decomposition proof
In a book on elementary number theory (pp 177-179, Underwood Dudley 2ed) I am having trouble understanding the following paragraph based on Erdos's proof using Binomial theorem:
"...each prime power …
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0
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1
answer
64
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Understanding part of derivation of Chebychev's Theorem
I cannot understand this result from pages 17–18 of Tenenbaum and Mendes's The Prime Numbers and Their Distribution on how the summation of $\frac{x\log(2)}{2^j}+O(\log(x))$ results in $2x \log(2) + O …
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