All Questions
Tagged with elementary-proofs analytic-number-theory
18 questions
32
votes
3
answers
8k
views
Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)
I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949).
One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$,
$$(1) \qquad\qquad \vartheta(x) \...
- 587
16
votes
4
answers
2k
views
What can be said about this double sum?
Question. Can this number be expressed in terms of classical values?
$$\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^{\frac32}}=1.056348517615643291\dots$$
UPDATE. I'm encouraged by Noam, Kevin and Igor's ...
- 43.2k
13
votes
3
answers
1k
views
At what point would an elementary generalization of Bertrand's Postulate be interesting?
I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...
- 1,049
7
votes
2
answers
636
views
How to use the Prime Number Theorem in order to prove Selberg's Formula?
I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory"
and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem.
This is one of the tasks ...
- 129
6
votes
1
answer
826
views
Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$
I was recent reading through Paul Erdos's classic elementary proof of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p_i$ represents the $...
- 1,049
6
votes
2
answers
788
views
How often does the Mertens function vanish?
It is well known that the Mertens function
$$M(x)=\sum _{n\leq x}\mu(n)$$
has infinitely many zeros, and this seems to be a short proof.
Are there known results about how often the Mertens function ...
- 587
6
votes
1
answer
835
views
Beauty of some numbers discovered by Ramanujan
I am a graduate PhD student and my topic is analytic number theory. I am also a mathematics teacher. I am planning to give a course to pupils in high school that motivates them to study arithmetic and ...
- 1,257
6
votes
0
answers
380
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
5
votes
1
answer
275
views
An elementary question about a sequence of numbers
Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that
$$\sum_{n=1}^\infty a_n \lambda_n^k=0 \ \ \text{ for all }\ \ k\geq ...
3
votes
1
answer
154
views
Arithmetical function comparable to sine function [closed]
I was wondering if there exists or can we construct (using known arithmetic functions) an arithmetical function that has the same behaviour of the function sine or comparable to it (I mean that ...
- 1,257
3
votes
0
answers
1k
views
Formula for $\pi$ involving exponents of Mersenne primes
Can someone provide a proof for the following claim?
$$\pi=\dfrac{S_0S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
- 2,661
2
votes
0
answers
65
views
Can this function satisfy Song conditions?
Let $r_{s,k}(n)$ be the number of representation of a naturel number $l$ as a sum of $s$ positive $k$-th powers.
Joung Min Song introduced some conditions to study asymtotic behavior of some positive ...
- 1,257
1
vote
1
answer
519
views
A new perspective on Lehmer's totient problem
Lehmer's totient problem asks if there are any composite integers $n$ with $\phi(n) \ | \ n-1$.
It is known that any such $n$ must be odd. It must also be a charmichael number.
Assume $n=4m+3$ then $\...
- 319
1
vote
2
answers
288
views
Implicit constant in Tenenbaum's result
In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5):
Let $g$ be a positive multiplicative function and ...
- 1,257
1
vote
1
answer
199
views
Effective estimate for this infinite product over Hecke eigenvalues
Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$ and let $\lambda_f(n)$ be the $n$-th normalized Fourier coefficient of $f.$ Can someone provide me with an ...
- 1,257
1
vote
0
answers
127
views
Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
- 1,257
1
vote
0
answers
118
views
Carry operations when adding two numbers [closed]
Let $x$ be a large positive real number and let $q\leq 2$ be a positive integer. It is known that for a positive integer $n,$ there exists a unique sequence $\left\{0\leq n_k\leq q-1\right\}_{k\geq 0}$...
- 1,257
1
vote
0
answers
104
views
Digits of sums of two integers [closed]
Let $q$ be a non-negative integer $\geq 2.$ For a non-negative integer $n$ It is known that there exixts a unique sequence of integer $0\leq n_k \leq q-1$ such that $$n=\sum_{k=0}^{+\infty} n_k q^k.$$
...
- 1,257