Questions tagged [analytic-number-theory]
Questions on the use of the methods of real/complex analysis in the study of number theory.
3,997
questions
3
votes
1
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views
$1^\alpha+2^\alpha+3^\alpha+\cdots+n^\alpha$
Let $\alpha>0$ and $m$ be a positive integers, use Euler's summation formula we can prove that there exists a constant $C$ such that
$$ \sum_{k=1}^nn^\alpha=\frac{n^{\alpha+1}}{\alpha+1}+\frac{n^...
- 915
1
vote
0
answers
46
views
Exercise 2 in Chapter 1 of Apostol's "Modular Functions and Dirichlet series in Number Theory"
I am reading the book in the title and I don't know how to prove the following (that is Exercise 2 in Chapter 1):
Suppose $f$ is an elliptic function (meaning that $f$ is meromorphic and there are two ...
- 1,136
2
votes
0
answers
53
views
Reconciling different ideal-theoretic definitions of Hecke Characters
I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as:
Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\...
- 787
0
votes
1
answer
34
views
Question about the proof of distribution of $\Omega(n)-\omega(n)$
On page 68 of the book Multiplicative Number Theory by Montgomery and Vaughan, in the last step of the proof of Theorem 1.16, we have the following:
Here we want to show that $d_k$ above is the $k$-...
- 521
1
vote
0
answers
53
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M/V Multiplicative NT : Theorem 11.3 and the Siegel zero
Two questions regarding Theorem 11.3 in the book of Montgomery & Vaughan Multiplicative Number Theory on the section "Case 4. Quadratic $χ$, real zeros.":
First the book supposes there ...
- 281
1
vote
1
answer
44
views
Clarification on an argument in Opera de Cribro
In the beginning of Chapter 24 of Opera de Cribro, I am not able to understand why $\chi(a)=-1$ is not permissible? Why would the set $\mathcal{A}$ be empty in that case?
Relevant extract from the ...
- 1,914
0
votes
0
answers
16
views
Estimates for the logarithmic derivative of $\Lambda(s,\chi)$
We have the following estimate about the logarithmic derivative of $\xi_0(s)=s(1-s)\dfrac{\zeta(s)}{\zeta_\infty(s)}$ for $s=\sigma+it$ and $\rho=\beta+i\gamma$:
$$
\frac{\xi_0'}{\xi_0}(s)=\sum_{\rho\...
- 117
0
votes
1
answer
37
views
Zeros of L function
If $\chi$ is imprimitive character mod $q$, we know that there exists a primitive character $\chi^*$ (mod $q^*$) which induces the character $\chi$ (mod $q$). Then we have
$$L(s,\chi) = L(s,\chi^*) \...
- 1,132
0
votes
0
answers
20
views
counting function into integration
Could someone please tell me how one write some counting function over the zeros of L function in terms of the Riemann Stieltjes integration wrt the the same counting function? I know Riemann ...
- 1,132
1
vote
0
answers
39
views
Counting solution to congruences
I want to count the $x, y \mod a$ and $r, s \mod b$ subject to the following conditions (defining $u, v, w, k$ which exist by the coprimality conditions)
$$(a, x, y) = 1$$
$$(b, r, s) = 1$$
$$ as+xr+...
- 1,285
1
vote
0
answers
39
views
Evaluation of sum of product of dirichlet characters
An Analytic number theory question:
Let, $\chi_{0,q_2}$ be the trivial character modulo $q_2$ and $rad(q)$ be the radical of $q$.
I need to estimate the summation below:
$$S_1=\sum\limits_{\substack{...
- 11
1
vote
0
answers
51
views
Bounding $\psi(x)-x$ outside a set of finite logarithmic measure
By a 1980 result of Gallagher, assuming the Riemann hypothesis, one has $\psi(x)-x = O(x^{1/2} (\log \log x)^2)$ outside a set of finite logarithmic measure. I'm wondering what the state-of-the-art ...
- 877
4
votes
1
answer
118
views
$3\frac{\zeta'(\sigma)}{\zeta(\sigma)}+4 \Re \frac{\zeta'(\sigma+i t)}{\zeta(\sigma+i t)}+\Re\frac{\zeta'(\sigma+2 i t)}{\zeta(\sigma+2it)}\leq0$
In our script it is used without proof that
For $\sigma>1$ and $t \in \mathbb{R}$
$$
3 \frac{\zeta^{\prime}(\sigma)}{\zeta(\sigma)}+4 \Re \frac{\zeta^{\prime}(\sigma+i t)}{\zeta(\sigma+i t)}+\Re\...
- 2,308
2
votes
2
answers
96
views
Probabilistic Goldbach conjecture under Cramér random model
The Cramér random model for the primes is a random subset ${{\mathcal P}}$ of the natural numbers with ${1 \not \in {\mathcal P}}, {2 \in {\mathcal P}}$, and the events ${n \in {\mathcal P}}$ for ${n=...
- 1,011
0
votes
1
answer
82
views
Estimation of the absolute value of the $n$th non-real zero of the Riemann zeta function
Recently, I have been studying the oringinal proof of the prime number theory by Hadamard. I didn't get it on the estimation of the absolute value of the $n$th non-real zero of the $\zeta$ function by ...
