All Questions
Tagged with analytic-number-theory dirichlet-series
159
questions
2
votes
1
answer
74
views
Dirichlet series and Laplace transform
Let $\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}$ be a Dirichlet series. It can be represented as a Riemann-Stieltjes integral as follows:
$$\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}=\int_1^\...
2
votes
1
answer
45
views
Perron's formula in the region of conditional convergence
I am a bit confused about the proof of Perron's formula. It states that for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ and real numbers $c > 0$, $c > \sigma_c$, $x > 0$ we have
$$\...
- 825
1
vote
0
answers
75
views
A question about Lemma 15.1 (Landau’s theorem for integrals) in Montgomery-Vaughan’s book
Lemma 15.1 in Montgomery-Vaughan’s analytic number theory book is Landau’s theorem for integrals. My question is, why is it necessary to have $A(x)$ bounded on every interval $[1,X]$? Doesn’t the ...
- 405
6
votes
0
answers
194
views
Proof of Theorem 1.1 of Analytic Number Theory by Iwaniec & Kowalski
I am not clear about the proof of Theorem 1.1 in the book `Analytic Number Theory' by the authors Iwaniec & Kowalski.
They say that if a multiplicative function $f$ satisfies $$\sum_{n\le x}\...
- 1,145
0
votes
1
answer
107
views
Proof of $\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx$ and $\limsup_{x\to\infty} \frac{\log|A(x)|}{\log x} = \sigma_c$
Theorem. Let $A(x) := \sum_{n\le x} a_n$. If $\sigma_c < 0$, then $A(x)$ is a bounded function, and $$\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx \tag{1}$$ for $\sigma > 0$. ...
- 14.2k
2
votes
1
answer
88
views
A question about Landau’s theorem for Dirichlet series and integrals
A well known theorem of Landau’s for Dirichlet series and integrals goes as follows (I copy the theorem almost exactly as it appears in Ingham’s Distribution of Prime Numbers, Theorem H in Chapter V, ...
- 405
0
votes
0
answers
62
views
"Mollifier" of the Dirichlet L-function
I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
0
votes
1
answer
67
views
Asymptotics for the number of $n\le x$ which can be written as the sum of two squares. Is Perron's formula applicable?
For all $n\ge 1$, let
$$
a_n = \begin{cases}
1\quad&\text{if $n$ can be written as the sum of two squares;}\\
0&\text{otherwise}
\end{cases}
$$
I am interested in $A(x):=\sum_{n\le x}a_n$.
...
- 8,421
1
vote
2
answers
101
views
Evaluate $L(1, \chi) = \sum_{n=1}^\infty \frac{\chi_5(n)}{n},$ for $\chi$ mod $5$
My HW question is:
Evaluate the series
$$L(1, \chi_5) = \sum_{n=1}^\infty \frac{\chi_5(n)}{n},$$
where $\chi_5$ is the unique nontrivial Dirichlet character mod $5$.
My work is:
\begin{align*}
...
- 3,187
2
votes
1
answer
116
views
Evaluate $L(1, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n}$ for $\chi$ mod $3$
Here is the homework question I am working on:
Evaluate (as a real number) the series
$$L(1, \chi_3) = \sum_{n=1}^\infty \frac{\chi_3(n)}{n},$$
where $\chi_3$ is the unique nontrivial Dirichlet ...
- 3,187
2
votes
1
answer
272
views
Dirichlet series with infinitely many zeros
Can a Dirichlet series have infinitely many zeros and be nonzero?
To be precise, by a Dirichlet series I mean a function of the form $s\mapsto \sum_{n\geq 1}\frac{a_n}{n^s}$ where the domain is the ...
- 165
4
votes
0
answers
74
views
Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?
It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
- 1,810
0
votes
0
answers
26
views
How to construct a Dirichlet series that cannot be analytically continued beyond its abscissa of absolute convergence?
If I want a power series $\sum_n a_n \, z^n$ that cannot be analytically continued anywhere beyond its disk of convergence $|z| < R$, then I can use a lacunary series, e.g., $\sum_n z^{2^n}$.
Are ...
- 1,755
2
votes
1
answer
113
views
Residue of a Dirichlet Series at $s=1$
I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
0
votes
0
answers
40
views
LCM sum with $\log $'s
If I want to evaluate $$\sum _{[r,r']\leq x}\log r\log r'$$ I could write it as an integral using Perron's formula, pick up a pole, and get a main term which involves looking at (the derivatives at $\...
- 1,662