4
votes
Accepted
Proof of Dirichlet's Theorem on Primes using $\sum_{\substack{p=1\\ p\equiv h\bmod k}}^\infty\frac{\ln p}{p^s}\sim\frac{1}{\varphi(k)(s-1)}$
The major aspects of the proof are implicit in most treatments. I don't know where this is written explicitly, but I'll sketch a proof of the claim and how this implies Dirichlet's Theorem. The main ...
- 90.7k
4
votes
Accepted
Direct proof for $\sum_{n\le x}\frac{\mu (n)}{n}=O(1)$
Clearly $\left|\sum_{n \leq x} \frac{\mu(n)}{n}\right| \leq 1$ for $x < 2$. For $x \geq 2$, we have that
$$\sum_{n \leq x} \frac{\mu(n)}{n} = \frac{1}{x} \sum_{n \leq x} \mu(n) \frac{x}{n} = \frac{...
- 3,539
3
votes
Accepted
Complex logarithm of zeta functions and L-functions
Here is a general theorem which can be used in to compute logarithms for products of functions:
Theorem. Assume that $f(s)=\prod_{n=1}^{\infty}f_n(s)$ is a product of complex-valued functions which is ...
- 2,405
3
votes
Accepted
Asymptotics for the number of $n\le x$ which can be written as the sum of two squares. Is Perron's formula applicable?
You still can use Perron's formula, but Hankel contour has to be used instead of closed rectangles because the the function in question is only one-valued in a cut plane. The resulting formula is
$$
A(...
- 6,373
2
votes
Can $f(z) = \sum_p \frac {z^p}{p^2}$ teach us about primes?
Today the prime number theorem is proved using the Dirichlet series $\sum \Lambda(n)/n^s$, where $\Lambda$ is the von Mangoldt function. This is the negative log derivative $-\zeta'(s)/\zeta(s)$, ...
- 43.2k
2
votes
Complex logarithm of zeta functions and L-functions
You ask a good question that is often misunderstood and not explained clearly in textbooks. In brief, the left side of your log equation is defined to be the right side and there is no need to refer ...
- 43.2k
2
votes
Accepted
Confusion about definition Petersson product
This is a good, natural question. The point is that the measure $\mu(z) = \frac{dx dy}{y^2}$ is the unique (up to multiple) Haar measure here. This measure is invariant under the isometries of the ...
- 90.7k
1
vote
Evaluate $\zeta(0)$
$\Gamma$ has a simple pole at $s=0$ with residue $1$, and so $\lim\limits_{s\to 0}s\Gamma(s)=1$. Thus:
$\lim\limits_{s\to 0}\frac{\pi^{\frac{s}{2}}}{(s-1)\Gamma(\frac{s}{2})}=\frac{1}{(-1)\cdot\infty}=...
- 37.3k
1
vote
If $\pi(x) = Li(x) + O(\ln^3(x) \sqrt x) $ is true, what does that say about the Riemann zeta zeros?
If $\zeta(s)$ has a nontrivial zero $\rho = \sigma_0 + it_0$ with $\frac{1}{2} < \sigma_0$, then
$$ \pi(X) - \mathrm{Li}(X) = \Omega_{\pm}(X^{\sigma_0}).$$
As this is a less common Landau-type ...
- 90.7k
1
vote
Accepted
Estimating $\pi(x)$ in terms of logarithmic integral
Since
$$
\pi(x)-\operatorname{Li}(x)=\int_{2^-}^x{\mathrm d[\theta(u)-u]\over\log u}={\theta(x)-x\over\log x}+\int_2^x{\theta(u)-u\over u(\log u)^2}\mathrm du,
$$
it suffices to show that the ...
- 6,373
1
vote
Accepted
Relationships between Riemann Xi and Jacobi Theta functions
There are two notations for theta functions. They are related by
$$
\theta_j(z\vert\tau)=\theta_j(z,e^{\pi i\tau}),\tag1
$$
so it is more correct to write $\theta_j(z|\tau)$ instead of $\theta_j(z,\...
- 6,373
1
vote
The inverse of a Dirichlet product is the Dirichlet product of the inverses of each function
Actually we can look at this question from the group theory perspective. If $a,b$ are elements of an abelian group $G$, then $(a*b)^{-1}$ is $a^{-1} * b^{-1}$. So we just need to prove that all ...
- 11
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