All Questions
Tagged with analytic-number-theory integration
72
questions
4
votes
1
answer
140
views
Closed form of $\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))}\, dx\,dy\,du\,dv\,dw$
I need closed form for the integral $$I:=\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))} \, dx\,dy\,du\,dv\,dw$$
Then $$0<I<\int_{(0,1)^5} \frac{1}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)...
- 928
2
votes
1
answer
192
views
solution verification: Is $K(s)$ holomorphic on $\Bbb C$?
Consider the Mellin integral
$$K(s)=\int_{1/2}^1 \zeta\bigg(-\frac{1}{\log x}\bigg)~x^{s-1}~dx $$
Where $\zeta(\cdot)$ is the Riemann zeta function defined for real $1/e<x<1.$ $K(s)$ is ...
- 754
4
votes
1
answer
297
views
Newman's Short Proof of Prime Number Theorem
I'm going through the paper of D. Zagier on Short Proof of Prime Number Theorem. There it says in V that
$\Phi(s)=\int_1^\infty \frac{d\vartheta(x)}{x^s}$ . Can someone please explain in details why ...
- 55
0
votes
0
answers
40
views
Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two be derived?
After experimenting with polynomial expansions for the Riemann $\Xi(t)=\xi\left(\frac12+it\right)$-function, I landed on these two equations:
with $M$ the KummerM confluent hypergeometric funcion and ...
- 3,191
3
votes
0
answers
288
views
Understanding Lemma 4.6. of Titchmarsh's book
The following is Lemma 4.6. in Titchmarsh's book The Theory of the Riemann Zeta-Function:
My questions:
1- (Red-underlined) WolframAlpha gives $\int_0^{\infty} \dfrac{e^{iu}}{\sqrt{u}}$ same as in ...
- 281
3
votes
1
answer
99
views
Confusion about definition Petersson product
I'm taking a course on modular forms, but my background in analysis is not that strong (I have taken complex analysis and measure theory before however). Therefore I'm a bit confused about the ...
2
votes
1
answer
136
views
Understanding definition of Adelic integral and calculate simple example
I'm trying to understand the definition the adelic integral given in Goldfeld and Hundley. It says that:
Suppose that $f =\prod_v f_v$ is a factorizable function, that $f_\infty$ is an integrable ...
- 469
4
votes
1
answer
90
views
Bounds for vertical integrals and Mellin transforms
I am trying to bound Mellin transforms. They look like
$$I(x) = \int_{(c)} f(w) \frac{x^w}{w} dw$$
where the integral is on the vertical line of abscissa $c$. Let's say that $f$ has vertical growth ...
- 567
2
votes
0
answers
129
views
Question about proof of the truncated Perron's formula dealing with bounds and convergence
I have a question about the proof of the truncated Perron formula in my analytic number theory lecture notes.
The Formula is given as follows: Let $x,c,T>0$ and suppose that $\sum_n |a_n|/n^c$ is ...
- 5,601
4
votes
0
answers
83
views
Differential-difference equation solution
I have the following set of differential-difference equations
$$
\begin{aligned}
& (s F(s))^{\prime}=f(s-1) \text { for } s>1 \\
& (s f(s))^{\prime}=F(s-1) \text { for } s>2
\end{aligned}...
- 41
8
votes
2
answers
485
views
Closed-form of $\int_0^\infty \frac{t^s}{(e^t-1)^z}dt$
I am looking for a closed form for the integral $$\int_0^\infty \frac{t^s}{(e^t-1)^z}dt$$ valid for $s,z$ being both complex numbers, hopefully using complex analysis. I have already evaluated this ...
- 301
1
vote
1
answer
93
views
Size of a Mellin integral [closed]
I am trying to estimate the size (in $X$) of
$$\int_{(1/2)} \frac{\Gamma(s)}{\Gamma(1/2-s)} X^{-2s} ds$$
where the integration is on the vertical line of real part $1/2+\varepsilon$. I would have ...
- 1,285
7
votes
1
answer
252
views
Transformation Identities of the $_2F_1$ function
From Wolfram Functions we have the following identities for the hypergeometric function $_2F_1$:
$$\begin{align}
_2F_1\left(a,c-b;c;\tfrac{z}{z-1}\right)&=(1-z)^a\,_2F_1(a,b;c;z)\tag1\\
_2F_1\left(...
- 17.3k
4
votes
2
answers
264
views
Unit square integral $\int_{0}^{1} \int_{0}^{1} \left( - \frac{\ln(xy)}{1-xy} \right)^{m} dx dy $
In an article by Guillera and Sondow, one of the unit square integral identities that is proved (on p. 9) is: $$\int_{0}^{1} \int_{0}^{1} \frac{\left(-\ln(xy) \right)^{s}}{1-xy} dx dy = \Gamma(s+2) \...
- 7,118
2
votes
0
answers
154
views
How is this indefinite integral a holomorphic function?
In an analytic number theory textbook we derived the formula
$$\zeta(s)=\frac{s}{s-1}-s\int_1^\infty \{x\}x^{-s-1}\mathrm{dx}$$
and the textbook says, that the integral on the right side converges ...
- 901