All Questions
Tagged with analytic-number-theory analysis
138
questions
-1
votes
1
answer
70
views
How to give this sum a bound?
Let $x,y\in\mathbb{Z},$ consider the sum below
$$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$
is there anything I could do to give this sum a ...
- 329
1
vote
2
answers
53
views
Is there an analytic continuation of the Legendre Chi function $\chi_2(z)$ for $z > 1$?
The Legendre Chi function $\chi_2(z)$ is define as
$$
\chi_2(z) = \sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)^2}
$$
for $-1 \le z \le 1$. But $z > 1$ the series diverges. For real value of $z$ is ...
- 20.8k
1
vote
0
answers
50
views
Explicit upper bound on $\pi(n)$ (Weak version of PNT)
Chebyshev first proved that there exist constants $a$, $b$ such that
$$ a \frac{n}{\log n} < \pi(n) < b \frac{n}{ \log n}.$$ The proof is well understood, and relies on elementary techniques.
...
- 57
1
vote
0
answers
58
views
disk of convergence of composition of p-adic functions
I watched a video that sketched out the regions of convergence for both the p-adic logarithm and the p-adic exponential functions. I thought about all this and asked myself:
How would I find the ...
- 754
0
votes
0
answers
74
views
Is $\phi_0$ equivalent to the Riemann hypothesis?
This is an extension (and more distilled version) of Extension of PDE's to critical strip, with new information. I am fairly sure that my constructions are an alternate description of the De Brujn ...
- 754
2
votes
1
answer
53
views
Value of a Sum linked with Beta Dirichlet Function and Zeta Function
Recently, I tried to calculate this double sum:
$$ F(s) = \sum_{(a,b) \in \mathbb{Z}^2 \backslash (0,0) } \frac{1}{(a^2 + b^2)^s}$$
For $ s \in \mathbb{C}, Re(s) > 1$
I think i found the value of ...
- 31
0
votes
0
answers
33
views
Bound for exponential
Let $\displaystyle f(x)= e^{-Ae^x} e^{Bx} \left( 1+ \mathcal{O} \left( e^{-x} \right)\right)$, where $A$ and $B$ are constants.
Claim: $|f(x)| = \mathcal{O}\left( \exp(-e^x)\right)$ as $x \rightarrow ...
- 177
3
votes
0
answers
90
views
A question on Beukers proof of irrationality of $\zeta(3)$
I am reading the paper A note on the Irrationality of ζ(2) and ζ(3) by F. Beukers.
In equation $(7)$, we have $$I_n=\int_{(0,1)^3}\frac{x^n(1-x)^ny^n(1-y)^nw^n(1-w)^n}{(1-(1-xy)w)^{n+1}} dx dy dw \tag{...
- 928
0
votes
1
answer
107
views
Proof of $\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx$ and $\limsup_{x\to\infty} \frac{\log|A(x)|}{\log x} = \sigma_c$
Theorem. Let $A(x) := \sum_{n\le x} a_n$. If $\sigma_c < 0$, then $A(x)$ is a bounded function, and $$\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx \tag{1}$$ for $\sigma > 0$. ...
- 14.2k
1
vote
1
answer
149
views
Question about Salem's integral equation reformulation of Riemann hypothesis
Consider an integral equation:
$$\int_{-\infty}^{+\infty}\frac{e^{-\sigma y}f(y)}{e^{e^{x-y}}+1}dy=0$$, where $\sigma\in(\frac{1}{2},1)$
Salem proved that this equation has no bounded solution other ...
- 375
1
vote
1
answer
136
views
Divergent sums of reciprocals of specific natural numbers [duplicate]
It is known that
$$\sum_{k=1}^n \frac1k \sim \ln(n)$$
and
$$\sum_{p \text{ prime}}^n \frac1p \sim \ln(\ln(n)),$$
with the property that the difference in each case converges to a constant (Euler–...
- 349
1
vote
1
answer
50
views
Removing a smooth weight
If I understand $$\sum _{n\leq x}a_n(1-n/x)\hspace {5mm}\text {or}\hspace {5mm}\sum _{n\leq x}a_n\log (x/n)$$ then I understand $$\sum _{n\leq x}a_n$$ by writing
$$\sum _{n\leq x}a_n=\frac {x+h}{h}\...
- 1,662
0
votes
1
answer
55
views
is there a notion of natural density over sets of integer with special structure?
Generally, the natural density of a set $A\subseteq \{1, \dots, n\}$, denoted by $d(A)$, is defined as $$d(A)= \lim_{n\to\infty} \frac{\mid A \mid }{n}$$
Now, I am wondering, is there a similar notion ...
- 197
12
votes
0
answers
351
views
Approximate the sum of a non $C^1(0,1)$ function by its integral
Consider the function $f: [0,1] \to \mathbb{C}$ defined by
$$
f(x)=\sum_{n=1}^{9} e^{2\pi n i x},
$$
so that
$$
|f(x)|=\bigg|\frac{\sin(9\pi x)}{\sin(\pi x)}\bigg|.
$$
I'm interested in approximating $...
- 656
1
vote
1
answer
160
views
Residues of $\zeta'(s)/\zeta(s)$ at non-trivial zero of the Riemann zeta function and the order of zeros
I don't understand something about the residues of $\zeta'(s)/\zeta(s)$.
I'm studying complex analysis using Stein's Complex Analysis. The zeros of the Riemann zeta function are $-2n$ ($n$ is natural) ...
- 11