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Results tagged with analytic-number-theory
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user 110603
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
1
vote
Error term of Fourier series for fractional part
There's some details in page 8 of
https://eudml.org/doc/278867
- 1,381
4
votes
Consecutive integers that are coprime to a given number
Unless I'm overlooking something (which is very very possible...) I think you can just use the Moebius function in the form
\[ \sum _{d|n}\mu (d)=\left \{ \begin {array}{ll}1&\text { if }n=1\\ 0&\text …
- 1,381
3
votes
Accepted
Ramanujan's type sum
If I'm not mistaken, the $h>0$ part of the sum is
\[ \sum _{d|q}d\mu (q/d)\sum _{h\leq q\atop {h\equiv a(d)}}\frac {1}{h}\]
and here the inner sum is
\[ \sum _{0\leq h\leq (q-a)/d}\frac {1}{hd+a}\leq …
- 1,381
1
vote
Upper bound for the integral over minor arcs of the exponential sum with prime omega functio...
Not an answer just a small comment: if the singular series coming from the problem of evaluating $$\sum _{n-n=0}h(n)h(n')$$ with the circle method is equal to $$\int _0^1\left |\sum_{n\leq N}h(n)e(n\b …
- 1,381
1
vote
Error term when truncating series for $1/\zeta(s)$
I'd have a lousy suggestion, but it's too long for a comment so here goes. Maybe it's the
wrong way to think with this problem, but
my first idea would be to look for an approximate functional equa …
- 1,381
4
votes
Distribution of $\{x/n^2\}$
I'm not sure but I'll point out two things which are hopefully somehow useful (and if they're not I'll just delete it later).
First, for the range close to $\sqrt x$: you have
\[ \int _{\epsilon \sqrt …
- 1,381
8
votes
Looking for some interesting complex integration contours
"I'm interested in examples where a relatively simple path is distorted by some epsilon to avoid a singularity": The proof of Theorem 5.2 in Montgomery and Vaughan's "Multiplicative Number Theory" see …
- 1,381
1
vote
Distribution of $\alpha n^2/q$ modulo $1$?
Lemma 1 of
http://www.numdam.org/item/ASNSP_1999_4_28_4_591_0.pdf
bounds $\{ d\sim D\text { with }||\alpha d^2||<D\} $ and gives a non-trivial result for $D\approx q$... not sure if this is enough to …
- 1,381
4
votes
Sum of $\frac{1}{(\delta_1,\delta_2)}$ with congruence restrictions
Your sum is
\begin{align*}
L(Q)&=\sum _{d\leq Q}\frac {1}{d}\sum _{\delta _1,\delta _2\leq Q/d\atop {a|d\delta _1\atop {b|d\delta _2\atop {(\delta _1,\delta _2)=1}}}}1=\sum _{dh\leq Q}\frac {\mu (h)}{ …
- 1,381
2
votes
Accepted
Bounding a number-theoretic integral
The integral looks something like $$\sum _{n=1}^\infty \frac {\Lambda (n)}{n^c}\int _1^Tt^{1/2-c}\cdot e(t-t\log (X/nt))\cdot dt\hspace {10mm}e(z)=e^{2\pi iz}.$$
The derivative of the phase is somethi …
- 1,381
8
votes
What is the intuition behind applying the Mellin Transform to prime distribution?
This is too thin an answer really, but it's too long for a comment - I expect there'll be much better answers soon enough but hopefully this is at least a start:
If you have any sum, you wonder what y …
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