Intereting Posts

Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?
solution of ordinary differential equation $x''(t)+e^{t^2} x(t)=0, for t\in$
What type of discontinuity is $\sin(1/x)$?
Normal subgroups and cosets
fields are characterized by the property of having exactly 2 ideals
Baire space homeomorphic to irrationals
Ways of coloring the $7\times1$ grid (with three colors)
A robot moves on 2-Dimensional grid. It starts out at (0,0) and is allowed to move in any of these four ways:
Why do differential forms and integrands have different transformation behaviours under diffeomorphisms?
If coefficients of the Quadratic Equation are in AP find $\alpha+\beta +\alpha\beta$.
The inverse of Lagrange's Theorem is true for finite supersolvable group.
Non-UFD integral domain such that prime is equivalent to irreducible?
$\int_X |f_n – f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
Derivative of $a^x$ from first principles
Integral related to a geometry problem

I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N \|a(n)\| \leq c_f \cdot N^{\frac{k+1}{2}} $.

Now, somehow, using the theorem that states $ \| \sum_{n=1}^N a(n) \| \leq c_f \cdot N^{\frac{k}{2}} \cdot logN $ we’ve got that there are a lot of sign changes in the coefficients. I can’t see how it follows, as it wasn’t stated anywhere that the first bound is tight.

Can someone explain how this result is obtained? And maybe give me some reference to literature on the topic (if exists), of advanced undergraduate – beginner graduate level?

Thanks in advance.

- Limit inferior of the quotient of two consecutive primes
- Would proof of Legendre's conjecture also prove Riemann's hypothesis?
- Logarithmic derivative of Riemann Zeta function
- Are these zeros equal to the imaginary parts of the Riemann zeta zeros?
- Constructing arithmetic progressions
- Derivative of the Riemann zeta function for $Re(s)>0$.

- Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$ using Fourier series
- Fourier series for $-x+\frac{1}{2}$
- “Counterexample” for a weaker version of Riemann–Lebesgue lemma
- Use Fourier series for computing $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$
- Rate of Fourier decay of indicator functions
- Riemann zeta function at odd positive integers
- Dirichlet's Divisor Problem
- Solving an integral coming from Perron's formula
- Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$
- Proving that the limit of an integral of a series exists

In lieu of another answer, I’d wager that the Iwaniec-Kowalski book “Analytic Number Theory” (AMS) would provide you with prototypes for many such arguments.

- Limits of recurrently defined sequences.
- Can a group have more subgroups than it has elements?
- How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
- Which topics of mathematics should I study?
- Removable singularities and an entire function
- prove that $(1 + x)^\frac{1}{b}$ is a formal power series
- Is the Subset Axiom Schema in ZF necessary?
- How is this linear 2nd-order ODE solved?
- How many trees in a forest?
- The Limit of $x\left(\sqrt{a}-1\right)$ as $x\to\infty$.
- How to find the area of any irregular shape?
- Why does an irreducible polynomial split into irreducible factors of equal degree over a Galois extension?
- Factoring Quadratics
- Matrix raised to a matrix
- Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $