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This isn't exactly along the lines you are suggesting, but Hildebrand does have a proof of the Prime Number Theorem which proceeds but estimating $M(x)$ using the large sieve inequality. See: A. Hilde …

It depends on what range of $\epsilon$ you are interested in. If the exponential sum is near its max possible size ($> \delta N$) then you are saying that your sequence correlates with an exponential …

Yes, Speiser's theorem is an if and only if. See Theorem 1 and "Corollary to Theorem 1" in Levinson and Montgomery's Zeros of the derivatives of the Riemann Zeta-function. Acta Math. 133 (1974), 49–65 …

When I was a graduate student, I worked out a more simple direct variant of this argument avoiding any use of matrics, bilinear forms, etc. It's so simple I can't imagine I was the first to do this. h …

As Joel mentions, this follows from the work of Tao and Ziegler. Alternatively, this can be directly deduced from the density of the primes and the known bounds on the Furstenberg–Sárközy theorem. Ind …

For even integer exponents, say $p=2k$ and $p \geq2$, the quantity is just the $k$-order additive energy of the set $S \subset \mathbb{Z}$ of non-zero Fourier coefficients. It is easy to see that this …

It seems what you are asking is "If we have a precise asymptotic for the number of elements of a set, can we solve binary additive problems involving that set?" The answer in general seems to be `no'. …

See "On the number of primes $p$ for which $p+a$ has a large prime factor." (Goldfeld, Mathematika 16 1969 23--27.) Using Bombieri-Vinogradov he proves, for a fixed integer $a$, that $$\sum_{p \leq x …

Estimates on these quantities are used in Elsholtz and Tao's work on the Erdos-Straus conjecture. See their paper "Counting the number of solutions to the Erdos-Straus equation on unit fractions" and …

Let me sketch a proof using Fourier analysis that I quite like, although perhaps not exactly what I would call the Hardy-Littlewood circle method. Let $r(n)$ denote the number of representations of $ …

J. Friedlander and H. Iwaniec, The polynomial $X^2+Y^4$ captures its primes. Ann. of Math. (2) 148 (1998), no. 3, 945--1040. see: http://www.ams.org/mathscinet-getitem?mr=1670069 http://www.jstor.o …

As Alexey has pointed out the problem can be reduced, via summation by parts, to understanding the asymptotic of Mertens' sum $$M(x) := \sum_{n\leq x} \mu(n).$$ Conditional on the Riemann hypothesis, …

Elliott has a result in this direction. Let $A>0$ and $a \geq 2$ a fixed integer. Furthermore, let $q \leq x^{1/3-\epsilon} $ be a large power of $a$. One then has that $$ \sum_{\substack{d \leq q^{-1 …

Zhang's strategy shows that any admissible tuple of size h contains at least 2 primes infinity often, as long as h is larger than some threshold (the primary theoretical thrust of the polymath project …

Updated answer: Here's an unconditional lower bound of exactly the right order $\sqrt{a}$, however I need $a$ to be comparable to $p$. It is known that (for any $a$) $ \frac{1}{p-1} \sum_{\chi} |S(\ …