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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.

3 votes
Accepted

Implications for large sums of roots of unity

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 …
Mark Lewko's user avatar
6 votes
Accepted

Sum of number of divisors function

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 …
Mark Lewko's user avatar
9 votes
Accepted

Other implications of Zhang's method

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 …
Mark Lewko's user avatar
11 votes
Accepted

short character sums averaged on the character

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(\ …
Mark Lewko's user avatar
14 votes

Primes $p$ for which $p-1$ has a large prime factor

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 …
Mark Lewko's user avatar
4 votes
Accepted

$L_p$ norms of $0-1$ exponential sums

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 …
Mark Lewko's user avatar
24 votes
Accepted

A question about Speiser's 1934 result on the Riemann hypothesis

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 …
Mark Lewko's user avatar
0 votes
Accepted

Minor Arc Estimates for an Exponential Sum for a Quadratic Polynomial Over the Primes

This result (or at least the method) probably goes back to Vinogradov or Davenport. For an explicit statement/proof of this result you can see, for instance, Theorem 1 in: J. Liu, T. Zhan, Estimation …
Mark Lewko's user avatar
2 votes
Accepted

Proof of the Friedlander–Iwaniec theorem

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 …
Mark Lewko's user avatar
2 votes

Partial sums of multiplicative functions

This might not be exactly what you have in mind, but it is an old result of Paley that for an infinite (but very sparse) set of real Dirichlet characters the inequality $max_{N} |\sum_{n}^{N} \chi(n) …
Mark Lewko's user avatar
3 votes

Arithmetic progressions without small primes

For a fixed a>0 it is known that there are infinitely many moduli q such that the least prime in the arithmetic progresion a mod q is at least >> q ln (q) lnln (q) ln ln ln ln (q) / (ln ln ln(q) )^2. …
Mark Lewko's user avatar
3 votes
Accepted

A question about the Beurling-Selberg majorant

Given such a $M$ note that $F(x)=M^2(x)$ will be (1) non-negative, (2) majorize $1_{[a,b]}$, and (3) has $\hat{F}$ supported in $[-2\delta,2\delta]$. Thus it will be a (possibly not optimal) solution …
Mark Lewko's user avatar
4 votes

Prime plus square equals prime

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 …
Mark Lewko's user avatar
6 votes

Schur's proof of Hilbert's inequality: streamlining?

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 …
Mark Lewko's user avatar
13 votes
Accepted

Reference and best bounds of $\sum_{n\leq x}\frac{\mu(n)}{n}$

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, …
Mark Lewko's user avatar

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