Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling

We prove three results on the density resp. local density and clustering of zeros of the Beurling zeta function $\zeta(s)$ close to the one-line $\sigma:=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. Theorem 4 provides a zero density estimate, which is a complement to known results for the Selberg class. Note that density results for the Selberg class rely on use of the functional equation of $\zeta$, which we do not assume in the Beurling context. In Theorem 5 we deduce a variant of a well-known theorem of Tur\'an, extending its range of validity even for rectangles of height only $h=2$. In Theorem 6 we will extend a zero clustering result of Ramachandra from the Riemann zeta case. A weaker result -- which, on the other hand, is a strong sharpening of the average result from the classic book \cite{Mont} of Montgomery -- was worked out by Diamond, Montgomery and Vorhauer. Here we show that the obscure technicalities of the Ramachandra paper (like a polynomial with coefficients like $10^8$) can be gotten rid of, providing a more transparent proof of the validity of this clustering phenomenon

21:27 WSPC/INSTRUCTION FILE LFunctions Correlations in Prime Number Distribution and L-function Zeros

A simple analysis of the gaps in primes shows an interesting correlation between neighbouring primes. Neighbouring primes are more likely to have differing remainders on being divided by 6 (the remainders can be 1 or 5). We give a heuristic argument for the observed correaltion. We apply the tool of rescaled range analysis to study the statistical properties

An annotated bibliography for comparative prime number theory

The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained. We encourage feedback on this manuscript (see the end of Section~1 for details).Comment: 98 pages; supersedes "Comparative prime number theory: A survey" (arXiv:1202.3408

On S\'ark\"ozy's theorem for shifted primes

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by $p-1$, $p$ prime. Then $|A| \ll N^{1 - c}$.Comment: 104 pages, submitted. Version 2 incorporates some minor corrections and adds an explicit estimate for the Gamma function, allowing for the possibility of an explicit computation of c using forthcoming zero-density estimates of Thorner and Zama

On multiplicative functions which are small on average

Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if \sum_{n2, for some $A>2$, then either $f(p)$ is small on average or $f$ pretends to be $\mu(n)n^{it}$ for some $t$.Comment: 51 pages. Slightly strengthened Theorem 1.2 and simplified its statement. Removed Remark 1.3. Other minor changes and corrections. To appear in Geom. Funct. Ana