arXiv subjects

János Pintz

Publications and source records attributed to János Pintz.

7 recordsLinked to original sources

arXiv2026

Oscillation of partial sums of the Möbius function and zeros of Riemann's zeta function

The oscillation of M(x), the partial sum of the Möbius function has been in the focus of researchers in the theory of primes since the famous conjecture of Mertens in 1905 (formulated in a weaker form by Stieltjes in 1885 in a letter to Hermite). The average order of the modulus of M(x) in an interval of type [0,Y] is clearly in connection with the distribution of primes. The author proved at the beginning of 1980's that this average tends unconditionally to infinity with Y. The present work shows that (similarly to the average order of the error term of the Prime Number Theorem) this average agrees with great accuracy for large values of Y with the largest error term of the Riemann-von Mangoldt prime number formula for the value Y.

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arXiv2025

Polynomial configurations in dense subsets of the prime lattice

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive relative upper density. We show that A contains all polynomial configurations of the form $x+P_0(y)v_0,\ldots, x+P_l(y)v_l$, for some $x$ in $\mathbb{Z}^d$ and $y$ in $\mathbb{N}$, which satisfy a certain non-degeneracy condition. We also obtain quantitative bounds on the size of such polynomial configuration, if $A$ is a subset of the first $N$ positive integers.

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arXiv2024

On the Density Hypothesis for the Selberg class

We prove a general zero density theorem on the Selberg class of functions. The result verifies the Density Hypothesis in the strip when the real part of the variable is at least 0.9 under the assumption that the degree of the function does not exceed 2. In the other cases we obtain a density theorem weaker than the Density Hypothesis.

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arXiv2024

New zero-density estimates for the Beurling $ζ$ function

In two previous papers the second author proved some Carlson type density theorems for zeroes in the critical strip for Beurling zeta functions satisfying Axiom A of Knopfmacher. In the first of these invoking two additonal conditions were needed, while in the second an explicit, fully general result was obtained. Subsequently, Frederik Broucke and Gregory Debruyne obtained, via a different method, a general Carlson type density theorem with an even better exponent, and recently Frederik Broucke improved this further, getting $N(σ,T) \le T^{a(1-σ)}$ with any $a>\dfrac{4}{1-θ}$. Broucke employed a new mean value estimate of the Beurling zeta function, while he did not use the method of Halász and Montgomery. Here we elaborate a new approach of the first author, using the classical zero detecting sums coupled with a kernel function technique and Halász' method, but otherwise arguing in an elementary way avoiding e.g. mean value estimates for Dirichlet polynomials. We will make essential use of the additional assumptions that the Beurling system of integers consists of natural numbers, and that the system satisfies the Ramanujan condition, too. This way we give a new variant of the Carlson type density estimate with similar strength as Turán's 1954 result for the Riemann $ζ$ function, coming close even to the Density Hypothesis for $σ$ close to 1.

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arXiv2015

A note on the distribution of normalized prime gaps

Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erdős conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the nonnegative part of the real line. He and independently G. Ricci proved in 1954-55 that the set J of limit points of the sequence {d_n/logn} has positive Lebesgue measure. The first and until now only concrete known element of J was proved to be the number zero in the work of Goldston, Yildirim and the present author. The author of the present note showed in 2013 (arXiv: 1305.6289) that there is a fixed interval containing 0 such that all elements of it are limit points. In 2014 it was shown by W.D. Banks, T. Freiberg and J. Maynard (arXiv: 1404.5094) that one can combine the Erdős-Rankin method (producing large prime gaps) and the Maynard-Tao method (producing bounded prime gaps) to obtain the lower bound (1+o(1))T/8 for the Lebesgue measure of the subset of limit points not exceeding T. In the present note we improve this lower bound to (1+o(1))T/4, using a refinement of the argument of Banks, Freiberg and Maynard.

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arXiv2015

On a conjecture of ErdŐs, Pólya and Turán on consecutive gaps between primes

Let p_n denote the sequence of all primes and let d_n=p_n-p_{n-1} denote the sequence of all gaps between consecutive primes. In 1948 Erdős and Turán showed that d_{n+1}-d_n changes sign infinitely often and together with Pólya asked for a necessary and sufficient condition that a fixed linear combination of r>1 consecutive prime gaps d_{n+i} i=(1,2,...r) should change sign infinitely often as n runs through the sequence of all natural numbers. They conjectured that the condition is that the non zero values among the coefficients of the linear combination cannot all have the same sign but they could prove only much weaker relations. In the present work this conjecture is proved by the aid of the Maynard-Tao method and other important ideas of the recent work of W.D. Banks, T. Freiberg and J.Maynard (see arXiv:1404.59094v2).

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arXiv2013

On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes

We work out the optimization problem, initiated by K. Soundararajan, for the choice of the underlying polynomial P used in the construction of the weight function in the Goldston--Pintz--Yildirim method for finding small gaps between primes. First we reformulate to a maximization problem on L^2[0,1] for a self-adjoint operator T, the norm of which is then the maximal eigenvalue of T. To find eigenfunctions and eigenvalues, we derive a differential equation which can be explicitly solved. The aimed maximal value is S(k)=4/(k+ck^{1/3}), achieved by the (k-1)st integral of x^{1-k/2}J_{k-2}(a_1\sqrt{x}), where a_1 is the first positive root of the (k-2)nd Bessel function J_{k-2} and as such, is asymptotically ck^{1/3} with a well-known constant c. As this naturally gives rise to a number of technical problems in the application of the GPY method, we also construct a polynomial P which is a simpler function yet it furnishes an approximately optimal extremal quantity, 4/(k+Ck^{1/3}) with some other constant C. In the forthcoming paper of J. Pintz (also to appear in the Turán-100 Memorial Volume by de Gruyter), it is indeed shown how this quasi-optimal choice of the polynomial in the weight finally can exploit the GPY method to its theoretical limits.

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