This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) J´anos Pintz, Supported by the National Research Development and Innovation Office, NKFIH, KKP 133819, ELKH Alfr´ed R´enyi Mathematical Institute H-1053 Budapest Realtanoda u. 13–15. Hungary and e-mail: pintz@renyi.hu

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**Keywords and phrases:** Riemann’s zeta function, density hypothesis, density theorems.

2020 Mathematics Subject Classification: Primary 11M26, Secondary 11M06.

## 1

The plausibility of the Riemann Hypothesis (RH) is supported (among others) by results of type (1.1)

valid for all σ > 1/2 as T → ∞. (1.1) was first shown by Bohr and Landau in 1914 ([BL1914]). The first estimate of type

was shown few years later by Carlson [Car1920] with

It was Hoheisel [Hoh1930] who first observed that such estimates lead to arithmetic consequences about the difference of consecutive primes. He proved the first approximation towards the famous conjecture that there exists always a prime between two consecutive squares. This conjecture was characterised by Landau [Lan1913] in his plenary talk at ICM1912 in Cambridge as one of the four main problems of the distribution of primes, besides the Riemann Hypothesis. Hoheisel [Hoh1930] could show (pn denotes the nth prime)

In the proof important role was played by Carlson’s density theorem (1.2)– (1.3).

Later it was realized that a uniform estimate of the form

yields (combined with a slightly better zero-free region than the classical one of de la Vall´ee Poussin)

In particular, the best possible estimate A = 2 would almost imply Landau’s conjecture. This is especially remarkable in light of the fact that even assuming the Riemann Hypothesis the best estimate we know is

(with C = 2 easily by the Riemann–Von Mangoldt explicit formula, with C = 1 by a deeper argument of Cram´er [Cra1921]).

This explains the significance of the Density Hypothesis (DH) which states

or, in a slightly weaker form, using the notation (1.2),

The Riemann Hypothesis clearly implies the Density Hypothesis. However, Ingham [Ing1937] showed that also the Lindel¨of Hypothesis (LH) (1.10)

implies DH.

In 1954 Tur´an [Tur1954] used his celebrated power-sum method [Tur1953], [Tur1984] to give a different proof of Ingham’s result that LH implies DH. In the same work he came very close to breaking the DH in the vicinity of the boundary line σ = 1. In the following we will use the notation:

His result was with a small constant c1 [Tur1954]

or with our new notation

It was 16 years later when G. Hal´asz and Tur´an succeeded to break the DH [HT1969], that is, to prove (in the refined form appearing as Theorem 38.2 of Tur´an’s book [Tur1984]) with the notation (1.1)–(1.2), (1.11)

The proof was based on

(i) Vinogradov’s estimate µ(1 − η) ≪ η 3/2,

(ii) Tur´an’s power-sum method;

(iii) a simple but ingenious idea of Hal´asz [Hal1968].

Soon after this Bombieri [Bom1971] gave a different proof using ideas from the large sieve coupled with (i) and (iii). Further, Montgomery has shown the DH for σ ≥ 9/10, i.e. η ≤ 1/10 ([Mon1969], [Mon1971]). Using another simple but ingenious idea Huxley [Hux1972] (with a refinement of Montgomery) reached the DH for η ≤ 1/6 and proved that

The DH was shown for larger and larger ranges, until improving the result σ > 11/14 of Jutila [Jut1977] Bourgain proved its validity [Bou2000] for

Another direction of the research was to give strong density estimates for small values of η (often especially for η → 0). In this direction the most important results were reached (in alphabetical order) by Bourgain, Ford, Heath-Brown, Huxley, Ivic, Jutila and Montgomery.

All these results use the large sieve and in some form the mentioned idea of Hal´asz, which needs η ≤ 1/4. This is also the reason that they cannot give improvements of the classical zero-density theorem of Ingham [Ing1940], A(σ) ≤ 3/(2 − σ), which is still the best today in the whole range

Recently I gave two alternative variants for proving the DH for small values of η. In [Pin2022] the goal was to reach a possibly simple proof while in [Pin2023] to show the strongest possible result both in specific ranges of η < 1/12 and for η → 0. Both proofs were based on Vinogradov’s method and Hal´asz’s idea. However, in the second [Pin2023] it was important to use recent deep results of Heath-Brown [Hea2017], further of Bourgain, Demeter and Guth [BGD2016].