Introduction:
This memo provides an overview of our ongoing research into a specific set of numbers associated with significant deviations in prime-related functions. These numbers, tentatively referred to as “numbers of type Skewes,” represent points where the normalized deviation of the Chebyshev psi function (( \psi(x) )) from its expected value ( x ) reaches new record highs.
Background:
The research stems from an exploration of the behavior of the function ( f(u) = \frac{\psi(e^u) – e^u}{\sqrt{e^u}} ), which measures the deviation of the Chebyshev psi function, normalized by the square root of ( e^u ). This function is significant for its relation to prime number distribution and has connections to concepts like Skewes’ number in number theory.
Methodology:
Our approach involves iterative computational analysis to identify numbers where ( f(u) ) achieves new record values. These numbers, primarily primes or prime powers, are identified using a Pari/GP script that computes ( f(u) ) up to a certain range (recently extended to 30 billion). The script flags numbers that cause ( f(u) ) to reach a peak, which are then classified as “numbers of type Skewes.”
Findings:
The sequence of numbers identified includes a mix of smaller primes and significantly larger numbers. Notably, the sequence extends far beyond traditionally studied ranges in prime number theory. The inclusion of very large numbers suggests that the observed behavior spans a broad spectrum and is not confined to smaller ranges. These numbers potentially represent points of substantial deviation in the distribution of primes and prime powers.
Significance:
The term “numbers of type Skewes” is proposed to describe these record-breaking points, drawing a parallel to the historical context of Skewes’ work. This designation aims to facilitate communication and research related to these numbers. Their study could reveal novel insights into the distribution of primes, particularly in relation to the Riemann Hypothesis and other deep conjectures in number theory.
Next Steps:
- Statistical Analysis and Theoretical Contextualization: To understand the full implications of these findings, a statistical analysis of the distribution of these numbers is crucial. Additionally, placing these results within the context of existing theoretical frameworks in number theory will be essential.
- Validation and Sharing: Presenting these findings to the mathematical community for validation and discussion is a key next step. Publishing results in academic journals and presenting at conferences could help establish the term and findings.
- Further Research: Continued exploration of these numbers, including extending the range of analysis and applying similar methods to other functions, will be vital. Collaboration with mathematicians specializing in analytic number theory could lead to deeper insights.
The prime numbers in the list of numbers of type Skewes are known at the OEIS and the literature as Riemann primes of type psi with index 1.
The first numbers of type Skewes and the values of (psi(x)-x)/sqrt(x) :
3 -0.69757799568699875729269075161497909582
4 -0.75754667510599984488514526008056057961
5 -0.40502142461274279065583051710744138753
7 -0.36274962227512085233818497771804968610
11 -0.23219103618098440552426666313754475516
13 -0.056895928961161321239626357807107470697
19 0.060946197190840935632358055423772602050
31 0.16352807100266185198887896038182042316
73 0.24520859243985257707739238066669123896
109 0.35050095971762715507655633937089709211
113 0.41266840051963261948338418061038541799
199 0.50655635969618563162553916745669122512
1621 0.51618543268822636777791672128521481900
1627 0.54980461532870745759351385928963642340
24107 0.57886318920547976569795139011286113925
24109 0.63094391465142741947288165170043931559
24113 0.67011340365793650238265828870965073874
24121 0.68346467727324404815081503489272393050
24137 0.71016147158360319084997844432208639214
30909301 0.71129419270932904142455127637306156943
30909343 0.71304557582085800908355283278329816537
30909421 0.71452550544079104382550469840396474033
30909673 0.71572746171381291751331541176299426946
110080309 0.71609382868753016624885314877592783683
110080337 0.71695472815934243092024523449419088583
110080351 0.71738517782785532107660209875156257420
110080361 0.71819688629264738078229069259929197761
110080391 0.71886715597723625109940232510195033198
110080403 0.71948823486187038149980056487662503382
110080693 0.72008465757914190559039178215409731128
110080703 0.72089636495657202842448797277688878425
110080769 0.72166501528256393271222338049378113273
110080787 0.72171420600213537642044798847880658872
110080801 0.72214465487789233593991632775918819269
110080837 0.72224303629074671025953371351184137277
———————————————————————–
Below is a list of the record-breaking values of f(u) found by a program in PARI/gp. There is an imperfect match with the list of numbers of type Skewes. For example, the multiple entries beginning with 10.0 are the natural logarithms of the five numbers 24107, 24109, 24113, 24121, 24137. The PARI/gp program uses an analytical approximation to the Chebyshev psi-function that is efficient but slightly off at very large numbers:
1.0988000000000000000000000000000000000 0.32169031645584100044323816308240891912
1.0989000000000000000000000000000000000 0.38249985940121805312177397838039551870
1.6097000000000000000000000000000000000 0.45913748606812260112519187368355802494
1.6098000000000000000000000000000000000 0.46516344103875699271035514792795001448
2.5652000000000000000000000000000000000 0.49682133850436916152173368892595959473
2.5653000000000000000000000000000000000 0.51098577231388510328428572542469224285
2.9447000000000000000000000000000000000 0.52949896562802775502012175116912022849
2.9448000000000000000000000000000000000 0.53574667756651358270582409499318017287
3.4343000000000000000000000000000000000 0.54681005513414559617476763884957607789
4.6916000000000000000000000000000000000 0.55461637231514226214162118922094856089
4.6917000000000000000000000000000000000 0.56169082702246342125850930572360400488
4.7276000000000000000000000000000000000 0.59295276915305349199241702863245305714
4.7277000000000000000000000000000000000 0.62178455819991316933296093755302472291
5.2935000000000000000000000000000000000 0.63361143217817049784872496988597856516
5.2936000000000000000000000000000000000 0.66518790921417411078369376715736226699
10.090700000000000000000000000000000000 0.67032193824855559158026910451019020291
10.090800000000000000000000000000000000 0.67129532553420596681088059462041055712
10.091500000000000000000000000000000000 0.68522911316871038988158281091003948470
10.091600000000000000000000000000000000 0.69299010386031804963617748630697999259
10.998700000000000000000000000000000000 0.69636591675504486571156852507075530620
10.998800000000000000000000000000000000 0.69848391258857375519741694813785662738
18.516700000000000000000000000000000000 0.70099065763218492868894905348748004438
18.516800000000000000000000000000000000 0.73307102453256856488036251769255853303
18.516900000000000000000000000000000000 0.75810919925897432941307155524197007579
18.517000000000000000000000000000000000 0.76909105673739364528122576417228947961
