We obtain bounds for the error term in the prime number theorem of the form
π
(
x
)
-
Li
(
x
)
≤
9.2211
x
log
(
x
)
exp
-
0.8476
log
(
x
)
for all
x
≥
2
,
as well as other classical forms, improving ...upon the various constants and ranges compared to those in the literature. The strength and originality of our methods come from leveraging numerical results for small
x
in order to improve both the asymptotic and numerical bounds one obtains. We develop algorithms and formulas optimizing the conversion of both asymptotic and explicit numerical bounds from the prime counting function
ψ
(
x
)
to both
θ
(
x
)
and
π
(
x
)
.
In this note we exhibit some large sets
Θ
x
⊂
{
1
,
2
,
…
,
⌊
x
⌋
}
such that the sum of the Möbius function over
Θ
x
is small and independent of
x
. We show that the existence of some of these sets ...are intimately connected with the existence of the alternating series used by Tschebyschef and Sylvester to bound the prime counter function
Π
(
x
)
.
Let
V
(
T
) denote the number of sign changes in
ψ
(
x
)
-
x
for
x
∈
1
,
T
. We show that
lim inf
T
→
∞
V
(
T
)
/
log
T
≥
γ
1
/
π
+
1.867
·
10
-
30
, where
γ
1
=
14.13
…
denotes the ordinate of the ...lowest-lying non-trivial zero of the Riemann zeta-function. This improves on a long-standing result by Kaczorowski.
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (
) − li(
). Whether a Skewes’ number is an integer is ...an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.
Restricted Partitions: The Polynomial Case Chernyshev, V. L.; Hilberdink, T. W.; Minenkov, D. S. ...
Functional analysis and its applications,
12/2022, Volume:
56, Issue:
4
Journal Article
Peer reviewed
We prove a restricted inverse prime number theorem for an arithmetical semigroup with polynomial growth of the abstract prime counting function. The adjective “restricted” refers to the fact that we ...consider the counting function of abstract integers of degree
whose prime factorization may only contain the first
primes (arranged in nondescending order of their degree). The theorem provides the asymptotics of this counting function as
. The study of the discussed asymptotics is motivated by two possible applications in mathematical physics: the calculation of the entropy of generalizations of the Bose gas and the study of the statistics of propagation of narrow wave packets on metric graphs.
In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy–Littlewood prime tuple conjecture, ...conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. We prove an asymptotic formula for the number of simultaneous prime polynomial values of n linear functions, in the limit of a large finite field.
In classical prime number theory several asymptotic relations are considered to be “equivalent” to the prime number theorem. In the setting of Beurling generalized numbers, this may no longer be the ...case. Under additional hypotheses on the generalized integer counting function, one can however still deduce various equivalences between the Beurling analogues of the classical PNT relations. We establish some of the equivalences under weaker conditions than were known so far.
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