Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 282–293. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 287–298. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Moser. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

The Function ζ̃_{−1}(s) and Formulas of the Second Generation for π(x) and R(x)

Jan Moser

Received February, 1997

INTRODUCTION

Recall that in the asymptotic formula

π(x) = ∫_{2}^{x} \frac{du}{ln u} + R(x),  (1)

using the Riemann theory of zeta-function, de la Vallee–Poussin obtained in 1885 the estimate

R(x) = O(xe^{−c\sqrt{ln x}}).  (2)

In this case, the Schmidt estimate

R(x) = O(\sqrt{x} ln x)  (3)

is a consequence of Riemann’s conjecture. However, 100 years passed since that time, but no essential result concerning (2), (3), at least R(x) = x^{0.999 999}, was obtained (see [2, p. 13] and Gel’fond’s remarks [1, p. 287]).

This situation is the basis

(a) for a more intensive study of the Riemann zeta-function,

(b) for constructing new models in which a proper analog of Riemann’s conjecture would be valid and proper formulas for basic functions would hold true.

In [4, 5], a model is studied which is based on a “part of the function ζ(s)”

ζ̃(s) = ζ̃(s; P, β) = ζ_{1}(s) + χ(s)ζ_{2}(s),

ζ_{1}(s) = ∏_{p≤P} ∑_{k=0}^{β} \frac{1}{p^{sk}} , ζ_{2}(s) = ∏_{p≤P} ∑_{k=0}^{β} \frac{1}{p^{(1−s)k}} ,

χ(s) = π^{s−1/2} \frac{Γ((1 − s)/2)}{Γ(s/2)}, s ≠ 2k + 1, k = 0, 1, 2, …, s = σ + it,

P = (ln P_{0})^{1−ε},  B = [(ln P_{0})^{2ε/3}],  P_{0} = \sqrt{\frac{T}{2π}} ,

where ε is an arbitrarily small positive number and p is a prime number.

THE FUNCTION ζ̃_{−1}(s) AND FORMULAS OF THE SECOND GENERATION FOR π(x) AND R(x) 283

Let

D = D(T, H, K) = \{s : σ ∈ ⟨−K, K⟩, t ∈ ⟨T, T + H⟩\},  K > 1, T > 0, H ≤ \sqrt{T} (4)

(the condition H ≤ \sqrt{T} is not essential and only express a tendency to localization), and let

D(Δ_{0}) = \{s : \frac{1}{2} + Δ_{0} ≤ σ ≤ K, t ∈ ⟨T, T + H⟩\},  Δ_{0} = \frac{A}{(ln P_{0})^{ε/2}} .  (5)

The following theorems were proved in [5].

Theorem A. For all sufficiently large T

ζ̃(s) ≠ 0, s ∈ D, σ ≠ \frac{1}{2},

i.e., an analog of Riemann’s conjecture is valid for ζ̃(s), s ∈ D, T → ∞ Theorem B. The function π(x), x ∈ ⟨2, P ⟩, is a solution to the integral equation

ln ζ̃(s) = s ∫_{2}^{P} \frac{Φ(x)}{x(x^{s} − 1)} dx − Φ(P) ln(1 − \frac{1}{P^{s}} ) + Ω_{1}(s),  (6)

(for any fixed s ∈ D(Δ_{0})), where

Ω_{1}(s) = ∑_{p≤P} ln(1 − \frac{1}{p^{s(β+1)}} ) + ln\{1 + \frac{χ(s)ζ_{2}(s)}{ζ_{1}(s)}\} = O(e^{−Aβ}).

Theorem C. The function R(x), x ∈ ⟨2, P ⟩, is a solution to the perturbed integral equation

ln ζ̃(s) = s ∫_{2}^{P} \frac{Φ(x)}{x(x^{s} − 1)} dx − Φ(P) ln(1 − \frac{1}{P^{s}} ) + Ω_{1}(s) + Ω_{2}(s),  (7)

where

Ω_{2}(s) = ∑_{m=0}^{∞} \frac{1}{m + 1} ∫_{2}^{P} \frac{dx}{x^{(m+1)s} ln x} = O(\frac{ln T}{T} ),  s ∈ D(Δ_{0}),  (8)

is a very small perturbation.

The problems of distribution of zeros of the function ζ̃(1/2 + it) were studied in [4]. However, equations (6), (7) for the functions π(x), R(x), respectively, have the following prop-

erties:

(c) they have a complicated structure, i.e., they contain only the terms ln ζ̃(s) and Ω_{1}(s),

(d) they are not applicable on the critical line σ = 1/2 (see (5)). In view of these circumstances, we shall consider a new model in this paper.

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

1. FUNCTION ζ̃_{−1}(s)

Actually, a whole class of functions of the type ζ̃(s) was introduced in [5]. For example, let

\begin{matrix}ζ̃_{−1}(s) = ∏_{p≤P}(1 − \frac{1}{p^{s}} ) + χ(s) ∏_{p≤P}(1 − \frac{1}{p^{1−s}} ) =\begin{matrix}∑ & ′ \\ n<P_{0} \end{matrix} \frac{μ(n)}{n^{s}} + χ(s)\begin{matrix}∑ & ′ \\ n<P_{0} \end{matrix} \frac{μ(n)}{n^{1−s}}, \\ s ≠ 2k + 1,  k = 0, 1, 2, …, \end{matrix}  (9)

where μ(n) is the Möbius function and

P = [(ln P_{0})^{1−ε}],  P_{0} = \sqrt{\frac{T}{2π}} .  (10)

Recall that the function ζ̃(s) turned out to be a “part of ζ(s)”, i.e., a part of the approximate functional equation for ζ(s). In this sense, the function ζ̃_{−1}(s) is not a “part of ζ(s)”, but, in turn, this function is constructed from the parts of the Euler product

\frac{1}{ζ(s)} = ∏_{p}(1 − \frac{1}{p^{s}} ), σ > 1.

Hence the function ζ̃_{−1}(s) preserves the relationship with the function ζ(s).

The following lemma holds true.

Lemma 1. For all sufficiently large T

ζ̃_{−1}(s) ≠ 0, s ∈ D, σ ≠ \frac{1}{2}

(see (4)), i.e., an analog of the Riemann conjecture is valid for ζ̃_{−1}(s), s ∈ D, T → ∞.

Proof. The proof of this assertion can be obtained by analogy with the proof of a similar theorem for the function ζ̃(s) given in [5]. However, in view of the independence of ζ̃_{−1}(s) of β this proof is simpler.

In the subsequent sections of this paper, we consider some properties of the model which is based on the function ζ̃_{−1}(s).

2. ZEROS OF THE FUNCTION ζ̃_{−1}(1/2 + it) AND THE FORMULA FOR π(x)

The properties of zeros of the function ζ̃_{−1}(s) on the critical line σ = 1/2, i.e., of zeros of the function

e^{iφ(t)}ζ̃_{−1}(\frac{1}{2} + it) = Z̃_{−1}(t; P),  t ∈ ⟨T, T + H⟩,  (11)

where

χ(\frac{1}{2} + it) = e^{−i2φ(t)},  φ(t) = \frac{t}{2} ln \frac{t}{2π} − \frac{t}{2} − \frac{π}{8} + O(\frac{1}{t}),  (12)

THE FUNCTION ζ̃_{−1}(s) AND FORMULAS OF THE SECOND GENERATION FOR π(x) AND R(x) 285

follow now from the formula (see (9), (12), cf. [4, (22)])

Z̃_{−1}(t; P) = 2B_{−1}(t; P) cos{φ(t) + C_{−1}(t; P)},  (13)
B_{−1}(t) = ∏_{p≤P} |1 − \frac{p^{−it}}{\sqrt{p}} | = exp(Re\{∑_{p≤P} ln(1 − \frac{p^{−it}}{\sqrt{p}} )\}),

C_{−1}(t) = ∑_{p≤P} arg(1 − \frac{p^{−it}}{\sqrt{p}} ) = Im\{∑_{p≤P} ln(1 − \frac{p^{−it}}{\sqrt{p}} )\}.

(14)

Consequently, the zero γ_{ν} = γ_{ν}^{−1}(P) of the function Z̃_{−1}(t, P) is defined by the equation (see (13))

φ(γ_{ν}) + C_{−1}(γ_{ν}) = πν + \frac{π}{2} ,  ν = 1, 2, ….  (15)

By analogy with case of the function ζ̃(s) (cf. [4, (11)]), we obtain from (15) the following lemma

Lemma 2. For the subsequent zeros ρ_{ν} = 1/2 + iγ_{ν}, ρ_{ν+1} = 1/2 + iγ_{ν+1} of the function ζ̃_{−1}(s) the asymptotic formula

γ_{ν+1} − γ_{ν} = \frac{π}{ln P_{0}} + O\{\frac{1}{(ln P_{0})^{3/2}} \},  γ_{ν}, γ_{ν+1} ∈ ⟨T, T + H⟩,  (16)

holds true as T → ∞.

Remark 1. It follows from (16) that the zeros of the function ζ̃_{−1}(s) are distributed locally in a regular way (cf. [4, (13)]). Hence, the zeros of the complicated transcendental equation

\begin{matrix}∑ & ′ \\ n<P_{0} \end{matrix} \frac{μ(n)}{\sqrt{n}} cos\{φ(t) − t ln n\} = 0

(see (9), (11)) are distributed locally in a regular way on the interval ⟨T, T + H⟩.

Furthermore, the following lemma holds.

Lemma 3. We have

− ∫_{2}^{P} Im\{\frac{ρ_{ν}}{x^{ρ_{ν}} − 1}\}\frac{π(x)}{x} dx + π(P) Im\{ln(1 − \frac{1}{P^{ρ_{ν}}} )\} = φ[t_{ν}(\frac{π}{2} )] − φ(γ_{ν}),  (17)

where γ_{ν} ∈ ⟨T, T + H⟩, H ≤ \sqrt{T}, and \{t_{ν}(π/2)\} is a sequence satisfying the condition

φ[t_{ν}(\frac{π}{2} )] = πν + \frac{π}{2}  (18)

(see [3]).

Proof. We write formula (15) in the form

Im\{∑_{p≤P} ln(1 − \frac{1}{p^{ρ_{ν}}} )\} = φ[t_{ν}(\frac{π}{2} )] − φ(γ_{ν})  (19)

(see (14), (18)). Next from the formula

ln(1 − \frac{1}{(n + 1)^{ρ_{ν}}} ) − ln(1 − \frac{1}{n^{ρ_{ν}}} ) = ρ_{ν} ∫_{n}^{n+1}\frac{dx}{x(x^{ρ_{ν}} − 1)}  (20)

we obtain

∑_{p≤P} ln(1 − \frac{1}{p^{ρ_{ν}}} ) = ∑_{n=2}^{P}\{π(n) − π(n − 1)\} ln(1 − \frac{1}{n^{ρ_{ν}}} )

= ∑_{n=2}^{P −1} π(n)\{ln(1 − \frac{1}{n^{ρ_{ν}}} ) − ln(1 − \frac{1}{(n + 1)^{ρ_{ν}}} )\} + π(P) ln(1 − \frac{1}{P^{ρ_{ν}}} )

= −ρ_{ν} ∫_{2}^{P} \frac{π(x) dx}{x(x^{ρ_{ν}} − 1)} + π(P) ln(1 − \frac{1}{P^{ρ_{ν}}} ).  (21)

Now, by virtue of (20), (21), formula (19) yields (17).

Remark 2. Formula (17) is a formula of a new type for π(x). It is constructed in accordance with the values of relatively simple functions

Im\{\frac{s}{x^{s} − 1}\},  Im\{ln(1 − \frac{1}{P^{s}} )\},  φ(t)

for terms of the well-defined sequences {t_{ν}(π/2)} and {γ_{ν}}.

3. A NEW FORMULA FOR R(x)

The following lemma holds.

Lemma 4. Let

U(x) = ∫_{2}^{x} \frac{du}{ln u},  x ∈ ⟨2, P ⟩.  (22)

Then we have

s ∫_{2}^{P} U(x) \frac{dx}{x(x^{s} − 1)} = U(P) ln(1 − \frac{1}{P^{s}} ) + ∑_{n=0}^{∞} \frac{1}{n + 1} ∫_{2}^{P} \frac{dx}{x^{(n+1)s} ln x}
= U(P) ln(1 − \frac{1}{P^{s}} ) + O(\frac{\sqrt{ln T}}{T} )  (23)

for σ ∈ ⟨1/2, K⟩ (cf. (8)).

Proof. Let

V (x) = V (x, s) = ∫_{2}^{x} \frac{dv}{v(v^{s} − 1)},  (24)

THE FUNCTION ζ̃_{−1}(s) AND FORMULAS OF THE SECOND GENERATION FOR π(x) AND R(x) 287

then

∫_{2}^{P} U(x)\frac{dx}{x(x^{s} − 1)} = U(P)V (P) − ∫_{2}^{P} \frac{V (x)}{ln x} dx.  (25)

Since the summation of relations (20) for ρ_{ν} → s, n = 2, …, P − 1, gives the formula

ln(1 − \frac{1}{P^{s}} ) = s ∫_{2}^{P} \frac{dx}{x(x^{s} − 1)} + ln(1 − \frac{1}{2^{s}} ),

we have

U(P)V (P) = U(P) ∫_{2}^{P} \frac{dx}{x(x^{s} − 1)} = \frac{U(P)}{s}\{ln(1 − \frac{1}{P^{s}} ) − ln(1 − \frac{1}{2^{s}} )\}.  (26)

Furthermore,

V (x) = ∫_{2}^{x} ∑_{n=0}^{∞} \frac{1}{v^{(n+1)s+1}} dv = \frac{1}{s} ∑_{n=0}^{∞} \frac{1}{n + 1}(\frac{1}{2^{(n+1)s}} − \frac{1}{x^{(n+1)s}} )

= −\frac{1}{s} ln(1 − \frac{1}{2^{s}} ) − \frac{1}{s} ∑_{n=0}^{∞} \frac{1}{(n + 1)x^{(n+1)s}}

(see (24)) and

−s ∫_{2}^{P} \frac{V (x)}{ln x} dx = U(P) ln(1 − \frac{1}{2^{s}} ) + ∑_{n=0}^{∞} \frac{1}{n + 1} ∫_{2}^{P} \frac{dx}{x^{(n+1)s} ln x}  (27)

(see (22)). Hence (see (25)–(27)),

s ∫_{2}^{P} U(x) \frac{dx}{x(x^{s} − 1)} = U(P) ln(1 − \frac{1}{P^{s}} ) + ∑_{n=0}^{∞} \frac{1}{n + 1} ∫_{2}^{P} \frac{dx}{x^{(n+1)s} ln x}.  (28)

Since

∫_{2}^{P} x^{−(n+1)s} \frac{dx}{ln x} = \frac{1}{(n + 1)s − 1}\{\frac{1}{2^{(n+1)s−1} ln 2} − \frac{1}{P^{(n+1)s−1} ln P} − ∫_{2}^{P} \frac{dx}{x^{(n+1)s} ln^{2} x}\}

= \frac{1}{(n + 1)s − 1}\{O(1) + O(\sqrt{P}) + O(\sqrt{P})\} = O\{\frac{\sqrt{P}}{(n + 1)T} \},

we have (see (10))

Ω(s, P) = ∑_{n=0}^{∞} \frac{1}{n + 1} ∫_{2}^{P} \frac{dx}{x^{(n+1)s} ln x} = ∑_{n=0}^{∞} O\{\frac{\sqrt{P}}{(n + 1)^{2}T} \} = O(\frac{\sqrt{ln T}}{T} ).  (29)

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

Now, by virtue of (29), (23) follows from (28). This completes the proof of Lemma 4. Setting π(x) = U(x) + R(x) (see (1), (22)) in the expression

W = − ∫_{2}^{P} \frac{ρ_{ν}}{x^{ρ_{ν}} − 1} \frac{π(x)}{x} dx + π(P) ln(1 − \frac{1}{P^{ρ_{ν}}} ),  (30)

we find by Lemma 4 (s = ρ_{ν}) that

W = − ∫_{2}^{P} \frac{ρ_{ν}}{x^{ρ_{ν}} − 1} \frac{R(x)}{x} dx + R(P) ln(1 − \frac{1}{P^{ρ_{ν}}} ) − ρ_{ν} ∫_{2}^{P} \frac{U(x)}{x(x^{ρ_{ν}} − 1)} dx + U(P) ln(1 − \frac{1}{P^{ρ_{ν}}} )

= − ∫_{2}^{P} \frac{ρ_{ν}}{x^{ρ_{ν}} − 1} \frac{R(x)}{x} dx + R(P) ln(1 − \frac{1}{P^{ρ_{ν}}} ) + O(\frac{\sqrt{ln T}}{T} ).  (31)

Thus, we have proved the following theorem (see (17), (30), and (31))

Theorem 1. We have

− ∫_{2}^{P} Im\{\frac{ρ_{ν}}{x^{ρ_{ν}} − 1}\}\frac{R(x)}{x} dx + R(P) Im\{ln(1 − \frac{1}{P^{ρ_{ν}}} )\} + O(\frac{\sqrt{ln T}}{T} ) = φ[t_{ν}(\frac{π}{2} )] − φ(γ_{ν}),

T → ∞,

(32)

for γ_{ν} ∈ ⟨T, T + H⟩, H ≤ \sqrt{T}.

4. RAPPROCHEMENT OF INTEGRAL EQUATIONS

From the lemma, we obtain

Corollary 1. The function π(x), x ∈ ⟨2, P ⟩, is a solution to the integral equation

− ∫_{2}^{P} Im\{\frac{ρ_{ν}}{x^{ρ_{ν}} − 1}\}\frac{Φ(x)}{x} dx + Φ(P) Im\{ln(1 − \frac{1}{P^{ρ_{ν}}} )\} = φ[t_{ν}(\frac{π}{2} )] − φ(γ_{ν})  (33)

(for any fixed γ_{ν} ∈ ⟨T, T + H⟩).

Next, we obtain the following corollary from Theorem 1.

Corollary 2. The function R(x), x ∈ ⟨2, P ⟩, is a solution to the perturbed integral equation

− ∫_{2}^{P} Im\{\frac{ρ_{ν}}{x^{ρ_{ν}} − 1}\}\frac{Φ(x)}{x} dx + Φ(P) Im\{ln(1 − \frac{1}{P^{ρ_{ν}}} )\} + Ω(ρ_{ν}, P) = φ[t_{ν}(\frac{π}{2} )] − φ(γ_{ν}) (34)

(for any fixed γ_{ν} ∈ ⟨T, T + H⟩), where

Ω(ρ_{ν}, P) = ∑_{n=0}^{∞} \frac{1}{n + 1} ∫_{2}^{P} \frac{dx}{x^{(n+1)ρ_{ν}} ln x} = O(\frac{ln T}{T} )  (35)

is a very small perturbation (cf. (8), (29)).

Remark 3. In the model based on the function ζ̃_{−1}(s), the rapprochement of the integral equations for π(x) and R(x) (equation (34) is obtained from (33) via a very small perturbation of (35)) can be observed in a “purer” form (compare with (c) in Introduction) than in the model based on the function ζ̃(s).

Recall that the functions π(x) and R(x) behave quite differently, namely, π(x) ∼ \frac{x}{ln x}, x → ∞, and R(x) changes sign infinitely many times as x → ∞ (Littlewood, 1914).

Remark 4. Obviously, there is a certain instability of solutions to the integral equations (33) and (34) which is inherent in the behavior of the functions π(x) and R(x) as x → ∞.

Remark 5. We can naturally assume that for any fixed γ_{ν} ∈ ⟨T, T + H⟩ and for the corresponding t_{ν}(π/2), there exists a set of solutions to the integral equation (33) of the form

Φ[x; γ_{ν}, t_{ν}(\frac{π}{2} ), P ], x ∈ ⟨2, P ⟩.

However, then the existing solution Φ(x) = π(x), x ∈ ⟨2, P ⟩, to the integral equation (33) must be regarded as a certain “singular solution”; the same is true of equation (34).

Question 1. Does the homogeneous integral equation

∫_{2}^{P} Im\{\frac{ρ_{ν}}{x^{ρ_{ν}} − 1}\}\frac{Φ(x)}{x} dx = Φ(P) Im\{ln(1 − \frac{1}{P^{ρ_{ν}}} )\}

have a nontrivial solution Φ(x) = Φ(x; γ_{ν}, P)?

5. “EXPLICIT FORMULAS” AND “IMPLICIT FORMULAS”

Recall that there exists a class of “explicit formulas” in the theory based on the function ζ(s). For example, the “explicit formula” of von Mangoldt

ψ_{0}(x) = \frac{ψ(x + 0) + ψ(x − 0)}{2} = x − ∑_{ρ} \frac{x^{ρ}}{ρ} − \frac{ζ′}{ζ} (0) − \frac{1}{2} ln(1 − \frac{1}{x^{2}} ), x > 1,

where ρ runs through all zeros of the function ζ(s) lying in the critical strip, is related to the Chebyshev function ψ(x) (cf. [6, p. 76, 77 and Subsection 7, p. 80]).

In this context, formulas (17) and (32) are called “implicit formulas.”

Remark 6. The “implicit formulas” (17), (32), which connect the functions π(x) and R(x) with any fixed critical zero ρ_{ν} = 1/2 + iγ_{ν}, γ_{ν} ∈ ⟨T, T + H⟩, of the function ζ̃_{−1}(s) and with the corresponding value of t_{ν}(π/2) are formulas of a new type and have no analogs in the theory based on the Riemann zeta-function.

It stands to reason that the Riemann formula

Π_{0}(x) = li x − ∑_{ρ} li x^{ρ} + ∫_{x}^{∞} \frac{du}{(u^{2} + 1)u ln u} − ln 2, x > 1,

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

where

Π_{0}(x) = \frac{Π(x + 0) + Π(x − 0)}{2}, Π(x) = π(x) + \frac{1}{2}π(x^{1/2}) + \frac{1}{3}π(x^{1/3}) + ⋯,

connects Π_{0}(x) with all zeros, namely, with an infinite series of zeros, of the function ζ(s), which lie in the critical strip σ ∈ (0, 1) (see [6, p. 81]).

Remark 7. “Implicit formulas” were obtained irrespective of the ordinary operations in a complex domain, namely, the operation of the displacement of the integration line and the residue theorem.

6. GENERALIZATION OF THE FUNDAMENTAL FORMULAS (17) AND (32)

Suppose now that {γ_{2ν}(α)} is a sequence satisfying the condition (cf. (15))

φ[γ_{2ν}(α)] + C_{−1}[γ_{2ν}(α)] = 2νπ + α,  α ∈ ⟨−π, π⟩  (36)

(the existence of {γ_{2ν}(α)} is a consequence of the increase of the function φ(t) + C_{−1}(t); cf. [4]). Since

\{γ_{2ν}(\frac{π}{2} )\} ∪ \{γ_{2ν}(−\frac{π}{2} )\} = \{γ_{ν}\},

{γ_{2ν}(α)} is a generalization of the sequence {γ_{ν}}.

Note that in the case t = γ_{2ν}(α), formula (13) yields

Z̃_{−1}[γ_{2ν}(α); P] = 2B_{−1}[γ_{2ν}(α); P] cos α.  (37)

Formula (36) gives the following results (cf. Lemmas 2 and 3 and Theorem 1).

Lemma 5. For the adjacent terms of the sequence {γ_{2ν}(α)} we have the asymptotic formula

γ_{2ν+2}(α) − γ_{2ν}(α) = \frac{2π}{ln P_{0}} + O\{\frac{1}{(ln P_{0})^{3/2}} \}, γ_{2ν}(α), γ_{2ν+2}(α) ∈ ⟨T, T + H⟩, T → ∞.

Lemma 6. We have

− ∫_{2}^{P} Im\{\frac{ρ_{2ν}(α)}{x^{ρ_{2ν}(α)} − 1}\}\frac{π(x)}{x} dx + π(P) Im\{ln(1 − \frac{1}{P^{ρ_{2ν}(α)}} )\} = φ[t_{2ν}(α)] − φ[γ_{2ν}(α)], (38)

where {t_{2ν}(α)} satisfies the condition

φ[t_{2ν}(α)] = 2νπ + α

(see [3]), and ρ_{2ν}(α) = 1/2 + iγ_{2ν}(α).

Theorem 2. We have

\begin{matrix}− ∫_{2}^{P} Im\{\frac{ρ_{2ν}(α)}{x^{ρ_{2ν}(α)} − 1}\}\frac{R(x)}{x} dx + R(P) Im\{ln(1 − \frac{1}{P^{ρ_{2ν}(α)}} )\} \\ + O(\frac{\sqrt{T}}{T} ) = φ[t_{2ν}(α)] − φ[γ_{2ν}(α)],  T → ∞. \end{matrix}  (39)

In addition, we note that the Corollary 3 follows from (37) (see (14)).

Corollary 3. For α ≠ ±π/2, the formula

ln\{\frac{Z̃_{−1}[γ_{2ν}(α); P]}{2 cos α}\} = − ∫_{2}^{P} Re\{\frac{ρ_{2ν}(α)}{x^{ρ_{2ν}(α)} − 1}\}\frac{π(x)}{x} dx + π(P) Re\{ln(1 − \frac{1}{P^{ρ_{2ν}(α)}} )\}

holds as well as the corresponding formula for the remainder R(x).

7. DISCRETE THEORY OF THE FUNCTIONS π(x) AND R(x)

Let

⟨m + 1, m + 2) = ⋃_{n=1}^{N} I_{mn}, I_{mn} = ⟨a_{mn}, b_{mn}), N = 1, 2, …,

and let

⟨2, P) = ⋃_{m=1}^{P −2} ⋃_{n=1}^{N} I_{mn} = ⋃_{k=1}^{K_{0}} \overline{I}_{k},  |\overline{I}_{k}| = \overline{b}_{k} − \overline{a}_{k},  (40)

be a subdivision of the interval ⟨2, P) into K_{0} = N(P − 2) nonoverlapping parts. Note that the formula

M_{0} ∼ \frac{1}{π} H ln P_{0}, T → ∞,

holds for the number M_{0} of zeros of the the function ζ̃_{−1}(1/2 + it), t ∈ ⟨T, T + H⟩, (see (16)). Suppose, for example, that H = \sqrt{T}, and let K_{0} satisfy the condition

K_{0} + 1 ≤ (1 − ε)\frac{1}{π} \sqrt{T} ln P_{0},

which, in turn, holds for

N ≤ (1 − 2ε)\frac{1}{π} \frac{\sqrt{T} ln P_{0}}{P − 2} = N_{0}.  (41)

Since π(x), x ∈ ⟨2, P ⟩ preserves the sign, we have

∫_{2}^{P} Im\{\frac{ρ_{ν}}{x^{ρ_{ν}} − 1}\}\frac{1}{x}π(x) dx = ∫_{2}^{P} J(x, γ_{ν})π(x) dx = ∑_{k=1}^{K_{0}} ∫_{\overline{I}_{k}} J(x, γ_{ν})π(x) dx

= ∑_{k=1}^{K_{0}} J(ξ_{k}^{ν}, γ_{ν}) ∫_{\overline{I}_{k}} π(x) dx = ∑_{k=1}^{K_{0}} J(ξ_{k}^{ν}, γ_{ν})|\overline{I}_{k}|π_{k} = − ∑_{k=1}^{K_{0}} 𝔸(ξ_{k}^{ν}, γ_{ν})π_{k},

where

ξ_{k}^{ν} = ξ(\overline{I}_{k}, γ_{ν}), π_{k} = π(m + 1),  \overline{I}_{k} ∈ ⟨m + 1, m + 2).  (42)

Suppose, furthermore, that \{γ_{ν(l)}\}_{l=1}^{K_{0}+1}, γ_{ν(l)} ∈ ⟨T, T + \sqrt{T} ⟩, is any fixed collection of zeros of the

function ζ̃_{−1}(1/2 + it). We associate the corresponding collection of formulas (17) with the system of linear equations

∑_{k=1}^{K_{0}+1} 𝔸[ξ_{k}^{ν(l)}, γ_{ν(l)}]x_{k} = 𝔹[γ_{ν(l)}],  l = 1, …, K_{0} + 1,  (43)

where

\begin{matrix}𝔸[ξ_{k}^{ν(l)}, γ_{ν(l)}] = −\frac{|\overline{I}_{k}|}{ξ_{k}^{ν(l)}} Im\{\frac{ρ_{ν(l)}}{(ξ_{k}^{ν(l)})^{ρ_{ν(l)}} − 1}\},  k = 1, …, K_{0}, \\ 𝔸[ξ_{K_{0}+1}^{ν(l)}, γ_{ν(l)}] = 𝔸[γ_{ν(l)}] = Im\{ln(1 − \frac{1}{P^{ρ_{ν(l)}}} )\}, \\ 𝔹[γ_{ν(l)}] = φ[t_{ν(l)}(\frac{π}{2} )] − φ[γ_{ν(l)}]. \end{matrix}  (44)

Therefore the collection of formulas (17) for π(x) yields

Corollary 4. For any fixed integer N ∈ ⟨2, N_{0}⟩ there exists an infinite number of systems of linear equations of the type (43) (since, in this case, there exists an infinite number of subdivisions (40)). Moreover, each of these systems has an integer-valued solution

\underbrace{π(2), …, π(2)}_{N}, \underbrace{π(3), …, π(3)}_{N}, …, \underbrace{π(P − 1), …, π(P − 1)}_{N}, π(P)  (45)

(see (42)).

This regularity admits the following interpretation. We fix any system (let it be the basic system from the infinite number of systems (43)), and consider each of the remaining systems to be a perturbation of the initial system.

Remark 8. The integer-valued solution (45) to the basic system is an invariant number (a fixed point) with respect to the above-mentioned infinite set of perturbations (transformations) of the basic system.

Remark 9. Thus, the augmented matrix of system (43) which is constructed from the elements of (44) has arithmetic properties, i.e., it generates integers.

Let h[𝔸(S, N)] be the rank of the matrix S (see (43)) for any fixed integer N ∈ ⟨2, N_{0}⟩.

Question 2. Is it true that

min_{S}{h[𝔸(S; N)]} = π(P)?

Remark 10. We can put the system of linear equation of the form (43) into correspondence with

(a) the collection of formulas (32) for R(x), but we have to use the second mean value theorem instead of the first one; R(x) decreases on the interval (m + 1, m + 2),

(b) the systems of generalized formulas (38) and (39) for π(x) and R(x) respectively.

8. INDEPENDENCE OF THE THEORY OF FUNCTIONS π(x) AND R(x) OF THE ZEROS OF THE FUNCTION ζ̃_{−1}(s)

The peculiarities of the model based on the function ζ̃_{−1}(s) are the following facts (A) An analog of the Riemann conjecture for ζ̃_{−1}(s) holds (see Lemma 1), but this fact was not

used (see Remark 7) in the proof of the basic formulas (17), (32), (43) and Corollary 4 for π(x) and R(x) on the crirical line. However, formulas (17), (32), and (43) are related to the critical zeros ρ_{ν} = 1/2 + iγ_{ν}, γ_{ν} ∈ ⟨T, T + H⟩ (a finite set) of the function ζ̃_{−1}(s).

(B) Formulas (38), (39), (43′), and Corollary 4′ (43′) and Corollary 4′, which are analogs of (43) and Corollary 4, (cf. Remark 10) are, in turn,

(a) not connected with the analog of the Riemann conjecture for ζ̃_{−1}(s),

(b) not connected even with the critical zeros ρ_{ν} of the function ζ̃_{−1}(s) (for α ≠ ±π/2),

(c) connected only with an infinite set (of the power of continuum) of the segments of sequences

\{γ_{2ν}(α)\}, α ∈ ⟨−π, π⟩, α ≠ ±\frac{π}{2} ,

which lie in the interval ⟨T, T + H⟩ of the critical line.

(C) What we said in (B) also holds for the integral equations (33′) and (34′) which are analogs of equations (33) and (34).

Recall that the theory of functions π(x) and R(x) based on the (complete) Riemann zetafunction is closely connected with the question concerning the distribution of zeros of the function ζ(s) in the critical strip (of course, there are no analogs of formulas (38), (39), and (43′) and equations (33′), and (34′) in this theory at all).

In the model based on the function ζ̃_{−1}(s), we have a completely opposite case, namely, by virtue of (A)–(C), we make

Remark 11. The theory of functions π(x) and R(x) based of formulas (38), (39), (43′) and

on the integral equations (33′) and (34′) is not connected with the zeros of the function ζ̃_{−1}(s) for α ≠ ±π/2.

Remark made during the correction. Using the formulas from this paper one can prove the orthogonality conditions on the finite closed intervals for the functions π(x) and R(x).

REFERENCES

1. Chebyshev, P. L., Teoriya chisel (Number Theory), Collection of Works, vol. 1, Moscow, Leningrad: Akad. Nauk SSSR, 1946.

2. Vinogradov, I. M., Metod trigonometricheskikh summ v teorii chisel (Method of Trigonometric Sums in Number Theory), Moscow: Nauka, 1980.

3. Mozer, Jan, New Consequences of the Riemann–Siegel Formula, Acta Arith., 1982, vol. 42, pp. 1–10.

4. Mozer, Jan, Prime Numbers and the “Random Walk” of Zeros of Certain Parts of the Riemann–Siegel Formula, Math. Slovaca, 1997, vol. 47, pp. 321–341.

5. Mozer, Jan, Riemann Conjecture for Some Parts of the Function ζ(s) and a New Formula for π(x), Acta Arith., 1997, vol. 80, no. 4, pp. 297–310.

6. Ingham, A. E., The Distribution of Prime Numbers, Cambridge: Cambridge Univ. Press, 1932.

Translated by S. Vakhromeev