# A minimal model for the role of the reaction rate on the initiation and
self-sustenance of curved detonations

Matei Ioan Petru Rădulescu∗, Aliou Sow
Department of Mechanical Engineering, University of Ottawa, Canada

###### Abstract

A minimal model for curved detonations is studied, illustrating the role of
the reaction rate on the detonation speed and its propagation limits. The
model is based on a simple extension of the minimal Fickett toy model for
detonations based on the kinematic wave equation. The use of a simple
depletion rate conditioned on the shock speed serves to illustrate its role in
the quasi-steady structure of curved waves and their initiation from a strong
blast wave. Calculations of strong initiation from a self-similar explosion
illustrate the various asymptotic regimes of the transition to self-sustenance
and their link to the steady wave structure. We recover the asymptotic regimes
of detonation formation suggested by He and Clavin and modelled in the context
of Detonation Shock Dynamics by Stewart and collaborators. Following an
analysis using the shock change equation, we identify a unique criterion that
permits to infer the critical energy for initiation from the competition
between energy release and geometric decay.

##  1 Introduction

The propagation of curved detonation waves is a central problem of detonation
theory. The response of detonation waves to lateral boundary conditions when
propagating in a weakly confined tube invariably leads to curved waves. It has
been established that an excess curvature leads to wave speed decrease, and
possible extinction. The problem has first been treated by Wood and Kirkwood
[1]. They recognized the presence of an internal sonic surface, along which
the rate of exothermicity balances the rate of energy withdrawal due to
divergence. This simultaneous sonic condition and zero net energy evolution at
the sonic surface is the so-called generalized Chapman-Jouguet condition. This
confirmed that flow divergence acts similarly to heat losses, confirming the
earlier findings of Zel’dovich and Kompaneets, who first demonstrated this
generalized CJ condition [2]. The state of the art on the problem of curved
detonations, as well as extensions to treat weakly non-steady effects are
summarized by Bdzil and Stewart [3, 4]. We provide experimental validation of
the models for gaseous detonations in our previous studies where constant
curvature and quasi-steady states were obtained in channels with exponentially
enlarging cross-sectional areas [5, 6, 7, 8].

In the present communication, we wish to illustrate this problem with a
minimal example. Instead of using the Euler equations describing the reactive
hydrodynamics of a reactive fluid, we instead use a reactive analogue problem
first suggested by Fickett [9, 10]. It consists of a kinematic wave equation
coupled with energy release. Using this toy model, we illustrate the role of
the reaction rate liberating energy on the condition for self-sustenance,
yielding the detonation speed - curvature relationship for quasi-steady
propagation. Variations upon this toy model has been very useful in studying
the stability and the non-linear dynamics of detonations [11, 12].

We also wish to study the role played by this relationship in the context of
critically initiated detonations from a strong self-similar blast wave, a long
standing problem in detonation theory [13]. The relative simplicity of the
model serves to illustrate and justify the model of He and Clavin [14],
subsequently refined by Kasimov and Stewart [15], Vidal [16] and Clavin and
Denet [17] to account for transient effects. The analysis of the dynamics are
cast in the modern treatment of this problem using the concept of
characteristics advocated by Kasimov and Stewart [15] and the shock change
equation closure for obtaining evolution equations for the front. We extend
the numerical study of Faria et al. on initiation of detonations in the
context of the toy Fickett model [18].

##  2 The Euler equations for quasi-1D or axi-symetric flow

Curved detonations can be described by the quasi-1D flow equations for
conservation of mass, momentum and energy, which read

$\displaystyle\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial
x}$ $\displaystyle=-\rho u\kappa$ (1) $\displaystyle\frac{\partial\rho
u}{\partial t}+\frac{\partial(\rho u^{2}+p)}{\partial x}$ $\displaystyle=-\rho
u^{2}\kappa$ (2) $\displaystyle\frac{\partial\rho e_{tot}}{\partial
t}+\frac{\partial(\rho ue_{tot}+pu)}{\partial x}$ $\displaystyle=-(\rho
ue_{tot}+pu)\kappa$ (3)

where $e_{tot}=e+\frac{1}{2}u^{2}$ is the total energy, which contains the
reference energies of each component, contributing to the net heat release. We
note that $\kappa$ is the rate of logarithmic area increase of the streamtube
$\kappa=\frac{1}{A}\frac{\mathrm{d}A}{\mathrm{d}x}=\frac{\mathrm{d}\ln
A}{\mathrm{d}x}$ and also the curvature of the front. These equations are also
those for cylindrical and spherical flows, with
$\kappa=\frac{1}{A}\frac{\mathrm{d}A}{\mathrm{d}x}=j/x$, with $j$=1 or 2 for
the cylindrical and spherical geometries respectively.

##  3 Fickett’s model with lateral divergence

Fickett’s model is an extension of the inviscid Burgers equation to model
reactive compressible flow. A single toy model equation instead of three
conservation laws for the Euler equations describes the compressible
hydrodynamics. In the absence of lateral divergence, the model is

$\frac{\partial\rho}{\partial t}+\frac{\partial p}{\partial x}=0$ (4)

where $p$ corresponds to the flux of the quantity for which $\rho$ is its
density. Fickett models the flux by:

$p=\frac{1}{2}\left(\rho^{2}+\lambda Q\right)$ (5)

where $Q$ is the energy release and $\lambda$ is a progress variable 0 in the
reactants and 1 in the products. In doing so, the model equation accounts for
the modification of the flux by the energy release. In the absence of the
reactive term, it is the inviscid Burgers equation. Fickett ascribes the
meaning of $p$ to something akin to pressure. This reflects the notion that
the flux is related to pressure in the real system. The mathematical structure
of this toy model retains the basic physics of the reactive Euler equations,
but it is much simpler to analyze its emerging dynamics. It thus serves very
well its main purpose of a toy model.

This simple partial differential equation has to be supplemented by another
dictating the rate of reaction. Fickett uses

$\frac{\partial\lambda}{\partial t}=r$ (6)

where $r$ is the reaction rate.

Our simple extension to treat curved detonations is to consider the wave
propagation in a channel of enlarging cross section $A(x)$. For such quasi-one
dimensional flow, a simple extension of Fickett’s model (9) is:

$\frac{\partial\rho}{\partial t}+\frac{\partial p}{\partial
x}=-\frac{1}{2}\rho^{2}\kappa$ (7)

This retains the structure of the reactive Euler equations, where a non-linear
term, usually the flux of the conserved quantity (except for momentum) appears
on the right hand side, multiplying the curvature.

In the following problem, we take the rate as

$r=D^{n}\left(1-\lambda\right)$ (8)

where $D$ is the detonation speed. This choice is motivated by the physical
picture where most detonations have a thin induction zone followed by a
hydrodynamic structure, which, one average, is two orders of magnitude longer.
By eliminating the induction zone, we also remove the pulsating instability
that would otherwise obscure the slow dynamics we wish to unravel. We allow
the sensitivity of the reaction rate on the shock strength to be described by
the exponent $n$. The same choice was made by Fickett in his discussion of
steady detonations with lateral divergence, but in which he chose a more
artificial model equation:

$\frac{\partial\rho}{\partial t}+\frac{\partial p}{\partial x}=-\kappa$ (9)

instead of our starting point (7).

##  4 Characteristic form

It is straightforward to cast our problem given by (6) and (7) in
characteristic form:

$\displaystyle\frac{\partial p}{\partial t}+\rho\frac{\partial p}{\partial x}$
$\displaystyle=\frac{1}{2}rQ-\frac{1}{2}\rho^{3}\kappa$ (10)
$\displaystyle\frac{\partial\lambda}{\partial t}$ $\displaystyle=r$ (11)

where the characteristics speeds are respectively $\rho$ and 0. We label the
characteristics in the first equation given by the paths satisfying
$dx/dt=\rho$ as C+ characteristics, by analogy to the physical system.

##  5 The generalized CJ condition based on characteristics

The role of these characteristics for the analogue problem has been discussed
by Radulescu and co-workers [11, 19, 20], in analogy to the physical
system[15]. For a steady traveling wave with speed $D$, characteristics from
the back feed into the lead shock, providing an amplification of the pressure
$p$ dictated by the rate of energy release in the first term on the right hand
side of (10) and a reduction of the pressure by the second term on the right
hand side of (10). The steady structure can thus be simply viewed as the
coherent amplification of forward facing waves by the energy release set up by
the shock and modulated by the geometrical terms. Figure 1 illustrates this
structure.

Figure 1: The structure of a steady travelling detonation wave and the network
of forward facing characteristics sustaining the lead shock.

The limiting characteristic is the characteristic where the flow is sonic in
the frame of the lead shock. It corresponds to the characteristic path
parallel to the lead shock, such that

$\rho_{*}=D\\\ $ (12)

For this characteristic to remain at constant speed, it requires that $p$ also
remains constant along its path. Since $\lambda$ is also a constant along this
path by the constraint provided by the reaction rate in the steady travelling
wave by virtue of (11), (10) requires its right hand side forcing to vanish:

$\frac{1}{2}r_{*}Q-\frac{1}{2}\rho_{*}^{3}\kappa=0$ (13)

The sonic condition (12) and vanishing of the net rate controlling flux
amplification (13) is the generalized Chapman-Jouguet condition.

Using the expressions for the rate (8) and the model flux (5), the balance
given by (13) gives the progress variable at the sonic surface, i.e.,

$\lambda_{*}=1-\frac{D^{3-n}\kappa}{Q}$ (14)

The generalized CJ condition thus limits the amount of energy release felt by
the shock to the fraction $\lambda^{*}$, which is less than unity. This is
given by the balance of the reaction rate and rate of loss at the sonic
surface.

##  6 The inner structure as a BVP

The detonation wave speed is determined from requiring that the internal
steady structure satisfies the post shock state and sonic state
simultaneously. The sonic state is known from above, i.e,
$(\rho=\rho_{*}=D,\lambda=\lambda_{*})$. The post-shock state without energy
evolution is obtained from the weak solution of the inviscid Burgers equation.

$D=\frac{[p]}{[\rho]}$ (15)

where the square brackets denote differences across the shock wave. Since we
take the strong shock approximation with $p=\rho=0$ in the upstream state, the
wave speed relation becomes:

$D=\frac{\frac{1}{2}\rho^{2}}{\rho}=\frac{1}{2}\rho$ (16)

For a given detonation speed $D$, the post-shock density is thus $\rho=2D$. At
the shock, $\lambda=0$.

The steady structure of the detonation is obtained by reverting to a wave-
fixed reference frame and requiring the structure to be steady. Let the new
coordinate system $(\zeta,t^{\prime})$ be related to the $(x,t)$ by:

$\displaystyle\zeta=x-\int^{t}_{0}D(\alpha)\mathrm{d}\alpha$ (17)
$\displaystyle t^{\prime}=t$ (18)

where $D$ is the shock speed. The derivatives appearing in (6) and (7)
transform accordingly as:

$\displaystyle\partial_{x,t}=\partial_{\zeta,t^{\prime}}\partial_{x,t}\zeta+\partial_{t^{\prime},\zeta}\partial_{x,t}t^{\prime}=\partial_{\zeta,t^{\prime}}$
(19)
$\displaystyle\partial_{t,x}=\partial_{\zeta,t^{\prime}}\partial_{t,x}\zeta+\partial_{t^{\prime},\zeta}\partial_{t,x}t^{\prime}=-D\partial_{\zeta,t^{\prime}}+\partial_{t^{\prime},\zeta}$
(20)

Using this coordinate transformation, (6) and (7) become:

$\displaystyle\frac{\partial\rho}{\partial t^{\prime}}$
$\displaystyle-D\frac{\partial\rho}{\partial\zeta}+\frac{\partial
p}{\partial\zeta}=-\frac{1}{2}\rho^{2}\kappa$ (21)
$\displaystyle\frac{\partial\lambda}{\partial t^{\prime}}$
$\displaystyle-D\frac{\partial\lambda}{\partial\zeta}=r$ (22)

The steady travelling wave solution corresponds to taking
$\frac{\partial}{\partial t^{\prime}}=0$. We are left with two ordinary
differential equations:

$\displaystyle-D\frac{\mathrm{d}\rho}{\mathrm{d}\zeta}+\frac{\mathrm{d}p}{\mathrm{d}\zeta}=-\frac{1}{2}\rho^{2}\kappa$
(23) $\displaystyle-D\frac{\mathrm{d}\lambda}{\mathrm{d}\zeta}=r$ (24)

These can be re-written as:

$\displaystyle\frac{\mathrm{d}\rho}{\mathrm{d}\zeta}=\frac{\frac{1}{2}Q\frac{r}{D}-\frac{1}{2}\rho^{2}\kappa}{\rho-D}$
(25) $\displaystyle\frac{\mathrm{d}\lambda}{\mathrm{d}\zeta}=-\frac{r}{D}$
(26)

The problem we thus wish to solve is a boundary value problem given by (25)
and (26), satisfying the boundary conditions at the shock and at the sonic
surface. Cast in this form, we recognize again the requirement for the
generalized CJ condition, requiring the simultaneous sonic flow (denominator
vanishing) and balance of exothermic and endothermic effects (numerator
vanishing) at the sonic surface. While mathematically it is equivalent with
what we already established above using the limiting characteristic criterion,
we prefer the latter, as being more physical.

The rate equation (26) readily integrates to

$\displaystyle\lambda=1-\exp\left(D^{n-1}\zeta\right)$ (27)

from which we can deduce the reaction zone length using (14).

The remaining ODE (25) can be manipulated by multiplying by $\rho-D$, dividing
by (26), and introducing the change of variables

$z=(\rho-D)^{2}$ (28)

such that $(\rho-D)\mathrm{d}\rho=\frac{1}{2}\mathrm{d}z$. It yields:

$\displaystyle\frac{\mathrm{d}z}{\mathrm{d}\lambda}=-Q+\frac{(D+\sqrt{z})^{2}\kappa}{D^{n-1}(1-\lambda)}$
(29)

At the shock, $z=D^{2}$, while $z=0$ at the sonic surface. Using (14), it can
be shown that the sonic locus corresponds to
$\frac{\mathrm{d}z}{\mathrm{d}\lambda}=0$. The sonic point is a degenerate
saddle point, as shown in Fig. 2. For given shock speed, trajectories for less
than critical values $\kappa$ approach the $z=0$ axis on the left of the sonic
point with finite slope. For values of $\kappa$ sufficiently large,
trajectories reach zero gradient prior to reaching the sonic point. The
critical value of $\kappa$ is obtained for trajectories reaching the $z=0$
axis with zero slope. This marks the eigenvalue $\kappa(D)$.

Figure 2: Phase space trajectory of the solution for fixed $D$ and varying
$\kappa$; red circle denotes the sonic point found numerically.

A closed form solution was not possible since the boundary value problem given
by (29) cannot be integrated in closed form. This is usually the case in all
problems of curved detonations. Instead, the BVP can be solved numerically
very simply by a shooting method, using the bevavior near the sonic point as
an indicator. Alternatively, linearization about the saddle point can
determine the initial direction for a backwards shooting method towards the
shock. For the present case, the change of variables from $\rho$ to
$(\rho-D)^{2}$ eliminated the blow-up at the sonic surface, rendering forward
or backward shooting methods equivalent. The change of variables (28) removing
the singularity at the sonic point was first identified by Fickett [10] while
studying a simpler analog problem. It was exploited further by Faria and his
collaborators for a model similar to ours [18] and generalized to the Euler
equations as well, simplifying the numerical determination of the inner
structure of detonations with losses [21].

Figure 3 shows the numerically obtained $D(\kappa)$ curve for different values
of $n$. For $n$ larger than 3, the curves display the characteristic turning
point at the maximum permissible curvature. This is in very good qualitative
agreement with the physical system.

##  7 Fickett’s simplest model for losses and divergence effects

It is worthwhile comparing our results with those of Fickett [10], who
investigated the structure of detonations given by (9):

$\frac{\partial\rho}{\partial t}+\frac{\partial p}{\partial x}=-\kappa$

Proceeding as we have illustrated above, the characteristic form of the
problem is given by

$\displaystyle\frac{\partial f}{\partial t}+\rho\frac{\partial p}{\partial x}$
$\displaystyle=\frac{1}{2}rQ-\rho\kappa$ (30)

instead of (10). From this equation we deduce the generalized CJ condition
$\rho_{*}=D$ and

$\lambda^{*}=1-2\kappa D^{1-n}$ (31)

instead of (14). The eigenvalue $\kappa$ is obtained again by requiring the
integral curve connecting the post shock state and the sonic point. With the
simpler loss term assumed here, we obtain

$\displaystyle\frac{\mathrm{d}z}{\mathrm{d}\lambda}=-Q+\frac{2\kappa}{D^{n-1}(1-\lambda)}$
(32)

instead of (29). The integral curve is readily integrated in closed form in
this case by separation of variables:

$\displaystyle\int_{D^{2}}^{0}\mathrm{d}z=\int_{0}^{\lambda^{*}}\left(-Q+\frac{2\kappa}{D^{n-1}(1-\lambda)}\right)\mathrm{d}\lambda$
(33)

yielding:

$D^{2}=\lambda^{*}Q+\frac{2\kappa}{D^{n-1}}\ln\left(1-\lambda^{*}\right)$ (34)

This shows clearly the two contributions to the velocity deficit in the
presence of curvature. The first term accounts for the incomplete energy
release, while the second accounts for the effect of curvature in modifying
the location of the sonic surface in relation to the shock.

With $\lambda^{*}$ given by (31), the final $D(\kappa)$ dependence is found to
satisfy the transcendental relation

$D^{2}=\left(1-2\kappa D^{1-n}\right)Q+2\kappa D^{1-n}\ln\left(2\kappa
D^{1-n}\right)$ (35)

Figure 3: Steady detonation speed dependence on curvature for different values
of $n$.

##  8 Direct initiation of spherical waves

We now turn to illustrating the utility of concepts used to obtain the steady
detonation curvature response in non -steady situations. We focus on
determining the critical conditions for detonation initiation from the decay
of a strong self-similar blast wave. This is the analogous problem to the
well-known direct initiation problem of detonation, where an initially strong
self-similar Taylor-Sedov blast wave decays in a reactive gas [13]. For
sufficiently large energy deposition, the blast decays but rapidly re-
amplifies to form a detonation. Below a critical value of energy deposition,
the blast does not re-amplify and a detonation is not initiated.

Figure 4: Shock density evolution for blasts of different strengths for $n=4$
and $Q=1$. Figure 5: Space time diagrams for the evolution of the density
field for $x_{s}=1$ ; characteristic trajectories are in red while $\lambda$
contours in black. Figure 6: Space time diagrams for the evolution of the
density field for $x_{s}=2$; characteristic trajectories are in red while
$\lambda$ contours in black. Figure 7: Space time diagrams for the evolution
of the density field for $x_{s}=4$; characteristic trajectories are in red
while $\lambda$ contours in black. Figure 8: The fields $\rho(x)$ (blue),
$\lambda(x)$ (green), $\rho_{s}(x)$ and the steady response (black) for
$x_{s}=1$ at a) $t=1$, b) $t=2$, c) $t=20$ and d) $t=400$; see also animations
as Supplemental Material. Figure 9: The fields $\rho(x)$ (blue), $\lambda(x)$
(green), $\rho_{s}(x)$ and the steady response (black) for $x_{s}=2$ at a)
$t=1$, b) $t=4$, c) $t=30$ and d) $t=400$; see also animations as Supplemental
Material.

The problem we wish to solve is the direct initiation of spherical waves
($j=2$), and the curvature is $\kappa=j/x=2/x$ in (7), re-written here:

$\frac{\partial\rho}{\partial
t}+\frac{\partial\left(\frac{1}{2}\rho^{2}+\frac{1}{2}\lambda
Q\right)}{\partial x}=-\frac{j}{x}\left(\frac{1}{2}\rho^{2}\right)$ (36)

At early times, a point source deposition of mass $m_{0}$ imposes a self-
similar shock decay analogous to the Taylor-Sedov blast wave resulting from a
point source ”energy” deposition. In the burgers case, the shock sphere
conserves the mass integral [22]. Taking the limit $\rho^{2}\gg Q$, the
problem is an inert one at early times. The inert point blast problem for the
Fickett equation admits a very simple self-similar solution [22]given by

$\displaystyle\rho=2D\frac{x}{x_{s}}$ (37)

where $x_{s}(t)$ is the position of the shock. For the spherical problem, the
shock speed decay is given in terms of the total mass $m_{0}$ deposited at the
center by

$\displaystyle D=\frac{m_{0}x^{-3}}{2\pi}$ (38)

The self-similar solution serves as our initial condition.

To explore the complex dynamics embedded in this seemingly simple problem, we
have first obtained numerical solutions to this reactive problem by an
explicit simple finite volume scheme discretizing the flux by the Roe method
[11]. The geometric and reaction terms were integrated by operator splitting.
As initial condition, we impose the self-similar inert solution with a shock
amplitude of $\rho_{s,0}=40$ and vary the location of the shock $x_{s,0}$ to
bracket the critical initiation conditions. We focus on the case $n=4$ and
take without loss of generality $Q=1$.

Figure 4 shows the evolution of the post shock density $\rho_{s}$ for
different blast intensities. Also shown is the steady state response obtained
above, reported here in terms of the radius of curvature. Recall that for a
strong Burgers shock with $\rho=0$ upstream, the post shock density is
$\rho_{s}=2D$. All blasts decay with the expected self-similar decay of
$x^{-3}$ until $\rho^{2}\sim Q$, when the energy release contribution slows
down the decay rate of the blast. For reference, the Chapman Jouguet solution
corresponds to $\rho_{s}=2D_{CJ}=2\sqrt{Q}$.

Depending on the magnitude of the blast, we recover two types of solutions.
The trajectories for blasts originating at a source radius of 4 or greater
(for the fixed strength imposed) decay below the steady response and quickly
re-amplify to form a self-sustained detonation. These asymptote towards the
quasi-steady self-sustained response. For blasts originating at a source
radius below 2, the decay is very long, with re-amplification occurring over
very long times, rapidly tending to very large distances as the blast is
reduced in strength. For example, the $x_{s}=1$ trajectory does not re-amplify
in our domain that is 200 long.

The critical behaviour thus appears to correlate very well with the passage of
the integral curves close to the steady state turning point. Our model is thus
in perfect accord with the results of numerical simulations in the reactive
Euler equations as well [23, 24], as well as Faria’s study of the analog
equations with an induction model [18], in spite of pulsating instabilities
masking the dynamics.

Figure 10: The fields $\rho(x)$ (blue), $\lambda(x)$ (green), $\rho_{s}(x)$
and the steady response (black) for $x_{s}=4$ at a) $t=2$, b) $t=10$, c)
$t=40$ and d) $t=100$; see also animations as Supplemental Material.

The results obtained further substantiate the He & Clavin criterion for direct
initiation [14], whereby the critical energy for initiation can be estimated
by determining which inert blast decay curve intercepts the critical turning
point of the steady-state response. Improvements for a better estimate were
suggested by Short & Bdzil [25] by using Korobeinikov’s blast wave solution
perturbed to account for energy release [26].

While the short answer is that the He & Clavin criterion appears to be valid
in our case, it is worthwhile to dig deeper for a dynamic justification. The
solution to the reactive Burgers problem provides a simpler setting, which
nevertheless incorporates the main physics. Figures 5,6 and 7 provide the
space-time diagrams of the evolution of three trajectories started at $x_{s}$
of 1, 2 and 4, bracketing the critical regime. On the space time diagrams of
the evolution of $\rho$ on the colour density plot, we have overlaid the
trajectories of the characteristics (red lines) and contours of the reaction
progress variable $\lambda$. These space time diagrams are augmented by the
individual profiles of $\rho(x,t)$ and $\lambda(x,t)$ of Figures 8, 9 and 10
and their animations available as supplemental material.

The main difference between the subcritical cases ($x_{s}=1,2$) and the
supercritical case ($x_{s}=4$) is the inversion of the density gradient while
passing on either side of the steady turning point. Recall that the density
also serves as the speed of forward facing characteristics. This can be
understood as follows. For the supercritical regime, the blasts starts out
with a linear positive density gradient imposed by the self-similar solution.
This gradient decays due to the growth of the blast wave (while remaining
linear in the inert solution). Nevertheless, the reactivity tends to steepen
this gradient. As the blast decays and the reactivity decreases, the delay of
forward facing characteristics to reach the front increases as the separation
between the shock and the bulk of the reaction zone increases and the
characteristics’ speed decrease. This flattens the gradient. The subcritical
case corresponds to the more rapid decay of the lead shock due to geometrical
blast decay than the the sustain of the front by pressure waves having
amplified while traversing the reaction zone. The gradient in this case gets
inverted. The situation is thus the competition between pressure waves
amplifying through the reaction zone reaching the shock and the shock decay
from geometrical and recession from the center of symmetry. For the super-
critical case, the gradient never gets inverted; it does not need to as it re-
initiates prior to the inversion of the gradient.

The gradient of the characteristic speed behind the shock is key here, as it
informs on the residence of the pressure waves in the reaction zone, hence
their amplification by virtue of (10) - see also [11, 19] for discussion. When
the gradient passes through zero, the characteristics become in phase with the
reactivity and build up pressure behind the lead shock. This picture of shock
dynamics being directly controlled by the geometrical effects, reactivity and
the gradient of characteristic speeds behind the front can be stated exactly
by projecting the reactive Burger’s equation (36) along the trajectory of the
shock $x_{s}(t)$. What amounts to using the so-called shock change equations
[27], the rate of change of the density behind the shock is written as

$\frac{\mathrm{d}\rho_{s}}{\mathrm{d}t}=\left(\frac{\partial\rho}{\partial
t}\right)_{s}+D\left(\frac{\partial\rho}{\partial x}\right)_{s}$ (39)

The subscript $s$ is to remind us that these are derivatives evaluated
immediately behind the shock. Since the shock jump is $\rho_{s}=2D$, we can
immediately write:

$\left(\frac{\partial\rho}{\partial
t}\right)_{s}=2\dot{D}-D\left(\frac{\partial\rho}{\partial x}\right)_{s}$ (40)

Likewise, the rate of change of the reaction progress variable following the
shock motion can be written as:

$\frac{\mathrm{d}\lambda_{s}}{\mathrm{d}t}=\left(\frac{\partial\lambda}{\partial
t}\right)_{s}+D\left(\frac{\partial\lambda}{\partial x}\right)_{s}$ (41)

At the shock, $\lambda$ is constant, so
$\frac{\mathrm{d}\lambda_{s}}{\mathrm{d}t}=0$. We also have the reaction rate
controlling the heat release via the time partial derivative via (8). We can
hence write:

$\left(\frac{\partial\lambda}{\partial
x}\right)_{s}=-\frac{1}{D}\left(\frac{\partial\lambda}{\partial
t}\right)_{s}=-\frac{r}{D}$ (42)

Using these projections, and the shock jump condition, we can re-write (36) as
the sought shock change equation:

$\frac{\dot{D}}{D}=\frac{1}{4}\frac{Q}{D^{2}}r-\frac{j}{x}D-\frac{1}{2}\left(\frac{\partial\rho}{\partial
x}\right)_{s}$ (43)

This provides a statement of the dynamics of the lead shock, given by the
influence of the heat release (first term on the RHS), the geometrical decay
(second term on the RHS) and the gradient of characteristic speeds behind the
front (third term on the RHS). It is not yet an evolution equation for the
front, since the gradient term needs closure, as it depends on reactivity and
the time-history of the solution interior.

Noting that $\frac{\dot{D}x_{s}}{D^{2}}=\frac{\mathrm{d}\ln D}{\mathrm{d}\ln
x}$, we can re-write this shock change equation as:

$\theta\equiv\frac{\mathrm{d}\ln D}{\mathrm{d}\ln
x}=\frac{1}{4}\frac{Q}{D^{2}}\frac{x}{D}r-j-\frac{1}{2}\frac{x}{D}\left(\frac{\partial\rho}{\partial
x}\right)_{s}$ (44)

In blast wave theory, $\theta$ is called the blast decay coefficient, and
serves to describe the local quasi-self-similar dynamics. On the basis of this
shock change equation, the qualitative description provided above can be put
on stronger footing. At early times, when the energy release term is
negligible, the two last terms control the dynamics. Approximating the
derivative in the last term by its linear profile provides the self-similar
decay. Subsequently, the decay of the blast wave is controlled by the change
in importance of the 1st and 3rd terms. The gradient requiring clusure
requires the knowledge of the gradients in the characteristic speeds behind
the front.

Without solving the problem, it is sufficient to seek the locus where the
characteristic speed gradient vanishes and the zero in blast decay
coefficient. This will be the locus where the reactivity has a strong positive
feedback on the gasdynamic amplification and amplification begins. In the
French expression frequently used by my colleague Ashwin Chinnayya to decribe
such dynamics: Ça Pousse!; which poorly translates to ”it starts to grow”, or
”it starts to self-propel”. For a lack of a better nomination, and since
Frenchmen like acronyms, this locus will be referred to as the ÇaP locus.

Adopting the rate given by (8), the ÇaP locus is given by

$D_{cap}=\left(\frac{4j}{xQ}\right)^{\frac{1}{n-3}}$ (45)

This locus is shown in Fig. 4. It captures very well the locus where the
subcritically initiated detonations start growing again. This locus separates
the slow dynamics attracted by the bottom branch of the steady-state response
curve and the top branch of fast dynamics.

Interestingly, the expression (45) also reflects the important role of the
sensitivity of the reaction rate to shock strength via the exponent $n$. For
$n$ greater than 3, the ÇaP locus flattens out with increasing $n$. This
signifies that the dynamics of the re-initiation in the subcritical regime
become increasingly slower, as the threshold to re-initiate involves ever
increasing travel distances to reach it. This again is in good accord with
existing phenomenology. For $n$ less than 3, the slope of the curve changes
sign and the criticality in direct initiation disappears, in good accord with
phenomenology and the absence of a turning point in the steady state response.

The existence of the ÇaP locus marking the critical re-initiation locus
prompts us to formulate a criterion à la He & Clavin and Kasimov & Stewart to
estimate the critical blast strength conducive to rapid re-initiation. Since
the solution requires connecting the trajectories of the inert blast wave with
the re-initiation ÇaP locus, by inspection of Fig. 4, this occurs in the
vicinity of the CJ solution. Formally matching them at the CJ solution
$D=D_{CJ}=\sqrt{Q}$, we obtain an estimate for the critical source strength:

$m_{3,crit}=2\pi 8^{3}Q^{\frac{4-3n}{2}}$ (46)

The dynamics modeled by our ÇaP criterion is close in spirit to the model put
forward by Kasimov & Stewart [15] and Vidal [16] for critically initiated
detonation waves modelled by weak curvature and non-steadiness. In their
model, which they do close approximately, they ascribe importance to the
dynamics of a limiting characteristic isolating the dynamics of the front from
the back. The description requires the notion of this limiting characteristic
to persist in the dynamics. For the subcritical case of Fig. 5, all
characteristics feed into the lead shock, albeit after a very long time. For
the re-initiating cases of Fig. 6 and Fig. 7, there is a characteristic that
never reaches the front present in the reaction zone. It would be of interest
to re-formulate the Kasimov and Stewart model for the simplified Fickett
problem presented here in order to assess its accuracy in capturing its
dynamics. Particularly, the model predicts a separatrix. It is not clear at
present how this separatrix connects to our ÇaP locus obtained on the basis of
the shock change equation applied at the shock.

##  9 Conclusion

In closing, the results obtained using our toy model illustrate well how the
reactivity and its sensitivity to the shock controls the speed of steady
curved waves and their initiation dynamics. The existence of a limiting
characteristic in the steady problem permits to establish the fraction of
energy release ”lost” after the sonic surface. This fraction increases with
increasing loss rate. Indeed, it is given by equating the loss rate to the
energy release rate at the sonic surface. The second contribution to the speed
deficit and turning point comes from the lengthening of the reaction zone
structure in the presence of losses. In the transient case of direct
initiation of a detonation from a strong self-similar blast wave, we have
shown how the reactivity competition with geometrical effects control the re-
initiation locus, which rapidly extends to very large distances once the
detonation decays in the subcritical regime. The importance of a favorable
characteristic speed gradient for re-initiation was emphasized. It would be of
great interest to verify whether the present non-steady description applies to
the transient case of the Euler equations.

## Acknowledgments

This work was supported by AFOSR grant FA9550-23-1-0214, with Dr. Chiping Li
as program monitor. The author also acknowledges financial support from NSERC
Discovery Grant ”Predictability of detonation wave dynamics in gases:
experiment and model development”.

## Supplementary material

Video animations of the results presented in Figs. 8, 9 and 10 are provided as
supplementary material.

————————————————————

## References

  * [1] W. W. Wood and J. G. Kirkwood, “Diameter Effect in Condensed Explosives. The Relation between Velocity and Radius of Curvature of the Detonation Wave,” _The Journal of Chemical Physics_ , vol. 22, no. 11, pp. 1920–1924, 12 1954\.
  * [2] I. B. Zeldovich and A. S. Kompaneets, _Theory of detonation_. Acadcemic Press, 1960.
  * [3] J. B. Bdzil and D. S. Stewart, “The dynamics of detonation in explosive systems,” _Annual Review of Fluid Mechanics_ , vol. 39, no. 1, pp. 263–292, 2007.
  * [4] J. Bdzil and D. Stewart, _Detonation Dynamics_ , ser. Shock Wave Science and Technology Reference Library. Springer, 2012, vol. 6, ch. Theory of Detonation Shock Dynamics.
  * [5] M. I. Radulescu and B. Borzou, “Dynamics of detonations with a constant mean flow divergence,” _Journal of Fluid Mechanics_ , vol. 845, pp. 346–377, 2018\.
  * [6] Q. Xiao and M. I. Radulescu, “Dynamics of hydrogen-oxygen-argon cellular detonations with a constant mean lateral strain rate,” _Combustion and Flame_ , vol. 215, pp. 437 – 457, 2020.
  * [7] ——, “Role of instability on the limits of laterally strained detonation waves,” _Combustion and Flame_ , vol. 220, pp. 410–428, 2020.
  * [8] F. Zangene, Q. Xiao, and M. Radulescu, “Critical diffraction of irregular structure detonations and their predictability from experimentally obtained $D$-$\kappa$ data,” _Proceedings of the Combustion Institute_ , vol. 39, no. 3, pp. 2935–2944, 2023.
  * [9] W. Fickett, “Detonation in miniature,” _American Journal of Physics_ , vol. 47, no. 12, pp. 1050–1059, 1979.
  * [10] ——, _Introduction to Detonation Theory_. Berkeley: University of California Press, 1985.
  * [11] M. I. Radulescu and J. Tang, “Nonlinear dynamics of self-sustained supersonic reaction waves: Fickett’s detonation analogue,” _Physical Review Letters_ , vol. 107, no. 16, p. 164503, 2011.
  * [12] A. R. Kasimov, L. M. Faria, and R. R. Rosales, “Model for shock wave chaos,” _Phys. Rev. Lett._ , vol. 110, p. 104104, Mar 2013.
  * [13] J. H. S. Lee, _Direct initiation of detonations_. Cambridge University Press, 2008, p. 297–372.
  * [14] L. He and P. Clavin, “On the direct initiation of gaseous detonations by an energy source,” _Journal of Fluid Mechanics_ , vol. 277, p. 227–248, 1994\.
  * [15] A. R. Kasimov and D. S. Stewart, “Asymptotic theory of evolution and failure of self-sustained detonations,” _Journal of Fluid Mechanics_ , vol. 525, p. 161–192, 2005.
  * [16] P. Vidal, “Critical slow dynamics of detonation in a gas with non-uniform initial temperature and composition: A large-activation-energy analysis,” _International Journal of Spray and Combustion Dynamics_ , vol. 1, no. 4, pp. 435–471, 2009.
  * [17] P. Clavin and B. Denet, “Analytical study of the direct initiation of gaseous detonations for small heat release,” _Journal of Fluid Mechanics_ , vol. 897, p. A30, 2020.
  * [18] L. M. Faria and A. R. Kasimov, “Qualitative modeling of the dynamics of detonations with losses,” _Proceedings of the Combustion Institute_ , vol. 35, no. 2, pp. 2015–2023, 2015.
  * [19] J. Tang and M. I. Radulescu, “Dynamics of shock induced ignition in Fickett’s model: Influence of $\chi$,” _Proceedings of the Combustion Institute_ , vol. 34, pp. 2035–2041, 2013.
  * [20] S. Lau-Chapdelaine, F. Zhang, and M. Radulescu, “Multiplicity of detonation regimes in systems with a multi-peaked thermicity,” _Combustion Theory and Modelling_ , vol. 0, no. 0, pp. 1–26, 2019.
  * [21] R. Semenko, L. Faria, A. Kasimov, and B. Ermolaev, “Set-valued solutions for non-ideal detonation,” _Shock Waves_ , vol. 26, p. 141–160, 2016.
  * [22] M. I. Radulescu, “Self-similar shock dynamics satisfying the inviscid burgers equation in planar, cylindrical and spherical problems,” arXiv:2311.09909 [physics.flu-dyn], Tech. Rep., 2023.
  * [23] L. He, “Theoretical determination of the critical conditions for the direct initiation of detonations in hydrogen-oxygen mixtures,” _Combustion and Flame_ , vol. 104, no. 4, pp. 401–418, 1996.
  * [24] S. D. Watt and G. J. Sharpe, “Linear and nonlinear dynamics of cylindrically and spherically expanding detonation waves,” _Journal of Fluid Mechanics_ , vol. 522, p. 329–356, 2005.
  * [25] M. Short and J. B. Bdzil, “Propagation laws for steady curved detonations with chain-branching kinetics,” _Journal of Fluid Mechanics_ , vol. 479, p. 39–64, 2003.
  * [26] V. Korobeinikov, _Problems of Point Blast Theory_. American Institute of Physics Melville, NY, 1991.
  * [27] M. Radulescu, “On the shock change equations,” _Physics of Fluids_ , vol. 32, no. 5, 2020.


# Some remarks on sum of Euler’s totient function

Es-said En-naoui
<EMAIL_ADDRESS>

###### Abstract

Euler’s totient function counts the positive integers up to a given integer
$n$ that are relatively prime to $n$. The aim of this article is to give a
result about the sum : $\sum\limits_{\underset{p|k}{k=1}}^{n}\phi(k)$ , for
every prime number $p$ .

Keywords : Euler’s totient , Dirichlet product, additive function ,
multiplicative function , Dirichlet series.

##  1 Introduction

Euler totient function $\phi$ is the function defined on the positive natural
numbers $\mathbb{N}^{*}$ in the following way: if $n\in\mathbb{N}^{*}$, then
we have :

$\phi(n)=\big{\\{}x\in\mathbb{N}^{*}:1\leq x\leq n\;,\;\gcd(x,n)=1\big{\\}}.$

Clearly $\phi(p)=p-1$, for any prime number $p$ and, more generally, if
$\alpha\in\mathbb{N}^{*}$, then $\phi(p^{\alpha})=p^{\alpha}-p^{\alpha-1}$.
This follows from the fact that the only numbers which are not coprime with
$p^{\alpha}$ are multiples of $p$ and there are $p^{\alpha-1}$ such multiples
$x$ with $1\leq x\leq p^{\alpha}$.

It is well-known that $\phi$ is multiplicative, i.e., if $m$ and $n$ are
coprime, then $\phi(mn)=\phi(m)\phi(n)$. If $n\geq 3$ and the prime
decomposition of $n$ is

$n=p_{1}^{\alpha_{1}}\ldots p_{s}^{\alpha_{s}},$

then from what we have seen

$\phi(n)=\prod_{i=1}^{s}(p_{i}^{\alpha_{i}}-p^{\alpha_{i}-1})=n\prod_{i=1}^{s}(1-\frac{1}{p_{i}}).$

.

The following property is simple but very useful in this paper.

######  Proposition 1.

If $p,m\in\mathbb{N}^{*}$, with $p$ prime, and $n=pm$, then
$\phi(n)=(p-1)\phi(m)$, if $\gcd(p,m)=1$, and $\phi(n)=p\phi(m)$, if
$\gcd(p,m)\neq 1$.

###### Proof.

If $\gcd(p,m)=1$, then we have

$\phi(n)=\phi(p)\phi(m)=(p-1)\phi(m).$

Now suppose that $\gcd(p,m)\neq 1$. We may write $m=p^{\alpha}m^{\prime}$,
with $\alpha\geq 1$ and $\gcd(p,m^{\prime})=1$. Thus

$\phi(n)=\phi(p^{\alpha+1})\phi(m^{\prime})=p^{\alpha}(p-1)\phi(m^{\prime}).$

However

$\phi(m)=p^{\alpha-1}(p-1)\phi(m^{\prime})$

and so

$\phi(n)=p\phi(m)$

. ∎

##  2 Calcul the sum $\sum\limits_{\underset{p|k}{k=1}}^{n}\phi(k)$

Let $x$ be a real number , then $\lfloor x\rfloor$ is defined to be the unique
integer $n$ satisfying $n\leq x<n+1$.This integer $\lfloor x\rfloor$ is called
the floor of $x$. Then we have rhis results :

######  Corollary 2.

Let $m$ be an integer, and let $x$ be a real number. Then, $m\leq x$ holds if
and only if $m\leq\lfloor x\rfloor$ holds.

###### Proof.

Recall that $\lfloor x\rfloor$ is the unique integer $n$ satisfying $n\leq
x<n+1$.
Thus, $\lfloor x\rfloor$ is an integer satisfying $\lfloor x\rfloor\leq
x<\lfloor x\rfloor+1$ .
If $m\leq x$ holds, then $m\leq\lfloor x\rfloor$ holds . Conversely, if
$m\leq\lfloor x\rfloor$ holds, then $m\leq x$ (because $m\leq\lfloor
x\rfloor\leq x$). Combining these two implications, we conclude that $m\leq x$
holds if and only if $m\leq\lfloor x\rfloor$ holds. Corollary 2 is thus
proven. ∎

######  Proposition 3.

Let $x\in\mathbb{R}$ and $m\in\mathbb{N}^{*}$. Then ,
$\big{\lfloor}\frac{\lfloor
x\rfloor}{m}\big{\rfloor}=\big{\lfloor}\frac{x}{m}\big{\rfloor}$.

###### Proof.

By the definition of $\lfloor x\rfloor$, that
$\big{\lfloor}\frac{x}{m}\big{\rfloor}$ is the unique integer $n$ satisfying
$n\leq\frac{x}{m}<n+1$.Thus, $\big{\lfloor}\frac{x}{m}\big{\rfloor}$ is an
integer satisfying
$\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq\frac{x}{m}<\big{\lfloor}\frac{x}{m}\big{\rfloor}+1$
.
But $m\in\mathbb{N}^{*}$ and thus $m\geq 1>0$ . Hence, we can multiply the
inequality $\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq\frac{x}{m}$ by m. We
thus obtain $m\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq x$. But the Corollary
2 (applied to $m\big{\lfloor}\frac{x}{m}\big{\rfloor}$ instead of $m$) shows
that $m\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq x$ holds if and only
$m\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq\lfloor x\rfloor$ holds (since
$m\big{\lfloor}\frac{x}{m}\big{\rfloor}$ is an integer). Thus,
$m\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq\lfloor x\rfloor$ holds (since
$m\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq x$ holds ).Dividing this
inequality by $m$, we obtain
$\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq\frac{\lfloor x\rfloor}{m}$. We can
now divide the inequality $\lfloor x\rfloor\leq x$ by $m$ (since $m>0$).We
thus obtain $\frac{\lfloor x\rfloor}{m}\leq\frac{x}{m}$.
Hence , $\frac{\lfloor
x\rfloor}{m}\leq\frac{x}{m}<\big{\lfloor}\frac{x}{m}\big{\rfloor}+1$.So we
have $\big{\lfloor}\frac{x}{m}\big{\rfloor}\leq\frac{\lfloor
x\rfloor}{m}<\big{\lfloor}\frac{x}{m}\big{\rfloor}+1$.In other
words,$\big{\lfloor}\frac{x}{m}\big{\rfloor}$ is an integer $n$ satisfying
$n\leq\frac{\lfloor x\rfloor}{m}<n+1$.But $\big{\lfloor}\frac{\lfloor
x\rfloor}{m}\big{\rfloor}$ is the unique integer $n$ satisfying
$n\leq\frac{\lfloor x\rfloor}{m}<n+1$ (because this is how
$\big{\lfloor}\frac{\lfloor x\rfloor}{m}\big{\rfloor}$ is defined).Hence, if
$n$ is any integer satisfying $n\leq\frac{\lfloor x\rfloor}{m}<n+1$, then
$n=\big{\lfloor}\frac{\lfloor x\rfloor}{m}\big{\rfloor}$.We can apply this to
$n=\big{\lfloor}\frac{x}{m}\big{\rfloor}$ (since
$\big{\lfloor}\frac{x}{m}\big{\rfloor}$ is an integer $n$ satisfying
$n\leq\big{\lfloor}\frac{x}{m}\big{\rfloor}<n+1$), and thus obtain
$\big{\lfloor}\frac{\lfloor
x\rfloor}{m}\big{\rfloor}=\big{\lfloor}\frac{x}{m}\big{\rfloor}$.Proposition 3
is proven . ∎

######  Proposition 4.

Let $n$ a positive integer and $p$ a number prime, put
$f(n)=\big{\lfloor}\frac{n}{p}\big{\rfloor}$ , Then we have :

$\forall k\in\mathbb{N^{*}}\;\;,\;\;\underbrace{fo\cdots
of}_{k\;\;times}(n)=\bigg{\lfloor}\frac{n}{p^{k}}\bigg{\rfloor}$ (1)

###### Proof.

we prove by induction on k that , then When $k=1$, it is clear that
$f(n)=\big{\lfloor}\frac{n}{p}\big{\rfloor}$.
Assume that $\underbrace{fo\cdots
of}_{k\;\;times}(n)=\bigg{\lfloor}\frac{n}{p^{k}}\bigg{\rfloor}$. Then we have
:

$\displaystyle\underbrace{fo\cdots of}_{k+1\;times}(n)=fo\underbrace{fo\cdots
of}_{k\;times}(n)=f\bigg{(}\frac{\big{\lfloor}\frac{n}{p^{k}}\big{\rfloor}}{p}\bigg{)}$

We let $x=\big{\lfloor}\frac{n}{p^{k}}\big{\rfloor}$ to substitute into
proposition 3, then We will find :

$\underbrace{fo\cdots
of}_{k+1\;times}(n)=\bigg{\lfloor}\frac{n}{p^{k+1}}\bigg{\rfloor}$

∎

Now after we give some property useful in this paper ,then , let n be a
positive integer ,and put :$\Psi(n)=\sum\limits_{k=1}^{n}\phi(k)$ , So for
every prime number $p$ we put :

$\Upsilon(n,p)=\sum\limits_{\underset{(k,p)=1}{k=1}}^{n}\phi(k)\;\;\;and\;\;\;\Delta(n,p)=\sum\limits_{\underset{p|k}{k=1}}^{n}\phi(k)$

Then we have :

$\Psi(n)=\Upsilon(n,p)+\Delta(n,p)$ (2)

######  Theorem 5.

For every positive integer $n\geq 2$, and a prime number $p$ , we have :

$\Delta(n,p)=(p-1)\sum_{\alpha=1}^{\big{\lfloor}\log_{p}(n)\big{\rfloor}}\Psi\bigg{(}\bigg{\lfloor}\frac{n}{p^{\alpha}}\bigg{\rfloor}\bigg{)}$
(3)

###### Proof.

we have :

$\displaystyle\sum\limits_{\underset{p|k}{k=1}}^{n}\phi(k)$
$\displaystyle=\phi(p)+\phi(2p)+\cdots+\phi(jp)\;\;,\;with\;\;j\leq\big{\lfloor}\frac{n}{p}\big{\rfloor}$
$\displaystyle=\sum_{j=1}^{\big{\lfloor}\frac{n}{p}\big{\rfloor}}\phi(jp)=\sum\limits_{\underset{p|j}{j=1}}^{\big{\lfloor}\frac{n}{p}\big{\rfloor}}\phi(jp)+\sum\limits_{\underset{(p,j)=1}{j=1}}^{\big{\lfloor}\frac{n}{p}\big{\rfloor}}\phi(jp)$

From the proposition 1 , we have $\phi(jp)=(p-1)\phi(j)$ if $(j,p)=1$ and
$\phi(jp)=p\phi(j)$ if $(j,p)>1$ then :

$\displaystyle\sum\limits_{\underset{p|k}{k=1}}^{n}\phi(k)$
$\displaystyle=p\sum\limits_{\underset{p|j}{j=1}}^{\big{\lfloor}\frac{n}{p}\big{\rfloor}}\phi(j)+(p-1)\sum\limits_{\underset{(p,j)=1}{j=1}}^{\big{\lfloor}\frac{n}{p}\big{\rfloor}}\phi(j)$
$\displaystyle=p\Delta\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}+(p-1)\Upsilon\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}$
$\displaystyle=p\Delta\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}+p\Upsilon\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}-\Upsilon\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}$
$\displaystyle=p\bigg{(}\underbrace{\Delta\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}+\Upsilon\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}}_{=\Psi\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor}\big{)}\;\;by\;\;\ref{equa1}}\bigg{)}-\Upsilon(\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}$
$\displaystyle=p\Psi\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor}\big{)}-\Upsilon\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}$

Since
$\Upsilon\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}=\Psi\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor}\big{)}-\Delta\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}$
by equality 2 , Then :

$\displaystyle\Delta(n,p)$
$\displaystyle=p\Psi\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor}\big{)}-\bigg{(}\Psi\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor}\big{)}-\Delta\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}\bigg{)}$

so that :

$\Delta(n,p)=(p-1)\Psi\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor}\big{)}+\Delta\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}$
(4)

In the equality 4, Substituting $n$ by $\big{\lfloor}\frac{n}{p}\big{\rfloor}$
, and by using the proposition 4 we have :

$\Delta\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}=(p-1)\Psi\bigg{(}\big{\lfloor}\frac{n}{p^{2}}\big{\rfloor}\bigg{)}+\Delta\bigg{(}\big{\lfloor}\frac{n}{p^{2}}\big{\rfloor},p\bigg{)}$
(5)

Now , Substituting this
$\Delta\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor},p\big{)}$ into $4$ we find
that:

$\Delta(n,p)=(p-1)\Psi\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor}\big{)}+(p-1)\Psi\big{(}\big{\lfloor}\frac{n}{p^{2}}\big{\rfloor}\big{)}+\Delta\big{(}\big{\lfloor}\frac{n}{p^{2}}\big{\rfloor},p\big{)}$
(6)

In the same way if we repeat this operation up to an integer $\alpha$ such
that $\big{\lfloor}\frac{n}{p^{\alpha}}\big{\rfloor}\leq 1$ finds this
equality :

$\displaystyle\Delta(n)$
$\displaystyle=(p-1)\Psi\big{(}\big{\lfloor}\frac{n}{p}\big{\rfloor}\big{)}+(p-1)\Psi\big{(}\big{\lfloor}\frac{n}{p^{2}}\big{\rfloor}\big{)}+(p-1)\Psi\big{(}\big{\lfloor}\frac{n}{p^{3}}\big{\rfloor}\big{)}+\cdots+\Psi\big{(}\big{\lfloor}\frac{n}{p^{\alpha}}\big{\rfloor}\big{)}$
$\displaystyle=(p-1)\sum_{\alpha=1}^{\big{\lfloor}\log_{p}(n)\big{\rfloor}}\Psi\big{(}\big{\lfloor}\frac{n}{p^{\alpha}}\big{\rfloor}\bigg{)}$

∎

######  Remark 6.

For every prime number $p$ we have :

$\frac{\Delta(pn,p)}{p-1}=\Psi(n)+\Delta(n,p).$

So for $p=2$ , we have :

$\Delta(2n,2)=\Psi(n)+\Delta(n,2).$

## References

  * [1] Lehmer D.H.,  On Euler’s totient function, Bull. Amer.  Math. Soc., 38, 745-751(1932).
  * [2] Childs L.N.,  A Concrete Introduction to Higher Algebra 3rd edition,  Springer 2008.
  * [3] Tom M. Apostol,  Introduction to Analytic Number Theory,  SpringerVerlag, New York, Inc, USA, 1976 .
  * [4] L. Dickson,  History of the theory of numbers. Vol. I: Divisibility and primality,  Chelsea Publishing Co., New York, 1966 .