text:
\section{Introduction}
Image exchanges represent a large amount of Internet usage nowadays. This trend goes hand in hand with privacy requirements since the transmission can be spied on public channels. Therefore, it has been proposed to encrypt these images in order to hide their content, making them visually confidential to unauthorized users. Some encryption methods have been specifically designed for images in order to preserve their format and their size and allowing their visualization after encryption. Allowing visualization is interesting to let users being able to see that an image is present, but its access is restricted. Moreover, selective encryption, which only encrypts a fraction of image information, allows us to visualize a level of details of the image as a function of the encrypted information~\cite{van2002techniques}. In addition, visualization may be authorized only on a certain part of the image. Encryption can be then done partially, for example only on human faces, for privacy concerns. In this context, partial encryption can be selective~\cite{rodrigues2006selective}. Nevertheless, for end users such as cloud platforms or image based social networks, encrypted images are not convenient to work with. Indeed, using a classical encryption scheme, the targeted platform is not able to decide whether an image respects the terms of usage or not. In particular, it cannot check its integrity as this is done in the clear domain~\cite{christlein2012evaluation}. In order to preserve privacy while enabling analysis in the encrypted domain, homomorphic encryption has been proposed. This approach can be used for SIFT detection for example~\cite{hsu2011homomorphic}. However, homomorphic encryption schemes are computationally intensive, which avoids complex operations from being carried out, and requires more storage.
On the contrary selective encryption is fast and does not expand the original image size. With such an approach, a part of the image content is encrypted, while the other one remains in clear, {\em i.e.} non-encrypted, and can be then analyzed. This could introduce a security breach and image content privacy is thus questionable.

\begin{figure}[!t]
\begin{center}
\resizebox{\columnwidth}{!}{
\input{diagram}
}
\end{center}
\vspace{-2mm}
\caption{Trade-off between privacy preservation and integrity check in the context of selectively encrypted image exchanges throughout a public server.}
\label{fig:context}
\end{figure}

In this paper, we study how it is possible to use the framework of selective encryption in order to reach a trade-off between privacy preservation and integrity check.
An illustration of an application scenario on a public server is depicted in Fig.~\ref{fig:context}. From original images, several bit-planes are encrypted, from most significant to least significant bits. A forensics analysis based on the extraction of SRMQ1 features is then conducted to detect if a selectively encrypted image has been tampered or not. In addition, a privacy evaluation is carried out in order to assess the visual confidentiality of a selectively encrypted image. This is done in terms of recognizability by predicting the image class.

The rest of this paper is organized as follows. Section~\ref{sec:method} describes our proposed approach to analyze the influence of selective encryption on both image forensics and privacy preserving mechanisms. Experimental results are presented in Section~\ref{sec:results}. Finally, the conclusion is drawn in Section~\ref{sec:ccl}.

\section{Proposed approach\label{sec:method}}
In this section, we describe our proposed approach to analyze the trade-off between privacy and tampering detection in the context of selective encryption. Selective encryption consists to encrypt the most significant bit-planes (MSB) of an image, while keeping the least significant bit-planes (LSB) in clear. In order to perform forensics on selectively encrypted images, we focus on the residual information of the image
Moreover, for privacy evaluation, we are interested in assessing the recognizability of a selectively encrypted image by automatically predicting the class of the content. Note that our approach is detailed for an application to gray level images, but can be easily extended to RGB color images.

\subsection{Selective encryption}
Let us consider a gray level input image of $m\times n$ pixels. Each pixel $p(i,j)$ from this image, $0\leq i < m$ and $0\leq j < n$, is made of 8 bits and defined as:
\begin{equation}
p(i,j)= \sum_{k=0}^{7}p^k(i,j)\times 2^{7-k},
\end{equation}
where $p^k(i,j)$ is the bit of index $k$.

One can note that the smaller the index $k$, the more significant the associated bit. For privacy requirements, the input image is encrypted in order to ensure the visual security of its content. Moreover, depending on the application, it should be interesting to be able to preserve a part of the image in clear. In this context, encryption is selectively performed. Only a fixed number $s$ of bit-planes are encrypted and the remaining $8-s$ ones are kept in clear. Encryption is then performed from the most ($k=0$) to the least ($k = s-1$) significant bit-plane to encrypt (from MSB to LSB). An encryption key is used as a seed for a cryptographically secure pseudo-random number generator to obtain a pseudo-random sequence of $s\times m\times n$ bits $b^k(i,j)$, with $0\leq k < s$. For each bit-plane to encrypt, each bit $p^k(i,j)$ is XOR-ed with the associated bit in the pseudo-random sequence to generate an encrypted bit $p^k_E(i,j)$:
\begin{equation}
p^k_E(i,j) = p^k(i,j) \oplus b^k(i,j).
\end{equation}

\subsection{Tampering detection using residuals}
Tampering detection aims to decide whether or not an image has been altered by local modification. Most common forgeries are cloning (copy/move from a single image) and splicing (copy/paste between several images). If typical image forensics techniques use as inputs the whole image to be analyzed, this strategy is not the best for encrypted images since the encryption adds a noise of strong magnitude. This noise could also alter the extraction of significant features for a classification as authentic or tampered.

In order to perform a forensics analysis from selectively encrypted images, all the encrypted bit-planes, of index \mbox{$0\leq k < s$}, should be discarded in a pre-processing step. According to the Kerckhoffs' principle, we can assume that the number $s$ of encrypted bit-planes is known.  Therefore, for each pixel $p_E(i,j)$, the encrypted bits are set to zero using bitwise shift operations to obtain a value $p_0(i,j)$:
\begin{equation}\label{eqn:zeros}
p_0(i,j) = (p_E(i,j) \ll s) \gg s.
\end{equation}

In doing so, only the non-encrypted least significant bit-planes are considered. One can note that these bit-planes should be the most relevant for the classification task because they are directly linked to the image residuals. Steganalysis domain falls within the search of weak signals in image residuals. Due to the intrinsic properties of traces left by image forgery, steganalysis approaches can be applied to image forensics~\cite{chierchia2014bayesian,cozzolino2014image}.

In this context, one of the most popular feature extractor is the Spatial Rich Model (SRM)~\cite{fridrich2012rich}. Because it uses the statistics of neighboring noise residuals, it is widely employed for steganalysis, but can be also used for tampering detection. Indeed, noise residuals correspond to high frequency components of an image. They capture the dependency changes due to the tampering operation, in both horizontal and vertical directions. The SRM begins by the computation of the residuals. During this step, the input image is filtered by several high-pass filters to generate residual images with different shapes and orientations. After that, a quantization and a truncation steps are performed. Finally, an output feature vector with 37,561~residuals is obtained, whatever the size of the input image. The main drawback of SRM is that it leads to a high computational complexity. Therefore, in order to deal with this issue, a simplified version called SRMQ1 can be used instead. With this feature extractor, the output feature vector only contains 12,753~residuals.

For classification, an implementation of ridge regression using Least Square Minimum-Residual (LSMR) optimization method is used, due to its
low computational complexity and low memory requirements~\cite{fong2011lsmr,cogranne2015ensemble}. Two classes are considered for classification: authentic, {\em i.e.} with no falsification, and tampered, when there are forgeries due to cloning or splicing operations.

\begin{figure*}[ht]
\centering
\includegraphics[width=\linewidth]{illu}
\vspace{-2mm}
\caption{Illustration using the luminance component of the original image {\em Au\_ani\_00001} from the CASIA2 database~\cite{dong2013casia}: first row)~Selectively encrypted images depending on the number $s$ of encrypted bit-planes, from MSB to LSB and with $1\leq s\leq 7$; second row)~Images obtained by setting to zero the $s$~encrypted bit-planes of associated selectively encrypted images (images were standardized for the recognizability task; this also allows a better visualization of the significant information for classification).\label{fig:ex}}
\end{figure*}

\subsection{Privacy evaluation of selectively encrypted images\label{subsec:privacy}}
The selective encryption allows us to hide some levels on details of the image. The proposed tampering detection relies on the LSB that are not encrypted. Using a non full encryption method could lead to privacy leak using the clear content of the image. Therefore, we aim to know what is the trade-off between the visual confidentiality, which assesses the privacy, and the tampering detection.

The assessment of the visual confidentiality of an image is a difficult task. Indeed, it is known that usual quality metrics, such as PSNR or SSIM, are not relevant for assessing a perceptual low quality. A low score does not point out if the image has just a low quality or if the content can not be recognized \textit{i.e.} if the encryption preserves the privacy. Some perceptual metrics based on subjective evaluations were proposed. In the context of privacy evaluation, the main drawback of these metrics is that they focus on perceptual quality. Recently, Hofbauer~{\it et al.}~\cite{HOFBAUER2021128} have proposed an encrypted image database with subjective recognition ground truth and analyzed the correlation of subjective scores with some state of the art metrics. They conclude that the evaluation of image quality and the evaluation of the content recognizability are two really different tasks and, therefore, the visual quality metrics should not be used to assess content recognizability.

We propose to evaluate the privacy of a selectively encrypted image by trying to automatically predict its class. Our assumption is that if an algorithm can automatically predict the content of an image, then the encrypted image is leaking some visual information\textit{ i.e.}.
Image content classification methods have achieved high performance thanks to convolutional neural networks (CNN). Therefore, we propose to assess the recognizability of selectively encrypted image by training a model which predicts the class of the image.

As for forensics analysis, the $s$ encrypted bitplanes should be set to zero using Eq.~\eqref{eqn:zeros}. This boils down to work directly on high frequencies and short intensity range. In order to exploit the low dynamic, images are standardized before to be passed as input to the model.

Finally, to assess the classification score we use the accuracy score which measures how much data have been well predicted. As visual confidentiality is inherently linked to recognizability, we propose to define a privacy index such as $1 - accuracy_\text{\em~recognizability}$. Indeed, the more easily the content of an image is recognizable, the lower the level of visual confidentiality: recognizability and privacy are consequently antagonist.

\begin{comment}
\renewcommand{\arraystretch}{1.25}
\begin{table*}[ht]
\centering
\caption{\label{tab:accuracy1} Accuracy for tampering detection using SRM and SRMQ1 as a function of the number of encrypted bitplanes (from MSB to LSB).}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Feature\\ extractor\end{tabular}} & \multicolumn{9}{c|}{Number $s$ of encrypted bitplanes}           \\ \cline{2-10}
                                                                             & 0    & 1    & 2    & 3    & 4    & 5    & 6    & 7    & 8    \\ \hline
\multicolumn{10}{|c|}{Without pre-processing}\\ \hline
SRM~\cite{fridrich2012rich}  &   0.91   &   0.80   &   0.75   &  0.68    &            0.65 &   0.62   &   0.59  &   0.54 & 0.50  \\ \hline
SRMQ1~\cite{fridrich2012rich} & 0.90 & 0.81 & 0.76 & 0.69  & 0.62 & 0.61 & 0.55 & 0.54 & 0.50 \\ \hline
\multicolumn{10}{|c|}{With encrypted bitplanes set to zero}\\ \hline
SRM~\cite{fridrich2012rich} &   0.91   &   0.89   &   0.88   &  0.86    &         0.85   &   0.80   &   0.78   &   0.72 & 0.50   \\ \hline
SRMQ1~\cite{fridrich2012rich} & 0.90 & 0.87 & 0.87 & 0.86 & 0.84 & 0.81 & 0.79 & 0.72 & 0.50 \\ \hline
\end{tabular}
\vspace{-2mm}
\end{table*}
\end{comment}

\renewcommand{\arraystretch}{1.25}
\begin{table*}[ht]
\centering
\caption{\label{tab:accuracy1} Accuracy for tampering detection using SRMQ1~\cite{fridrich2012rich} as a function of the number of encrypted bitplanes (from MSB to LSB).}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Feature\\ extraction\end{tabular}} & \multicolumn{9}{c|}{Number $s$ of encrypted bitplanes}           \\ \cline{2-10}
                                                                             & 0    & 1    & 2    & 3    & 4    & 5    & 6    & 7    & 8    \\ \hline
                                                                             Without pre-processing & 0.90 & 0.81 & 0.76 & 0.69  & 0.62 & 0.61 & 0.55 & 0.54 & 0.50 \\ \hline
                                                                            With encrypted bitplanes set to zero & 0.90 & 0.87 & 0.87 & 0.86 & 0.84 & 0.81 & 0.79 & 0.72 & 0.50 \\ \hline
\end{tabular}
\vspace{-2mm}
\end{table*}

\renewcommand{\arraystretch}{1.25}

\begin{table*}[ht]
\centering
\caption{\label{tab:accuracy} Accuracy for the recognizability task as a function of the number of encrypted bitplanes (from MSB to LSB).}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Image\\ database\end{tabular}} & \multicolumn{9}{c|}{Number $s$ of encrypted bitplanes}           \\ \cline{2-10}
                                                                             & 0    & 1    & 2    & 3    & 4    & 5    & 6    & 7    & 8    \\ \hline
CASIA2~\cite{dong2013casia}  & 0.76 & 0.68 & 0.56 & 0.45 & 0.36 & 0.29 & 0.25 & 0.26  & 0.14 \\ \hline
Intel~\cite{intel}  & 0.93 & 0.89 & 0.82 & 0.73 & 0.60 & 0.52 & 0.45 & 0.49  & 0.17 \\ \hline
Cifar10~\cite{krizhevsky2009learning} & 0.87 & 0.68 & 0.52 & 0.37 & 0.23 & 0.13 & 0.09 & 0.10  & 0.10 \\ \hline
\end{tabular}
\vspace{-2mm}
\end{table*}

\section{Experimental results\label{sec:results}}
In this section, we present experimental results assessing the feasibility of combining forensics and privacy requirements for digital images. First, we provide an illustration of selectively encrypted images and standardized images in order to visualize the high frequency information. We then describe the training and the classification results obtained for the tampering detection and the recognizability tasks considering selectively encrypted images. Finally, we discuss the trade-off between tampering detection and privacy.

\subsection{Examples of selectively encrypted images}
In Fig.~\ref{fig:ex}, we first present the luminance component of the original image {\em Au\_ani\_00001} from the CASIA2 database~\cite{dong2013casia}. As an illustration, in the first row of the figure, we display selectively encrypted images obtained by encrypting $s$ bit-planes of this image, from MSB to LSB and with $1\leq s\leq 7$ (from left to right). One can notice that as soon as at least two bit-planes are encrypted, it is visually difficult to recognize the original image content. Indeed, in this example, distinguishing the silhouette of the zebra is not an easy task. In the second row, the presented images have been obtained by setting to zero the $s$~encrypted bit-planes of the selectively encrypted images and by performing a classical image standardization. Even after this process, the content of the original image is unrecognizable when at least five bit-planes are encrypted. This kind of images are taken as input of the CNN for the recognizability task. Moreover, they illustrate the significant information for classification in both forensics analysis and privacy evaluation tasks.

\subsection{Forensics analysis}
The CASIA2 database~\cite{dong2013casia} consists of authentic and tampered images (cloned or spliced) on JPEG or TIFF formats and with a size between $240\times 160$ and $900\times 600$~pixels~\cite{dong2013casia}. One can note that: 1)~tampered images have been generated using a subset of authentic images, and 2)~several tampered images have been issued from the same authentic images. In order to remove this bias in the construction of the database, we have randomly picked 1,000~authentic images and 1,000~tampered images in the full database making sure that there is no overlap between images, {\em i.e.} an image content only appears one time. Then, we have designed eight associated databases of selectively encrypted images, by encrypting between 1~to~8~bit-planes from MSB to LSB. After that, each of them has been processed separately. Into each database, images have been split into two balanced subsets with as many authentic images as tampered images: 80\%~of the images have been used for training and the remaining 20\%~for test. As feature extractor, we have used SRMQ1~\cite{fridrich2012rich}. In Table~\ref{tab:accuracy1}, we present the accuracy scores obtained during the test phase as a function of the number of encrypted bit-planes. First of all, on clear images ({\em i.e.} without encryption), we can see that the accuracy is equal to 0.90 even using a feature extractor as simple as SRMQ1.
To put this result in perspective with the state of the art, one of the best performing method~\cite{rao2016deep} uses CNN to achieve 0.97 accuracy, using 1:6 train to test ratio. Moreover, we can see that the pre-processing step consisting in discarding the encrypted bit-planes is relevant. If the performances are still quite good for $s=1$ and $s=2$ without pre-processing, the accuracy score falls significantly as soon as three bit-planes are encrypted. With encrypted bit-planes set to zero, what is particularly interesting is that, even with a reduced number of bit-planes in clear, accuracy remains high. Indeed, it is higher than 0.80 considering at least three bit-planes in clear and remains higher than 0.70 with only one or two bit-planes in clear. Therefore, with a very small amount of information on high frequencies, the tampering detection task is possible.
Note that the results obtained using SRMQ1 are comparable with those achieved using SRM, which highlights that the simplified version of SRM can be used in practice.

\subsection{Recognizability}

The recognizability of image content is assessed by automatically predicting the image class.

The CASIA2 database also provides coarse categories for image content: animals, architecture, art, character, indoor, nature, plants, text and sec. We choose to use the $7,491$~authentic images of the CASIA2 database for this task because authentic images are well labeled and do not contain falsification on which the model may focus. The number of images is relatively small thus, we propose to use the VGG11~\cite{SimonyanZ14a} network pre-trained on ImageNet~\cite{imagenet_cvpr09} as our baseline model. The database is randomly split into two subsets with 80:20 ratio for train and test. Images are cropped at their center to a size of $224 \times 224$ pixels to be passed as input of the model. The model is fine tuned using the train set, it converges quickly and it is stopped before overfitting. The model can predict CASIA2 classes with an accuracy of $0.76$ on the test set. This task is difficult because classes are not well defined and there are some overlap. Nevertheless, it shows that the model is able to predict CASIA2 classes on clear images.

In order to see if the content is still recognizable after the encryption of the $s$ most significant bitplanes, the baseline model is fine tuned using the same training set in which images are selectively encrypted. As we have to consider that the number $s$ is known, the best case for image classification is to work directly on the clear bits of the image. Therefore, image pixels are transformed using Eq.~\eqref{eqn:zeros}. In practice, as we want to standardize the model inputs, it is sufficient to apply the left shift operation $p_E(i,j) \ll s$ and then standardize images using classical image standardization.  We also perform these results on the selectively encrypted dataset. With $s=1$, the accuracy of the recognizability task is only of $0.37$, and for $s>1$, the accuracy is close to $0.14$. Indeed, the model does not directly converge toward the extraction of features that do not rely on the $s$ encrypted bits. Thus, it tends to classify all images into the most common class, \textit{i.e.} the ``animal" class which represents 14\% of the base. The fine tuning and testing phases have been independently done for $s \in \{0,7\}$, where $s=0$ means the image is in clear. The obtained results are reported in Table~\ref{tab:accuracy}. We also present results we have obtained using the Intel image classification~\cite{intel} and the Cifar10~\cite{krizhevsky2009learning} databases which were designed for image classification. The total images in each class is balanced. Intel image classification database contains $17,034$ images of  $150 \times 150$ pixels ($14,0034$ for train and $3,000$ for test) separated into $5$ classes: ``sea", ``mountain", ``buildings", ``forest", ``street" and ``glacier". CIFAR10 database is composed of $6,000$ images of $32 \times 32$ pixels ($5,000$ for train and $1,000$ for test) belonging to one of $10$ classes: ``airplane", ``automobile", ``bird", ``cat", ``deer", ``dog", ``frog", ``horse", ``ship" and ``truck". The recognizability task performs better on the Intel database because its classes are well separated, whereas in CIFAR10 there are classes that are close such as ``birds" and ``plane" or ``automobile" and ``truck". Note that the trend observed on the CASIA2 database is firmly established.

\subsection{Trade-off between tampering detection and privacy}
In Fig.~\ref{fig:tradeoff}, we illustrate the trade-off between tampering detection accuracy and privacy, as a function of the number $s$ of encrypted bit-planes. These results were obtained using images from the CASIA2 database. On the one hand, we can see that, from one to five encrypted bit-planes, the tampering detection accuracy is very good (higher than 0.8). On the other hand, the privacy index (computed from recognizability accuracy, as explained in Section~\ref{subsec:privacy}), is higher than 0.5 as long as at least three bit-planes are encrypted. This means that the classification rate is low for recognizability, {\em i.e.} the class of the image is mis-predicted on average. Therefore, this highlights that an interesting trade-off for combining tampering detection and privacy is achieved for three to five encrypted bit-planes. In particular, when five bit-planes are encrypted, tampering detection accuracy is equal to 0.81 and the privacy index is equal to 0.71. Fig.~\ref{fig:ex} illustrates the fact that it is very difficult to visually recognize the content of the selectively encrypted image when five bit-planes are encrypted, even by considering the associated standardized image. Moreover, depending on the application, it can be interesting to favor one of the other classification task (integrity check {\em vs} visual confidentiality).

\vspace{-2mm}
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\columnwidth]{Accuracy_Tampering_Privacy.pdf}
\vspace{-2mm}
\caption{Trade-off between tampering detection accuracy and privacy as a function of the number $s$ of encrypted bitplanes (from MSB to LSB).\label{fig:tradeoff}}
\end{figure}

\section{Conclusion\label{sec:ccl}}
In this paper, we have performed an analysis on selectively encrypted images to observe the trade-off between tampering detection and privacy. We have shown that SRMQ1 features can be used for a forensics analysis of selectively encrypted images. Moreover, privacy has been assessed experimentally by measuring the recognizability of an image content after encryption using a CNN. According to our experiments, an accuracy of more than $80\%$ for tampering detection is achieved when $s=0$ to $s=5$ bitplanes are encrypted, whereas the visual confidentiality is ensured as soon as $s=3$ bitplanes are encrypted.

In future work, we are interested by improving the classification performances during the forensics analysis using more specific tools, as those used for non-encrypted images. These approaches often rely on deep learning. Therefore, they may require using a larger database than CASIA2. It also could be interesting to investigate the tampered areas localization too, as done in clear~\cite{wu2019mantra}. Moreover, being able to identify the integrity threat from visually confidential image content should be relevant. Regarding the recognizability task, instead of only predicting the image class, we are planning to take an interest in object detection (its localization and its class) in protected images. Consequently, a subjective validation, involving human evaluation, should also be conducted.

\section*{Acknowledgment}
This work has been funded in part by the French National Research Agency (ANR-18-ASTR-0009), ALASKA project: https:// alaska.utt.fr, and by the French ANR DEFALS program (ANR-16- DEFA-0003).

\bibliographystyle{IEEEtran}


text:
\section{Introduction}

\noindent
A sequence $\{a_n\}$ is \emph{log-concave} if it satisfies
\[a_{n}^2 \. - \. a_{n-1}\hskip.03cm a_{n+1} \. \ge \. 0 \quad \text{for all $n$\hskip.03cm.}\]
Notable examples of log-concave sequences are the binomial coefficients,
the Stirling numbers, the Bessel numbers, etc.~(see~\cite{B,LW,S} for
well-written surveys).
Despite the interest in log-concave sequences and the detailed asymptotics
of the partition function in combinatorics, the log-concavity of $p(n)$
for all $n>25$ remained an open problem
(see Section~\ref{sec:final}).  Note that it fails for all odd values $n \leq 25$.

\begin{thm}\label{log concave theorem}
Sequence~$p(n)$ is log-concave for all $n>25$.
\end{thm}

This intuition behind the theorem comes from explicit calculations for small~$n$,
and the Hardy--Ramanujan asymptotic formula \cite{HR}
\begin{equation} \label{HR asymptotics}
 p(n) \. \sim \. \frac{1}{4\sqrt{3}\hskip.03cm n} \,\, e^{\pi \sqrt{\frac{2}{3}\hskip.03cm n}} \, \ \quad \text{as } \ n\to\infty\..
 \end{equation}
The function on the r.h.s.~of~(\ref{HR asymptotics}) is easily shown to be log-concave,
but without guaranteed error bounds there is no way of knowing precisely when the
asymptotic formula dominates the calculation.
Our proof is based on the estimates by Lehmer~\cite{L1,L2} which provide improved explicit guaranteed
error estimates valid for all~$n$. We use these to prove the theorem for all $n\geq 2600$,
and check log-concavity for smaller values using {\sc Mathematica}.

\smallskip

We turn to the following two recent conjectures by William Chen which partially motivated our
study, see~\cite[pp.~117--121]{C}.

\begin{con}[Chen]\label{conj1}  For all $\hskip.03cm n>1$, we have
\begin{equation}\label{eq:1}
\frac{p(n-1)}{p(n)}\left(1 + \frac{1}{n}\right) \,>\, \frac{p(n)}{p(n+1)}\,.\end{equation}
\end{con}

\begin{con}[Chen] \label{conj2} For all \hskip.03cm $n>m>1$, we have
\begin{equation}\label{eq:2}
p(n)^2 \. - \. p(n-m)\. p(n+m) \, \ge \, 0 \..
\end{equation}
\end{con}

Note that inequality~\eqref{eq:1} goes in the opposite
direction to log-concavity.  We fully establish Conjecture~\ref{conj1} in Theorem~\ref{t:1} and show
that the $(1+1/n)$ term can be further sharpened to $(1+O(n^{-3/2}))$, see
Theorem~\ref{conjecture 1}.  The inequality~\eqref{eq:2} is sometimes called
\emph{strong log-concavity} when extended to all~$m\ge 1$;
we prove Conjecture~\ref{conj2} in Theorem~\ref{t:3}.

\smallskip

The rest of the paper is structured as follows.  We first prove Theorem~\ref{log concave theorem}
in Section~\ref{proof section}. In a short but technical Section~\ref{sec:decay} we show that
the proof of Theorem~\ref{log concave theorem} gives sharp results on the asymptotic rate
of decay of log-concavity.  These are then used to prove of Chen's conjectures
(sections~\ref{s:chen} and~\ref{s:strong}), and Sun's strongly related conjecture
(Section~\ref{sec:Sun}). We conclude with final remarks in Section~\ref{sec:final}.

\bigskip

\section{Proof of Theorem~\ref{log concave theorem}} \label{proof section}

\noindent
The log-concavity is equivalent to
\[ \log a_{n-1} \. - \. 2\log a_n \. + \. \log a_{n+1} \. \leq \. 0 \quad \ \text{for all~$n$\.,}\]
which explains the name.

Recall the Rademacher's convergent series for~$p(n)$, based on the Hardy--Ramanujan asymptotic series
\cite{R} (see also e.g.~\cite{An,Ha}). Let $\hskip.03cm \mu(n) = \frac{\pi}{6}\hskip.03cm \sqrt{24\hskip.03cm n-1}$, which we abbreviate to~$\mu$ whenever
convenient.  We have
\begin{equation}\label{HR p(n)}
p(n) \, = \, \frac{\sqrt{12}}{24\hskip.03cm n-1} \sum_{k=1}^N A_k^{\ast}(n)\left[\left( 1 - \frac{k}{\mu}\right) e^{\mu/k}+\left( 1 + \frac{k}{\mu}\right) e^{-\mu/k}\right] \. + \. R_2(n,N)\hskip.03cm,
\end{equation}
where $A_k^\ast(n)$ is a complicated arithmetic function that will not be needed in its full generality for our analysis; see \cite{L1, L2} for the complete definition.  We shall only need that $A_1^\ast(n) = 1$ and $A_2^\ast(n) = (-1)^n/\sqrt{2}$ for all positive $n$.  Recall also Lehmer's error bound \cite{L1}
\begin{equation}
|R_2(n,N)| < \frac{\pi^2 N^{-2/3}}{\sqrt{3}} \left[ \left(\frac{N}{\mu}\right)^3 \sinh\frac{\mu}{N} + \frac{1}{6} - \left(\frac{N}{\mu}\right)^2\right],
\end{equation}
valid for all positive $n$ and~$N$.
  Define for all $n\geq 1$
\begin{equation}
T(n) :=   \frac{\sqrt{12}}{24\hskip.03cm n-1} \left[\left( 1 - \frac{1}{\mu}\right) e^{\mu} + \frac{(-1)^n}{\sqrt{2}}e^{\mu/2}\right].
\end{equation}
  The function $T(n)$ contains the three largest individual contributions to the sum,
  and is a refinement of the right-hand side of Equation~(\ref{HR asymptotics}).
  Define also the remainder term
\begin{equation}
R(n) :=  \frac{d}{\mu^2} \left[ \left( 1 + \frac{1}{\mu}\right) e^{-\mu} - \frac{(-1)^n}{\sqrt{2}}\frac{2}{\mu}e^{\mu/2} + \frac{(-1)^n}{\sqrt{2}}\left(1 + \frac{2}{\mu}\right)e^{-\mu/2}\right]  + R_2(n,2),
\end{equation}
where $d =  \frac{\pi^2}{6 \sqrt{3}}$\hskip.03cm.  We may rewrite Equation~\eqref{HR p(n)} as
\begin{equation}
p(n) \. = \. T(n) \. + \. R(n)\hskip.03cm.
\end{equation}

\begin{lemma}\label{lemma 0}{
Suppose $f(x)$ is a positive, increasing function with two continuous derivatives
for all $x> 0$, and that $f'(x)>0$ and decreasing, and $f''(x)<0$ is increasing for all $x>0$.   Then
\[  f''(x-1) < f(x+1)-2f(x) + f(x-1) < f''(x+1) \quad \text{for all \ $x>1$}\hskip.03cm. \]
}\end{lemma}

\begin{lemma}\label{lemma T(n)} Let
\[T_1(n) :=2 \hskip.03cm \log \hskip.03cm T(n) \. - \. \log \hskip.03cm T(n-1) \. - \. \log \hskip.03cm T(n+1).\]
Then, for all \hskip.03cm $n \geq 50$, we have
\ignore{\begin{equation}
\frac{24\hskip.03cm \pi}{(24(n+1)-1)^{3/2}} \. - \. \frac{4 \times 288}{(24(n-1)-1)^2} <  T_1(n) \, < \, \frac{24\pi}{(24(n-1)-1)^{3/2}} \. + \. \frac{4 \times 288}{(24(n+1)-1)^2} \..
\end{equation}}
\begin{equation}\label{T bound}
\frac{24\hskip.03cm \pi}{(24(n+1)-1)^{3/2}} \. - \. \frac{3}{n^2} <  T_1(n) \, < \, \frac{24\hskip.03cm \pi}{(24(n-1)-1)^{3/2}} + e^{-C \hskip.03cm \sqrt{n} \hskip.03cm /\hskip.03cm 10},
\end{equation}
where $C = \pi\hskip.03cm \sqrt{\frac{2}{3}}$.
\end{lemma}

\begin{proof}
We prove a stronger result, that for all $n\geq 2$,
    we have
\[\log T(n) \, = \, \log d \. +\. \mu \. + \. \log(\mu-1) \. - \. 3 \log \mu \.
+ \. \log\left(1 + \frac{(-1)^n e^{-\mu/2} \mu^3}{d \sqrt{2} (\mu-1)}\right).\]
  By Lemma \ref{lemma 0},
\begin{equation}\label{lower bound} T_1(n) >  -\mu''(n+1) - (\log(\mu(n+1)-1))'' + 3 \log(\mu(n-1))''+ G(n),\end{equation}
where $G(n)$ is the expression with the more complicated logs.
  This simplifies to
\[ T_1(n) \, > \, \frac{24\pi}{(24(n+1)-1)^{3/2}} \, + \, \frac{288\pi(-3+\pi\sqrt{24(n+1)-1})}{(24(n+1)-1)^{3/2}(-6+\pi\sqrt{24(n+1)-1})^2} \]
\[ - \ \frac{3 \times 288}{(24(n-1)-1)^2} \, + \, G(n)\.,\]
valid for all $n\geq 2$.  Let $\hskip.03cm x_n =  \frac{(-1)^n e^{-\mu/2} \mu^3}{d \sqrt{2} (\mu-1)}$.
Using $\hskip.03cm \log(1-|x|) \leq -\log(1-|x|)$ for all $|x|<1$, and $\log(1-x) \geq -x / (1-x)$ for $0<x<1$, we have
\[ G(n) \. > \. 4 \hskip.03cm \log \left( 1-|x_{n+1}|\right) \. > \.
\frac{-4|x_{n+1}|}{1-|x_{n+1}|} \. > \. -e^{-C \hskip.03cm \sqrt{n} \hskip.03cm / \hskip.03cm 10} \hskip.03cm , \quad \text{for all $\hskip.03cm n \geq 50$}\hskip.03cm.\]
For the upper bound, similarly note that
\[ G(n) \. < \. -4 \log( 1 - |x_{n-1}|) \. < \. \frac{4|x_{n-1}|}{1-|x_{n-1}|} \. < \. e^{-C \hskip.03cm \sqrt{n} \hskip.03cm / \hskip.03cm 10},  \quad \text{for all $\hskip.03cm n \geq 50$}\hskip.03cm, \]
as desired.
\end{proof}

\begin{lemma}\label{log rate} Let $y_n = |R(n)|/T(n)$.  Then
\begin{equation}\label{log equation} \log \left[ \frac{(1-y_n)(1-y_n)}{(1+y_{n-1})(1+y_{n+1})}\right]
\. > \. -e^{-C \hskip.03cm \sqrt{n} \hskip.03cm / \hskip.03cm 10}, \quad \text{for all $\hskip.03cm n\geq 10$}\hskip.03cm.
\end{equation}
\end{lemma}
\begin{proof}
Simply note that for all \hskip.03cm $n\geq 2$, we have
\begin{equation}
|R(n)| \. < \. 1 + \frac{16}{\mu^3} \hskip.03cm e^{\mu/2}\.,
\end{equation}
and
\begin{equation}
0 \. < \. \frac{|R(n)|}{T(n)} \. < \. e^{-C \hskip.03cm \sqrt{n} \hskip.03cm / \hskip.03cm 10}\..
\end{equation}
The result easily follows. \end{proof}

\begin{prop}\label{almost proof prop}
Let $\hskip.03cm p_2(n) = 2\log p(n) - \log p(n-1) - \log p(n+1)$.  Then,
for all $\hskip.03cm n \geq 2600$, we have
\begin{equation}\label{p bound}
\frac{1}{(24\hskip.03cm n)^{3/2}} \. < \. p_2(n) \. < \. \frac{2}{n^{3/2}}\..
\end{equation}
\end{prop}

\begin{proof}
We start by writing a double strict inequality for~$p(n)$,
\[ T(n)\hskip.03cm \left( 1 - \frac{|R(n)|}{T(n)}\right) \. < \. p(n) \. < \. T(n)\hskip.03cm \left( 1 + \frac{|R(n)|}{T(n)}\right),
\quad \text{for all \ $n\geq 1$}\hskip.03cm .\]
We then take logs and apply the Lemmas, and obtain
\begin{equation}\label{more explicit p bound}
\frac{24\hskip.03cm\pi}{(24(n+1)-1)^{3/2}} \. - \. \frac{3}{n^2} \. - \. 2\hskip.03cm e^{-C \hskip.03cm \sqrt{n} \hskip.03cm / \hskip.03cm 10}
\, < \, p_2(n) \, < \, \frac{24\hskip.03cm\pi}{(24(n-1)-1)^{3/2}} \. + \. 2\hskip.03cm e^{-C \hskip.03cm  \sqrt{n}\hskip.03cm / \hskip.03cm 10}\..
\end{equation}
Now the result easily follows by a direct calculation.
\end{proof}

\bigskip

\section{Rate of decay}\label{sec:decay}

\noindent
The rate $O(n^{3/2})$ in Lemma \ref{lemma T(n)} is asymptotically sharp.
Based on the asymptotics of~$T(n)$, we have the following corollary.

\begin{cor}\label{rate theorem}
Let $D(n) = 2 \log p(n) - \log p(n-1) - \log p(n+1)$.   We have
\[\lim_{n\to\infty} \. D(n) \. \frac{4\hskip.03cm n^{3/2}}{C} \, = \. 1, \quad \text{where} \ \ C = \pi \sqrt{\frac{2}{3}}\..\]
\end{cor}

A plot of exact values for $n$ up to 2000 is given in Figure \ref{rate figure}.  This asymptotic rate comes from the Taylor Series of $\mu \sim 2c\sqrt{n}$ starting with the second derivative term.
\begin{figure}
\includegraphics[height=5cm]{RateOfConvergence.pdf}
\caption{A plot of \. $n^{\frac32}\hskip.03cm D(n)/\bigl(\frac{\pi}{\sqrt{24}}\bigr)$,  for \hskip.03cm $n=1,\ldots, 2000$.}
\label{rate figure}\end{figure}

A refinement of the rate in Theorem \ref{rate theorem} to higher order terms follows easily as long as we also include the higher order terms of $-2\log(\mu)$.  Since $(2\log(\mu))'' \sim O(n^{-2})$, these terms did not appear in the first-order asymptotic expansion, but higher order expansions must take them into account.

Let
\[h_1(x) \. = \. \frac{4}{C\hskip.03cm x^{3/2}}\quad \ \, \text{and}  \quad h_2(x) \. =  \. \frac{-288}{(-1+24 x)^2}\.. \]
Fix an integer $k\ge 0$.  Denote by $L_k^+(n)$ the first $k$ even terms in the Taylor series expansion of $h_1(n+1)+h_2(n+1)$ evaluated at $x_0 = n$, $k=0,1,\ldots$.  Similarly, denote by $L_k^-(n)$ the first $k$ even terms in the Taylor series expansion of $h_1(n+1)$ evaluated at $x_0 = n$, plus the first $k-1$ even terms in the Taylor series expansion of $h_2(n+1)$ evaluated at $x_0 = n$, $k=1,2,\ldots$, with $L_0^-(n) = h_1(n)$.

\begin{cor}   For every integer \hskip.03cm $k\ge 0$, we have
\[ D(n) \. \sim \. L_k^+(n)\hskip.03cm, \quad D(n) \. \sim \. L_k^-(n)\quad \text{as \,$n\to \infty$}\hskip.03cm. \]
\end{cor}

\bigskip

\section{Reverse inequality}\label{s:chen}

\noindent
In this section, we establish and further strengthen Conjecture~\ref{conj1}.

\begin{thm}[formerly Chen's Conjecture~\ref{conj1}]\label{t:1}  For all \hskip.03cm $n>1$, we have
\begin{equation}\label{eq:rev}
\frac{p(n-1)}{p(n)}\left(1 + \frac{1}{n}\right) \,>\, \frac{p(n)}{p(n+1)}\,.\end{equation}
\end{thm}

\begin{proof}
We can rewrite Equation~\eqref{eq:rev}  as
\[ p_2(n) \. < \. \log\left(1 + \frac{1}{n}\right). \]
By Proposition \ref{almost proof prop}, for all \. $n\geq 2600$ we have
\[ p_2(n) \. < \. \frac{2}{n^{3/2}} \. < \. \frac{1}{n+1} \. < \.
\log\left(1 + \frac{1}{n}\right). \]
We then check numerically in {\sc Mathematica} that
Equation~\eqref{conjecture 1 eq} holds for all positive $n< 2600$.
\end{proof}

We next strengthen Theorem~\ref{conjecture 1} to the correct rate.

\begin{thm}\label{conjecture 1} For all \hskip.03cm $n> 6$, we have
\begin{equation}\label{conjecture 1 eq}
\frac{p(n-1)}{p(n)}\left(1 + \frac{240}{(24n)^{3/2}}\right) \, > \, \frac{p(n)}{p(n+1)}\..\end{equation}
\end{thm}
\begin{proof}
The theorem follows similarly from the proof above, since for all \hskip.03cm $n \geq 22$, we have
\[p_2(n) \. < \. \frac{2}{n^{3/2}} \, < \, \frac{240}{(24n)^{3/2} + 240} \, < \, \log\left( 1 + \frac{240}{(24n)^{3/2}}\right).\]
\end{proof}

\begin{conjecture}\label{conj3} For all \hskip.03cm $n\geq 45$, we have
\[\frac{p(n-1)}{p(n)}\left(1 + \frac{\pi}{\sqrt{24} \. n^{3/2}}\right) > \frac{p(n)}{p(n+1)}\,.\]
\end{conjecture}

We have checked this conjecture in {\sc Mathematica} for values of $n \leq 8000$, with violations occurring every even number less than~45.  Lemma~\ref{lemma T(n)} comes close to a proof, but one would have to check carefully the error involved in replacing the term $24(n-1)-1$ with $24\hskip.03cm n$, as well as the exponential contribution.\footnote{Added in proof: Conjecture 4.3 was proved by
W.Y.C. Chen et al. in~\cite{CWX}.}

\bigskip

\section{Strong log-concavity}\label{s:strong}

\noindent
In this section, we establish Conjecture~\ref{conj2} on strong log-concavity of the partition function.

\begin{thm} [formerly Chen's Conjecture~\ref{conj2}]
\label{t:3}
For all \hskip.03cm  $n >m>1$, we have
\begin{equation}
p(n)^2 \. - \. p(n-m)\hskip.03cm p(n+m) \. > \. 0\hskip.03cm.
\end{equation}
\end{thm}

\begin{proof}
First, recall that log-concavity implies strong log-concavity (see e.g.~\cite{Sa})
\[ a_k \hskip.03cm a_\ell \. \leq \. a_{k+i}\hskip.03cm a_{\ell-i}\., \]
for all $0 \leq k \leq \ell \leq n$ and $0 \leq i \leq k - \ell$.
Take $k=n-m$, $\ell = n+m$, and~$i = m$, to obtain:
\begin{equation}
p(n)^2 \. - \. p(n-m)\hskip.03cm p(n+m) \. > \. 0 \quad \text{for all} \ \ n > 25+m\hskip.03cm.
\end{equation}
To resolve the remaining cases, we use the following classical bounds which hold for all $m\ge 1$~:
$$e^{C \sqrt{m}} \, > \, p(m) \, > \,
\frac{e^{2\sqrt{m}}}{2\pi m \hskip.03cm e^{1/6m}}\,.
$$
The first inequality was given in~\cite{E}, and the second is from Section~2 of~\cite{HR}.  Next,
observe that
\begin{align*}
p(n-m) < p(25) <2000 & \quad \text{for all \, $1 \leq n-m \leq 25$\hskip.03cm.}
\end{align*}
Hence, for all $1 \leq n-m \leq 25$, we have
\[ p(n)^2 \.\geq \. p(m+1)^2 \. \geq\. p(25)\hskip.03cm p(25+m) \. \geq \. p(n-m)\hskip.03cm p(n+m)\.,\]
and thus it suffices to prove the middle inequality for all $m \geq 2$.  But this is equivalent to
\[ 2 \log p(m+1) - \log p(25) - \log p(25+m) \. > \. 0\hskip.03cm, \]
and by the inequalities above it is sufficient to establish for which values of $m$ the following holds:
\[ 4 \sqrt{m+1} - 2 \log(m+1) - \frac{1}{3(m+1)} - 2 \log(2\pi) - \log 2000 - C \sqrt{m+25} \. > \. 0\hskip.03cm.\]
This is easily seen to hold for $m \geq 300$, and a calculation in {\sc Mathematica} easily establishes the remaining cases.
\end{proof}

\bigskip

\section{Sun's conjecture}\label{sec:Sun}
In \cite{Sun}, a similar conjecture is stated for $q(n) := p(n) / n$.
We prove it here.

\begin{thm}[formerly Sun's Conjecture]
The sequence  $\{q(n)\}_{n \geq 31}$ is log-concave.
\end{thm}
\begin{proof}
Let $q_2(n) = 2\log q(n) - \log q(n-1) - \log q(n+1)$.  Then
\[ \log q_2(n) = \log p_2(n) - \log\left(\frac{n^2}{(n-1)(n+1)}\right). \]
By Lemma~\ref{lemma 0} with $f(x) = \log x$, we have
\[ \frac{1}{(n+1)^2} < \log\left(\frac{n^2}{(n-1)(n+1)}\right) < \frac{1}{(n-1)^2}. \]
We simply need to modify Equation~\ref{more explicit p bound} to include this new error, i.e.,
\begin{equation}\label{q bound}
\frac{24\hskip.03cm\pi}{(24(n+1)-1)^{3/2}} \. - \. \frac{3}{n^2} \. - \. 2\hskip.03cm e^{-C \hskip.03cm \sqrt{n} \hskip.03cm / \hskip.03cm 10}
\, -\frac{1}{(n+1)^2}\, < \, q_2(n) \, < \, \frac{24\hskip.03cm\pi}{(24(n-1)-1)^{3/2}} \. + \. 2\hskip.03cm e^{-C \hskip.03cm  \sqrt{n}\hskip.03cm / \hskip.03cm 10}\..
\end{equation}
The direct calculation is the same as in the proof of Lemma~\ref{almost proof prop}, which establishes
the log-concavity of $q(n)$ for $n\geq 2600$.  Since Sun checked numerically the log-concavity
of $q(n)$ for $n \leq 10^5$, see~\cite{Sun}, this completes the proof.
\end{proof}

\bigskip
\section{Final remarks}\label{sec:final}

\subsection{The log-concavity conjecture}\label{ss:fin-folk}
Let us mention that although the conjecture behind Theorem~\ref{log concave theorem}
seems to be folklore, we were unable to locate it in the literature.
The problem was revived by Chen~\cite{C}, who checked it for all $\hskip.03cm n\le 8000$,
by Janoski (see the discussion below in~\S\ref{ss:fin-jan}), and most recently by Burde on \hskip.03cm
{\it MathOverflow}.\footnote{See \hskip.03cm {\tt http://tinyurl.com/kkc6fwf}\hskip.03cm.}
After this paper was finished, we learned that it was also independently conjectured
by Andrews and Hirschhorn.\footnote{Personal communication.}  Most recently, a
different but somewhat related investigation was made in~\cite{BO}.

\subsection{Historical background}\label{ss:fin-hist}
The classical Hardy--Ramanujan formula is
\begin{equation}\label{HR full}
p(n) \. = \. \frac{\sqrt{12}}{24\hskip.03cm n-1} \sum_{k=1}^N A_k^{\ast}(n)\left[\left( 1 - \frac{k}{\mu}\right) e^{\mu/k}\right] \. + \. R_1(n,N)\hskip.03cm.
\end{equation}
They derived the explicit forms of the $A_k^{\ast}(n)$ for $k=1,\ldots, 18$ and all $n$ as a linear combination of cosines~\cite{HR}.  This allowed them to compute expansions for various small values of $n$ that were checked by MacMahon.  In particular, two terms of their series give
\begin{equation}
p(n) \, \sim \, \frac{1}{2\pi \sqrt{2}} \.\frac{d}{dn} \left(\frac{e^{C \lambda_n}}{\lambda_n} \right) \. + \. \frac{(-1)^n}{2\pi} \. \frac{d}{dn}\left(\frac{e^{\frac{1}{2}C \lambda_n}}{\lambda_n}\right) \. + \. O(e^{D \sqrt{n}}),
\end{equation}
where
\[  \qquad C \. = \. \pi \hskip.03cm \sqrt{\frac{2}{3}}\ , \qquad \lambda_n \. = \. \sqrt{24\hskip.03cm n-1}
\quad \ \ \text{and any} \ \quad D \hskip.03cm > \. \frac{C}{3} \..\]
This series was also extended to include $N = O(\sqrt{n})$ terms with an error of $O(n^{-1/4})$.  However,
it was later shown by Lehmer~\cite{L0} that this series diverges if extended to infinity.
The first term of the series easily suffices to establish log-concavity for sufficiently
large~$n$ (see~\S\ref{ss:fin-hist}), but there is no natural way to make these bounds explicit.
Rademacher modified the formula and produced a convergent series in~\cite{R}.
Most recently, this series used by Janoski~\cite[$\S 1.2$]{J}, to claim the proof of Theorem~1.
Unfortunately, this proof has a serious logical flaw (see below).

This relation has since been interpreted in the popular literature as
\[ p(n) = \frac{e^{C\hskip.03cm \sqrt{n}}}{4 \sqrt{3}\, n} \left( 1 + O(n^{-1/2})\right). \]
It follows then, that one cannot base any heuristic interpretation of log-concavity
on this simplest form, as it has an error that is larger than the asymptotic
rate of decay given by Theorem~\ref{rate theorem}.

One can, however, embrace the \emph{entire} first-order term of the Hardy-Ramanujan series, and obtai
\[ p(n) = \frac{e^{C\hskip.03cm \sqrt{n-\frac{1}{24}}}}{4 \sqrt{3}\, \left(n-\frac{1}{24}\right)}\left(1 - \frac{1}{C \sqrt{n-\frac{1}{24}}}\right) \left( 1 + O(e^{-\mu/2})\right). \]
The relative error term above decreases exponentially, and hence it easily follows that $\{p(n)\}$ is \emph{asymptotically log-concave},
i.e., that~$\{p(n)\}$ is log-concave for all $n>n_0$, for some \emph{unspecifiable}~$n_0$.\footnote{See e.g.,~Speyer's calculation in \. {\tt http://tinyurl.com/nyq2zrn} }  Without a concrete error bound in the form of a strict inequality, one cannot in general obtain the result.

The \emph{Rademacher series} for $p(n)$ is a convergent series of the form
\[ p(n) \, = \, \frac{\sqrt{12}}{24\hskip.03cm n-1} \, \sum_{k=1}^\infty \. A_k^{\ast}(n)\hskip.03cm \left[\left( 1 - \frac{k}{\mu}\right) e^{\mu/k} \hskip.03cm +\hskip.03cm \left( 1 + \frac{k}{\mu}\right) e^{-\mu/k}\right]\hskip.03cm. \]
This series was shown to converge by Rademacher.  Equation \eqref{HR p(n)} is the truncation of this series to $N$ terms, and Rademacher provided a \emph{strict} inequality on the absolute value of the error term $R_2(n,N)$ for all $n$ and $N$.  This was later improved by Lehmer, and we have used his estimates even though in principle Rademacher's estimates would have been sufficient.

An important part of the proof of Theorem~\ref{log concave theorem} is the form of the error.
 Rademacher's upper bound can be written for some positive constant $c'$ as
\[|R_2(n,N)| \. < \. c' e^{\mu/N} \frac{\sqrt{N}}{\mu} \quad \text{for all \. $n\hskip.03cm, \hskip.03cm N\geq 1$\hskip.03cm.}\]
Lehmer's upper bound can be written for some other positive constant $c''$ as
\[|R_2(n,N)| \. < \. c'' e^{\mu/N}\frac{N^3}{\mu^3} \quad \text{for all \. $n\hskip.03cm, \hskip.03cm N\geq 1$\hskip.03cm.}\]
Thus, neither Rademacher's nor Lehmer's upper bounds for $N=1$ provide a sharp enough bound to have a relative error $o(n^{-3/2})$.  In particular, the exponent $\mu/N$ implies that we must take $N \geq 2$ in order to obtain an exponentially decaying error term relative to the first term.  This demonstrates that one cannot obtain even a proof of asymptotic log-concavity using this error analysis with $N=1$.

\subsection{Janoski's thesis}\label{ss:fin-jan}
Recently, Janoski in her thesis~\cite{J} claimed to have a complete proof of log-concavity, for all~$n>25$.
Unfortunately, there is a serious technical error early in the proof, invalidating the argument.
Specifically, Janoski writes on page~9 the following inequality:
$$\frac{1}{\pi \sqrt{2}} \. \sum_{k=3}^{C\sqrt{t}} \. A_k(n) \sqrt{k} \. d(n,k) \, > \, C \sqrt{t} \hskip.03cm \frac{\sqrt{3}}{\pi \sqrt{2}} \. A_3(n) \. d(n,3),$$
where
$$t \. = \. n\hskip.03cm - \hskip.03cm \frac{1}{24}\., \qquad  A_k(n) \. = \. A_k^\ast(n)/\sqrt{k}\., \quad \text{and} \quad d(n,k) \. = \.
\left(\frac{\sinh C\sqrt{t}/k}{\sqrt{t}} \right)'.
$$
It seems, the author assumes that the functions $A_k(n)$ are positive and monotonic in~$k$.  We have checked this inequality for various small values of~$n$, and have found several counterexamples, such as $n$ equal to $27, 36, 87$ or~$744$.  Furthermore, the calculation bounding the absolute error $R_2(n,N)$ is incorrect. The rest of the analysis in~\cite{J} is based on bounding from below this faux lower bound term, which has a relative error $O(e^{-2\mu/3})$ (though not in the form of a strict inequality!), and so naturally the numerical results would appear to confirm the entire result.

\subsection{Computer calculations}
All of the calculations for small values of~$n$ mentioned in the proof above are trivial, if tedious.
The calculations were done with {\sc Mathematica}, which has a built-in partition function.
Let us note, however, that strictly speaking, the use of computer is not necessary to follow
and verify the proofs of the Lemmas, as there are comprehensive tables of the partition function,
see e.g.,~\cite{G}.

\subsection{Combinatorial proof}
It is natural to ask for a direct combinatorial proof of
Theorem~\ref{log concave theorem}, e.g.,~in the style of proofs in~\cite{Bo,HS,Kr,Sa}.
Unfortunately, the previous attempts and the fact that log-concavity fails for
small~$n$, are very discouraging.   It seems, even the simplest results for
the partition function are difficult to prove directly (cf.~\cite{A}).

Note, however, that for related sequences such as \emph{convex compositions}
(or \emph{stacks}), the asymptotic formulas are similar to~\eqref{HR asymptotics},
and thus not sharp enough for the analysis as in this paper (see~\cite{Br,Wr}).
A combinatorial proof is the best current hope for proving asymptotic
log-concavity of such sequences.

\subsection{Another Chen's conjecture}
Note that Theorem~\ref{t:3} says that the sequences $\{p(m\hskip.03cm n + a)\}$ are
log-concave, for all fixed $m > a \ge 0$. In~\cite{C}, Chen also makes
a related conjecture, that for all $a>b$ we have
\[p(a\hskip.03cm n)^2 \. - \. p(a\hskip.03cm n - b\hskip.03cm n)\. p(a\hskip.03cm n + b\hskip.03cm n) \. > \. 0\quad
\text{whenever \ $a\hskip.03cm n, \. b\hskip.03cm n \hskip.03cm \in \mathbb{N}$\hskip.03cm.} \]
Of course, this is a weak version of Conjecture~\ref{conj2}.
Now that Theorem~\ref{t:3} is proved, it also follows.

\subsection{Another Sun's conjecture}
Let $r(n) := \sqrt[n]{p(n)/n}$.  In~\cite{Sun} (see also~\cite{Sun1}),
Sun conjectures that $\{r(n)\}$
is log-concave for $n\ge 60$.  Unfortunately, the calculation for $r(n)$
does not follow immediately from our lemmas, since there is no simple lemma
analogous to Lemma~\ref{lemma 0}.  However, it is easy to see that
such result only requires a more careful bounding of elementary functions;
we hope the reader will continue this investigation.  Let us mention also
that in the same paper, Sun makes similar conjectures for partitions
into distinct part, supported by numerical evidence.

\vskip.8cm

\subsection*{Acknowledgements}
The authors would like to thank William Chen, Christian Krattenthaler,
Karl Mahlburg, Bruce Rothschild, Bruce Sagan for helpful remarks and suggestions.
The authors are grateful to Richard Arratia for introducing us to the problem,
to David Moews for bringing the work of Lehmer to our attention, and to
Janine Janoski for informing us about the status of~\cite{J}.
The second author was partially supported by the~NSF.

\vskip.9cm

{\footnotesize

redpajama_set_name: RedPajamaArXiv