We discuss cetain identities
involving $\mu(n)$ and $M(x) = \sum _{n \leq x}\mu (n)$,
the functions of M\"obius and Mertens.
These allow calculation of $M(N^d)$,
for $d=1,2,3,\ldots\ $, as a sum of
$O_d \left( N^d(\log N)^{2d - 2}\right)$ terms, each a product of the
form $\mu(n_1) \cdots \mu(n_r)$ with $r\leq d$ and
$n_1, \ldots , n_r\leq N$.
We prove a more general identity in which
$M(N^d)$ is replaced by $M(g,K)=\sum_{n\leq K}\mu(n)g(n)$, where
$g(n)$ is an arbitrary totally multiplicative function, while
each $n_j$ has its own range of summation, $1,\ldots , N_j$.
This is not new, except perhaps in that
$N_1,\ldots , N_d$ are arbitrary, but our
proof (inspired by an identity of E.~Meissel, 1854) is new.
We are mainly interested in the case $d=2$, $K=N^2$, $N_1=N_2=N$,
where the identity has the form
$M(g, N^2) = 2 M(g,N) - {\bf m}^{\rm T} A {\bf m}$,
with $A$ being the $N\times N$~matrix of elements
$a_{mn}=\sum _{k \leq N^2 /(mn)}\,g(k)$, while
${\bf m}=(\mu (1)g(1),\ldots ,\mu (N)g(N))^{\rm T}$.
Our results in Sections~2 and~3 of the paper
assume that $g(n)$ equals $1$ for all $n$.
The Perron-Frobenius theorem applies in this case:
we find that $A$ has one large positive eigenvalue,
approximately~$(\pi^2 /6)N^2$,
with eigenvector approximately ${\bf f} = (1,1/2,1/3,\ldots ,1/N)^{\rm T}$,
and that, for large $N$, the
second-largest eigenvalue lies in $(-0.58 N, -0.49 N)$.
Section~2 includes estimates for the traces
of~$A$ and~$A^2$ (though, for ${\rm Tr}(A^2)$, we omit part of the proof).
In Section~3 we discuss ways to approximate ${\bf m}^{\rm T} A {\bf m}$,
using the spectral decomposition of $A$, or (alternatively) Perron's formula:
the latter
approach leads to a contour integral involving the Riemann zeta-function.
We also discuss using the identity
$A = N^{2\,} {\bf f}^{\,} \!{\bf f}^T -
\textstyle{1\over 2} {\bf u} {\bf u}^T + Z$,
where ${\bf u} = (1,\ldots ,1)^{\rm T}$ and $Z$ is the $N\times N$ matrix
of elements $z_{mn} = - \psi(N^2 / (mn))$,
with $\psi(x)=x - \lfloor x\rfloor - \textstyle{1\over 2}$.

We give the version of Hua's method for the estimation of non"=com\-ple\-te ra\-ti\-onal tri\-go\-nometric sums. These estimates are non-trivial one for sums
with lengths exceeding a square root of length the complete sum.

Seien $a$ und $q$ zwei teilerfremde, positive, ganze Zahlen. In 1944 bewies
Y. Linnik, dass die kleinste Primzahl in einer arithmetischen Progression
$mod$ $q$ kleiner als $C q^L$ ist mit positiven Konstanten $C$ und $L$.
Aufbauend auf einer Arbeit von Heath-Brown beweisen wir, dass $L=5$ zulässig ist.

Let $\alpha_m$ and $\beta_n$ be two sequences of real numbers supported on $[M, 2M]$ and $[N, 2N]$ with $M = X^{1/2 - \delta}$ and $N = X^{1/2 + \delta}$. We show that there exists a $\delta_0 > 0$ such that the multiplicative convolution of $\alpha_m$ and $\beta_n$ has exponent of distribution $\frac{1}{2} + \delta-\varepsilon$ (in a weak sense) as long as $0 \leq \delta < \delta_0$,    the sequence  $\beta_n$ is Siegel-Walfisz and both sequences $\alpha_m$ and $\beta_n$ are bounded above by divisor functions. Our result is thus a general dispersion estimate for ``narrow'' type-II sums. The proof relies crucially on Linnik's dispersion method and recent bounds for trilinear forms in Kloosterman fractions due to Bettin-Chandee. We highlight an application related to the Titchmarsh divisor problem.

The paper is devoted to investigation of unique solvability of the Dirichlet variational
problem associated with integro-differential sesquilinear form
????[????, ????] =
Σ︁
????∈????
???????? [????, ????], (*)
where
???????? [????, ????] =
Σ︁
|????|=|????|=????
∫︁
Ω
????(????)2???????? ????????????(????)????(????)(????) ????(????)(????)????????,
Ω — a bounded domain in the euclidian space ???????? with a closed (???? − 1)-dimensional boundary
????Ω, ????(????), ???? ∈ Ω, — a regularized distance from a point ???? ∈ Ω to ????Ω, ???? — a multi-index,
????(????)(????) — a generalized derivative of multi-index ???? of a function ????(????), ???? ∈ Ω, ????????????(????) — bounded
in Ω complex-valued functions, ???? ⊂ {1, 2, . . . , ????} and ???????? , ???? ∈ ????, — real numbers. It is assumed
that ???? ∈ ????. A degeneracy of coefficients of the differential operator associated with the form
(*), is said to be coordinated if there exist a number ???? such that ???????? = ???? + ???? − ???? for all ???? ∈ ????.
Otherwise it is called uncoordinated.
The variational Dirichlet problem associated with the form (*) in the case of coordinated
degeneracy of coefficients is well studied in many papers, where it is also assumed that the form
(*) satisfies a coercivness condition. It should be mentioned that the case of uncoordinated
degeneracy of the coefficients is fraught with some technical complexities and it was only
considered in some separate papers. In this case with the aid of embedding theorems for spaces
of differentiable functions with power weights leading forms ???????? [????, ????], ???? ∈ ????2 ⊂ ????, are separated
and it is proved that solvability of the variational Dirichlet problem is generally depends on the
leading forms.
We consider the case of uncoordinated degeneracy of coefficients of the operator under
investigation and, in contrast to previously published works on this direction, it is allowed that
the main form (*) does not obey coerciveness condition.