This is an expository article. It is a collection of some important results on the meanvalue of$$\left| {\zeta (\frac{1}{2} + it)} \right|.$$

The effect of aspect ratio on the meridional circulation of a homogeneous fluid is analyzed. Aspect ratio is allowed to range between zero and unity. Relationships between possible horizontal and vertical length scales are obtained by length scale analysis as well as by solving an idealized problem. It is found that whenE^1/2 ≪ Z ≪ E^1/2/δ, whereE is the Ekman number, the stream lines are closely packed near the sidewall within a thickness ofO(E^1/2). The effect of stratification is unimportant within this depth range. In the depth rangeE^1/2/δ ≪ Z ≪ 1/Eδ the vertical boundary layer in which the streamlines are packed is ofO(EZδ)^1/3. WhenZ ≫ 1/Eδ it is shown that the circulation decays algebraically with depth as 1/Z.

For suitable functionsH = H(T) the maximum of¦(ζ(σ + it))^z¦ taken overT≤t≤T + H is studied. For fixed σ(1/2≤σ≤l) and fixed complex constantsz “expected lower bounds” for the maximum are established.

With some applications in view, the following problem is solved in some special case which is not too special. LetF(s) =Σ_n^∞=1a_nλ_n^−s be a generalized Dirichlet series with 1 =λ₁ <λ₂ < …,λ_n≤Dn, andλ_n+1 -λ_n ≥D^{− 1}λ_n+1^{− a} where α>0 andD(≥ 1) are constants. Then subject to analytic continuation and some growth conditions, a lower bound is obtained for$$(1/H)\int {_O^H |} F(it)|^2 dt$$. These results will be applied in other papers to appear later.

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples.

Theorem A.Let 0<θ<1/2and let {a_n}be a sequence of complex numbers satisfying the inequality$$\left| {\sum\limits_{m = 1}^N {a_m - N} } \right| \leqslant \left( {\frac{1}{2} - \theta } \right)^{ - 1} $$for N = 1,2,3,…,also for n = 1,2,3,…let α_nbe real and ¦α_n¦ ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function$$\sum\limits_{n = 1}^N {a_n \left( {n + \alpha _n } \right)^{ - s} } = \zeta \left( s \right) + \sum\limits_{n = 1}^\infty {\left( {a_n \left( {n + \alpha _n } \right)^{ - s} - n^{ - s} } \right)} $$in the rectangle (1/2-δ⩽σ⩽1/2+δ,T⩽t⩽2T) (where 0<δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T₀(θ,δ)a large positive constant.

Theorem B.In the above theorem we can relax the condition on a_n to$$\left| {\sum\limits_{m = 1}^N {a_m - N} } \right| \leqslant \left( {\frac{1}{2} - \theta } \right)^{ - 1} N^0 $$ and ¦a_N¦ ≤ (1/2-θ)^-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,T⩽t⩽2T)is > C(θ,δ) Tlog T(log logT)^-1.The upper bound for the number of zeros in σ⩾1/3+δ,T⩽t⩽2T) isO(T)provided$$\sum\limits_{n \leqslant x} {a_n } = x + O_s \left( {x^2 } \right)$$for every ε > 0.

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functionsF(a,b;c;x), c ≤a + b, near the singularityx = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions ona, b > 0 such that 2F(−a,b;a +b;r²) < (2−r²)F(a,b;a +b;r²) holds for all r∈(0,1).

For an abelian group 𝐺, the Davenport constant $D(G)$ is defined to be the smallest natural number 𝑘 such that any sequence of 𝑘 elements in 𝐺 has a non-empty subsequence whose sum is zero (the identity element). Motivated by some recent developments around the notion of Davenport constant with weights, we study them in some basic cases. We also define a new combinatorial invariant related to $(\mathbb{Z}/n\mathbb{Z})^d$, more in the spirit of some constants considered by Harborth and others and obtain its exact value in the case of $(\mathbb{Z}/n\mathbb{Z})^2$ where 𝑛 is an odd integer.

Given a prime number 𝑙, a finite set of integers $S=\{a_1,\ldots,a_m\}$ and 𝑚 many 𝑙-th roots of unity $\zeta^{r_i}_l,i=1,\ldots,m$ we study the distribution of primes 𝑝 in $\mathbb{Q}(\zeta_l)$ such that the 𝑙-th residue symbol of $a_i$ with respect to 𝑝 is $\zeta^{r_i}_l$, for all 𝑖. We find out that this is related to the degree of the extension $\mathbb{Q}\left(a^{\frac{1}{l}}_1,\ldots,a^{\frac{1}{l}}_m\right)/\mathbb{Q}$. We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from $S=\{a_1,\ldots,a_m\}$. This latter argument enables one to describe the degree $\mathbb{Q}\left(a^{\frac{1}{l}}_1,\ldots,a^{\frac{1}{l}}_m\right)/\mathbb{Q}$ in much simpler terms.