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

A generalization of the Riemann zeta-function which has the form$$\zeta _\alpha (s) = \prod\limits_p {\frac{1}{{1 - p^{ - s} + (p + a)^{ - 3} }}} $$ is considered. Analytical properties with respect to s and asymptotic behaviour whena → ∞ are investigated. The correspondingL-function is also discussed. This consideration has an application in the theory ofp-adic strings.

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.

Some very precise results (see Theorems 4 and 5) are proved about thea-values of thelth derivative of a class of generalized Dirichlet series, forl≥l_o =l_o(a) (l_o being a large constant). In particular for the precise results on the zeros ofζ⁽¹⁾(s) —a (a any complex constant andl≥l_o) see Theorems 1 and 2 of the introduction.