• K Ramachandra

Articles written in Proceedings – Mathematical Sciences

• Mean-value of the Riemann zeta-function on the critical line

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

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.

• On the frequency of Titchmarsh’s phenomenon for ζ(s)-VIII

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.

• Proof of some conjectures on the mean-value of Titchmarsh series — III

With some applications in view, the following problem is solved in some special case which is not too special. LetF(s) =Σn=1anλn−s be a generalized Dirichlet series with 1 =λ1 &lt;λ2 &lt; …,λnDn, andλn+1 -λnD− 1λn+1− a where α&gt;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.

• On the zeros of a class of generalised Dirichlet series-XIV

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&lt;θ&lt;1/2and let {an}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 αnbe real and ¦αn¦ ≤ C(θ)where C(θ) &gt; 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,Tt⩽2T) (where 0&lt;δ&lt;1/2)isC(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided TT0(θ,δ)a large positive constant.

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

• On the zeros ofζ(1)(s)a (on the zeros of a class of a generalized Dirichlet series — XVII)

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

• # Proceedings – Mathematical Sciences

Volume 132, 2022
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