| Summary: | In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting point of a novel approach which, in a series of companion papers, yields a formal proof of the Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> behaviour of the modified Hurwitz zeta function <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>ζ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>α</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>∈</mo> <mi mathvariant="bold">C</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mo>∞</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, which is also presented here.
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