This question assumes the following definitions.
(1) $\quad a(n)=\sum\limits_{d|n}\mu(d)\,\mu\left(\frac{n}{d}\right),\qquad \frac{1}{\zeta(s)^2}=\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}\qquad$(see OEIS A007427)
(2) $\quad f(m)=2-\frac{1}{2}\sum\limits_{n=1}^m a(n)$
(3) $\quad g(m)=\sum\limits_{n=1}^m\frac{a(n)}{n}$
(4) $\quad h(m)=-\frac{1}{2}\sum\limits_{n=1}^m a(n)$
Question (1): Can it be proven that the functions $f(m)$ and $h(m)$ both have an infinite number of integer zeros?
Question (2): Can it be proven that $\underset{m\to\infty}{\text{lim}}\ g(m)=0$?
The following plot illustrates $f(m)$ (blue) where the zeros of $f(m)$ are highlighted in red.
Figure (1): Illustration of $f(m)$ (blue) where the zeros of $f(m)$ are highlighted in red
The evaluations of $f(m)$ and $g(m)$ seem to be related. When $f(m)=0$, $g(m)$ tends to evaluate very close to zero as illustrated in the following figure where the evaluation of $g(m)$ at the zeros of $f(m)$ is highlighted in red.
Figure (2): Illustration of $g(m)$ (blue) highlighted at the zeros of $f(m)$ in red
The following plot illustrates $h(m)$ (blue) where the zeros of $h(m)$ are highlighted in red.
Figure (3): Illustration of $h(m)$ (blue) where the zeros of $h(m)$ are highlighted in red
The evaluations of $h(m)$ and $g(m)$ also seem to be related. When $h(m)=0$, $g(m)$ seems to evaluate mostly negative as illustrated in the following figure where the evaluation of $g(m)$ at the zeros of $h(m)$ is highlighted in red. The first exception to this is at $h(1546)=0.000941679$.
Figure (4): Illustration of $g(m)$ (blue) highlighted at the zeros of $h(m)$ in red
The integer values of $m\le 10^6$ for which $f(m)=0$ are as follows:
$\{25,29,68,86,126,133,141,309,628,638,844,847,848,1543,1544,2639,2640,3053,3067,3070,3074,3814,4604,4607,4608,4622,4627,4630,4634,4649,4681,6014,6036,6067,6070,6078,9156,9223,9224,9265,9268,11372,11377,11434,15124,15125,26641,27581,27591,27592,27603,27692,27735,27736,33755,34124,34125,43967,43968,81829,81836,81837,82077,82153,88978,88981,88998,89002,89012,89015,89016,89037,89044,89049,89196,89220,89223,89224,89230,89318,90690,90701,90722,90729,90918,90922,90967,90968,91149,91153,91198,108700,109677,110418,110421,110499,110500,110507,111343,111344,111521,131046,131269,131277,131294,131708,131755,131765,132105,163164,163294,163298,163301,163305,163340,163778,163850,163908,163930,163937,200402,200409,200412,200423,200424,200538,206413,206450,206588,206602,206727,206728,212902,214495,214496,214676,214677,214719,214720,214748,214753,214771,214799,214800,214890,214911,214912,215897,247315,247322,247325,247357,261167,261168,261229,261233,261251,261252,261262,261291,261316,261970,261973,261985,262029,262436,262979,262980,263242,263351,263352,263406,263421,263431,263432,263558,263609,264015,264016,264037,264041,264047,264048,264161,265458,265471,265472,265475,269034,269037,269066,269094,269234,269330,269357,272196,272199,272200,272206,272420,272450,272471,272472,272478,272714,272724,273063,273064,273102,273114,273204,273251,273283,273308,273315,273410,273413,273420,392444,392445,392459,392502,392509,392548,396564,396617,396662,396666,396726,494387,494402,494409,496551,496552,496634,496652,496706,496972,498543,498544,498547,498562,498733,498751,498752,498755,498769,498812,499521,499549,499603,511317,511383,511384,511417,511474,511491,511549,511553,511556,511599,511600,511735,511736,511788,515525,515562,515580,515583,515584,515601,515607,515608,515669,515757,515782,515803,515813,516012,516015,516016,516036,516039,516040,516043,516050,516051,516086,516318,516323,516351,516352,516358,516369,516377,516397,516415,516416,548297,556493,556498,556502,643311,643312,643343,643344,643391,643392,643438,643445,643612,643938,644713,644716,644738,779777,779870,779885,780027,780031,780032,780081,780099,780102,780814,780907,782474,782477,782481,782529,782539,786332,786505,790772,790775,790776,790779,790906,791638,791642,791646,791700,791714,967473,967534,967978,968099,968123,968276,968279,968280,968302,968881,969245,969246,969251,969359,969360,973276\}$
The integer values of $m\le 10^6$ for which $h(m)=0$ are as follows:
$\{6,49,52,70,74,77,85,124,125,132,140,308,315,618,630,641,1546,1556,2612,2615,2616,2665,2676,3052,3081,3089,3781,3813,3817,4606,4610,4621,4626,4639,4640,4679,4680,4683,6017,6035,9209,9222,9229,9267,11374,11375,11376,11433,15119,15120,15123,15127,15128,26646,27694,27734,27761,33761,34123,34669,34676,55363,56119,56120,56131,81831,81832,81835,82155,88980,89001,89011,89036,89043,89047,89048,89195,89198,89222,89237,89321,89324,90692,90693,90700,90717,90721,90727,90728,90917,90921,90924,90966,91190,91197,108292,108703,108704,108877,109676,110494,110498,110506,111513,111516,111523,131268,131276,131293,131707,131731,131764,163163,163166,163293,163297,163300,163339,163852,163859,163903,163904,163907,163932,163935,163936,200411,200485,200530,200540,206449,206459,206597,206726,206730,212813,212814,212901,214494,214502,214581,214747,214763,214798,247314,247317,247321,247324,247327,247328,261181,261242,261254,261261,261315,261975,261976,262004,262094,262431,262432,262978,262989,263078,263183,263184,263198,263262,263266,263269,263409,263430,263434,263557,263621,263954,264014,264031,264032,264033,264036,264039,264040,264043,264159,264160,265439,265440,268489,269014,269021,269029,269036,269097,269100,269289,269332,269342,269359,269360,271943,271944,271947,272209,272449,272470,272706,272709,272716,272723,273106,273113,273206,273214,273245,273249,273250,273268,273278,273419,392443,392447,392448,392454,392458,392547,392569,393124,393125,396590,396615,396616,396665,396729,494396,494397,494401,496554,496633,496651,496654,496737,496791,496792,496971,498749,498750,498767,498768,498814,499556,499602,511298,511299,511476,511490,511551,511552,511787,511907,511934,511938,512095,512096,515524,515599,515600,515606,515989,516014,516035,516049,516089,516095,516096,516284,516325,516329,516361,516396,516404,548292,548299,556217,556355,556492,556505,643257,643342,643346,643394,643429,643614,643940,644665,644711,644712,644715,779703,779704,779975,779976,780026,780083,780084,780091,780812,780813,780906,780909,782441,782476,782479,782480,782531,786334,790774,790905,791761,966906,967471,967472,967526,967533,967955,968122,968300,968301,968433,968874,968875,968879,968880,969227,969249,969250,973282\}$